QuantumFieldTheoryI - uniba.sksophia.dtp.fmph.uniba.sk/~mojzis/qft.pdf · QuantumFieldTheoryI Martin Mojˇziˇs. ... We chose An Introduction to Quantum Field Theory by Peskin and

Quantum Field Theory I

Martin Mojzis

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

1 Introductions 11.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . 81.1.3 Scattering amplitude . . . . . . . . . . . . . . . . . . . . . 111.1.4 cross-sections and decay rates . . . . . . . . . . . . . . . . 121.1.5 A redundant paragraph . . . . . . . . . . . . . . . . . . . 15

1.2 Many-Body Quantum Mechanics . . . . . . . . . . . . . . . . . . 171.2.1 Fock space, creation and annihilation operators . . . . . . 171.2.2 Important operators expressed in terms of a+i , ai . . . . . 221.2.3 Calculation of matrix elements — the main trick . . . . . 281.2.4 The bird’s-eye view of the solid state physics . . . . . . . 31

1.3 Relativity and Quantum Theory . . . . . . . . . . . . . . . . . . 391.3.1 Lorentz and Poincare groups . . . . . . . . . . . . . . . . 401.3.2 The logic of the particle-focused approach to QFT . . . . 451.3.3 The logic of the field-focused approach to QFT . . . . . . 47

2 Free Scalar Quantum Field 492.1 Elements of Classical Field Theory . . . . . . . . . . . . . . . . . 49

2.1.1 Lagrangian Field Theory . . . . . . . . . . . . . . . . . . 492.1.2 Hamiltonian Field Theory . . . . . . . . . . . . . . . . . . 54

2.2 Canonical Quantization . . . . . . . . . . . . . . . . . . . . . . . 562.2.1 The procedure . . . . . . . . . . . . . . . . . . . . . . . . 562.2.2 Contemplations and subtleties . . . . . . . . . . . . . . . 68

3 Interacting Quantum Fields 733.1 Naive approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.1.1 Interaction picture . . . . . . . . . . . . . . . . . . . . . . 753.1.2 Transition amplitudes . . . . . . . . . . . . . . . . . . . . 783.1.3 Cross-sections and decay rates . . . . . . . . . . . . . . . 93

3.2 Standard approach . . . . . . . . . . . . . . . . . . . . . . . . . . 983.2.1 S-matrix and Green functions . . . . . . . . . . . . . . . . 983.2.2 the spectral representation of the propagator . . . . . . . 1043.2.3 the LSZ reduction formula . . . . . . . . . . . . . . . . . . 107

3

Preface

These are lecture notes to QFT I, supplementing the course held in the winterterm 2005. The course is just an introductory one, the whole scheme being

QFT Ibasic concepts and formalism

scalar fields, toy-models

QFT IIelectrons and photons

quantum electrodynamics

Renormalizationpitfalls of the perturbation theory

treatment of infinities

Introduction to the Standard Modelquarks, leptons, intermediate bosons

quantum chromodynamics, electroweak theory

Initially, the aim of this text was not to compete with the textbooks availableon the market, but rather to provide a set of hopefully useful comments andremarks to some of the existing courses. We chose An Introduction to QuantumField Theory by Peskin and Schroeder, since this seemed, and still seems, to bethe standard modern textbook on the subject. Taking this book as a core, theoriginal plan was

• to reorganize the material in a bit different way

• to offer sometimes a slightly different point of view

• to add some material1

Eventually, the text became more and more self-contained, and the resem-blance to the Peskin-Schroeder became weaker and weaker. At the presentpoint, the text has very little to do with the Peskin-Schroeder, except perhapsthe largely common notation.

1Almost everything we have added can be found in some well-known texts, the mostimportant sources were perhaps The Quantum Theory of Fields by Weinberg, Diagrammar byt’Hooft and Veltman and some texts on nonrelativistic quantum mechanics of many-particlesystems.

Chapter 1

Introductions

Quantum field theory is

• a theory of particles

• mathematically ill-defined

• the most precise theory mankind ever had

• conceptually and technically quite demanding

Mainly because of the last feature, it seems reasonable to spend enough timewith introductions. The reason for plural is that we shall try to introduce thesubject in couple of different ways.

Our first introduction is in fact a summary. We try to show how QFT isused in practical calculations, without any attempt to understand why it is usedin this way. The reason for this strange maneuver is that, surprisingly enough,it is much easier to grasp the bulk of QFT on this operational level than toreally understand it. We believe that even a superficial knowledge of how QFTis usually used can be quite helpful in a subsequent, more serious, study of thesubject.

The second introduction is a brief exposition of the nonrelativistic many-particle quantum mechanics. This enables a natural introduction of many basicingredients of QFT (the Fock space, creation and annihilation operators, cal-culation of vacuum expectation values, etc.) and simultaneously to avoid thedifficult question of merging relativity and quantum theory.

It is the third introduction, which sketches that difficult question (i.e. merg-ing relativity and quantum theory) and this is done in the spirit of the Wein-berg’s book. Without going into technical details we try to describe how thenotion of a relativistic quantum field enters the game in a natural way. The maingoal of this third introduction is to clearly formulate the question, to which thecanonical quantization provides an answer.

Only then, after these three introductions, we shall try to develop QFTsystematically. Initially, the development will concern only the scalar fields(spinless particles). More realistic theories for particles with spin 1/2 and 1 arepostponed to later chapters.

1

2 CHAPTER 1. INTRODUCTIONS

1.1 Conclusions

The machinery of QFT works like this:

• typical formulation of QFT — specification of a Lagrangian L• typical output of QFT — cross-sections dσ/dΩ

• typical way of obtaining the output — Feynman diagrams

The machinery of Feynman diagrams works like this:

• For a given process (particle scattering, particle decay) there is a welldefined set of pictures (graphs, diagrams). The set is infinite, but thereis a well defined criterion, allowing for identification of a relatively smallnumber of the most important diagrams. Every diagram consists of severaltypes of lines and several types of vertices. The lines either connect vertices(internal lines, propagators) or go out of the diagrams (external legs). Asan example we can take

• Every diagram has a number associated with it. The sum of these numbersis the so-called scattering amplitude. Once the amplitude is known, it isstraightforward to obtain the cross-section — one just plugs the amplitudeinto a simple formula.

• The number associated with a diagram is the product of factors corre-sponding to the internal lines, external lines and the vertices of the dia-gram. Which factor corresponds to which element of the diagram is thecontent of the so-called Feynman rules. These rules are determined by theLagrangian.

• The whole scheme is Lagrangian↓

Feynman rules↓

Feynman diagrams↓

scattering amplitude↓

cross-section

• Derivation of the above scheme is a long and painful enterprise. Surpris-ingly enough, it is much easier to formulate the content of particular stepsthan to really derive them. And this formulation (without derivation1) isthe theme of our introductory summary.

1It is perhaps worth mentioning that the direct formulation (without derivation) of theabove scheme can be considered a fully sufficient formulation of the real content of QFT. Thispoint of view is advocated in the famous Diagrammar by Nobel Prize winners r’Hooft andVeltman, where ”corresponding to any Lagrangian the rules are simply defined”

1.1. CONCLUSIONS 3

1.1.1 Feynman rules

The role of the Lagrangian in QFT may be a sophisticated issue, but for thepurposes of this summary the Lagrangian is just a set of letters containing theinformation about the Feynman rules. To decode this information one has toknow, first of all, which letters represent fields (to know what the word fieldmeans is not necessary). For example, in the toy-model Lagrangian

L [ϕ] =1

2∂µϕ∂

µϕ− 1

2m2ϕ2 − g

3!ϕ3

the field is represented by the letter ϕ. Other symbols are whatever but notfields (as a matter of fact, they correspond to space-time derivatives, mass andcoupling constant, but this is not important here). Another example is theLagrangian of quantum electrodynamics (QED)

L[ψ, ψ,Aµ

]= ψ (iγµ∂µ − qγµAµ −m)ψ − 1

4FµνF

µν − 1

2ξ(∂µA

µ)2

where Fµν = ∂µAν − ∂νAµ and the fields are ψ, ψ and Aµ (the symbol γµ

stands for the so-called Dirac matrices, and ξ is called a gauge parameter, butthis information is not relevant here).

Now to the rules. Different fields are represented by different types of lines.The usual choice is a simple line for ϕ (called the scalar field), the wiggly linefor Aµ (called in general the massless vector field, in QED the photon field) anda simple line with an arrow for ψ and ψ (called in general the spinor field, inQED usually the electron-positron field). ϕ Aµ ψ ψ

The arrows are commonly used for complex conjugated fields, like ψ and ψ(or ϕ∗ and ϕ, if ϕ is complex2). The arrow orientation is very important for ex-ternal legs, where different orientations correspond to particles and antiparticlesrespectively (as we will see shortly).

Every line is labelled by a momentum (and maybe also by some other num-bers). The arrows discussed above and their orientation have nothing to do withthe momentum associated with the line!

The Feynman rules associate a specific factor with every internal line (prop-agator), line junction (vertex) and external line. Propagators are defined by thepart of the Lagrangian quadratic in fields. Vertices are given by the rest of theLagrangian. External line factor depends on the whole Lagrangian and usually(but not necessarily) it takes a form of the product of two terms. One of themis simple and is fully determined by the field itself, i.e. it does not depend onthe details of the Lagrangian, while the other one is quite complicated.

2Actually, in practice arrows are not used for scalar field, even if it is complex. The reasonis that no factors depend on the arrows in this case, so people just like to omit them (althoughin principle the arrows should be there).


vertices

For a theory of one field ϕ, the factor corresponding to the n-leg vertex is3

n-leg vertex = i∂nL∂ϕn

∣∣∣∣ϕ=0

δ4 (p1 + p2 + . . .+ pn)

where pi is the momentum corresponding to the i-th leg (all momenta are un-derstood to be pointing towards the vertex). For a theory with more fields, likeQED, the definition is analogous, e.g. the vertex with l,m and n legs corre-sponding to ψ, ψ and Aµ-fields respectively, is

(l,m, n) -legs vertex = i∂l+m+nL

∂Alµ∂ψ

m∂ψn

∣∣∣∣∣fields=0

δ4 (p1 + p2 + . . .+ pl+m+n)

Each functional derivative produces a corresponding leg entering the vertex.For terms containing space-time derivative of a field, e.g. ∂µϕ, the derivative

with respec to ϕ is defined as4

∂

∂ϕ∂µϕ× something = −ipµ × something + ∂µϕ× ∂

∂ϕsomething

where the pµ is the momentum of the leg produced by this functional derivative.

Clearly, examples are called for. In our toy-model given above (the so-calledϕ3-theory) the non-quadratic part of the Lagrangian contains the third powerof the field, so there will be only the 3-leg vertex = i

∂3

∂ϕ3

(− g

3!ϕ3)δ4 (p1 + p2 + p3) = −igδ4 (p1 + p2 + p3)

In our second example, i.e. in QED, the non-quadratic part of the Lagrangianis −ψqγµAµψ, leading to the single vertex = i

∂3(−qψγµAµψ

)

∂ψ∂ψ∂Aµ

δ4 (p1 + p2 + p3) = −iqγµδ4 (p1 + p2 + p3)

and for purely didactic purposes, let us calculate the vertex for the theory withthe non-quadratic Lagrangian −gϕ2∂µϕ∂

µϕ = i∂4

∂ϕ4

(−gϕ2∂µϕ∂

µϕ)δ4 (p1 + p2 + p3 + p4)

= −ig ∂3

∂ϕ3

(2ϕ∂µϕ∂

µϕ− 2iϕ2pµ1∂µϕ)δ4 (p1 + p2 + p3 + p4)

= −ig ∂2

∂ϕ2

(2∂µϕ∂

µϕ− 4iϕpµ2∂µϕ− 4iϕpµ1∂µϕ− 2ϕ2pµ1p2,µ)δ4 (. . .)

= −i4g ∂∂ϕ

(−ipµ3∂µϕ− ipµ2∂µϕ− ϕp2p3 − ipµ1∂µϕ− ϕp1p3 − ϕp1p2) δ4 (. . .)

= 4ig (p1p2 + p1p3 + p1p4 + p2p3 + p2p4 + p3p4) δ4 (p1 + p2 + p3 + p4)

3We could (should) include (2π)4 on RHS, but we prefer to add this factor elsewhere.4 ∂∂ϕ∂µϕ is by definition equal to −ipµ, and the Leibniz rule applies to ∂

∂ϕ, as it should

apply to anything worthy of the name derivative.

1.1. CONCLUSIONS 5

propagators

Propagators are defined by the quadratic part of the Lagrangian, they are neg-ative inverses of the 2-leg vertices with the satisfied δ-function (the δ-functionis integrated over one momentum)5

propagator = −(i∂2L∂ϕ2

∣∣∣∣ϕ=0,p′=−p

)−1

For complex fields one uses ∂2L/∂ϕ∗∂ϕ, definitions for other fields are similar.The examples below are more than examples, they are universal tools to be

used over and over. The point is that the quadratic parts of Lagrangians arethe same in almost all theories, so once the propagators are calculated, they canbe used in virtually all QFT calculations.

The quadratic part of the scalar field Lagrangian is 12∂µϕ∂

µϕ − 12m

2ϕ2,leading to ∂2L/∂ϕ2|ϕ=0,p′=−p = −p.p′ −m2|p′=−p = p2 −m2, i.e. =

i

p2 −m2

The quadratic part of the spinor field Lagrangian is ψ (iγµ∂µ −m)ψ, leadingto ∂2L/∂ψ∂ψ|fields=0,p′=−p = γµpµ −m, i.e. =

i

γµpµ −m=

i (γµpµ +m)

p2 −m2

where we have utilized the identity (γµpµ −m) (γµpµ +m) = p2−m2, which atthis stage is just a God-given identity, allowing to write the propagator in thestandard way with p2 −m2 in the denominator.

Finally, for the massless vector field the quadratic Lagrangian is− 14FαβF

αβ−12ξ (∂αA

α)2leading to6,7,8 ∂2L/∂Aµ∂Aν |fields=0,p′=−p = (1− 1

ξ )pµpν − p2gµν =

i(1− 1

ξ

)pµpν − p2gµν

=−i(gµν − (1− ξ) pµpν/p

2)

p2

Surprisingly enough, this is almost everything one would ever need as to thepropagators. In the Standard Model, the spinor propagator describes quarksand leptons, the massless vector propagator describes photon and gluons, thescalar propagator describes the Higgs boson. The only missing propagator is themassive vector one, describing the W± and Z0 bosons. This can be, however,worked out easily from the Lagrangian − 1

4FαβFαβ + 1

2m2AαA

α (the result is−i(gµν − pµpν/m2

)(p2 −m2)−1, the derivation is left as an exercise).

5This definition does not include the so-called iε-prescription, more about this later.6For L = 1

2

[

(

∂αAβ

) (

∂βAα)

−(

∂αAβ

) (

∂αAβ)

− 1ξ(∂αAα)

(

∂βAβ)

]

one has

∂L∂Aµ

= −i(

pαgµβ∂

βAα − pαgµβ∂

αAβ − pαgαµ

ξ∂βA

β)

= −i(

pα∂µAα − pα∂αAµ − pµ

ξ∂βA

β)

∂2L∂Aµ∂Aν

= −pαp′µgαν + pαp′αgµν + 1ξpµp′βg

βν = p.p′gµν − p′µpν + 1ξpµp′ν

7To find the matrix inverse toMλµ(1−ξ−1)pλpµ−p2gλµ one may either make an educatedguess M−1

µν = Agµν +Bpµpν (there is nothing else at our disposal) and solve for A and B, orone may simply check that [(1− ξ−1)pλpµ − p2gλµ]

(

−gµν + (1− ξ) pµpν/p2)

= gλν p2.

8Let us remark that without the term 12ξ

(∂αAα)2 the propagator would not exist, since

the 2-leg vertex would have no inverse. Two specific choices of the parameter ξ are known asthe Feynman gauge (ξ = 1) and the Landau gauge (ξ = 0).


external legs

The factor corresponding to an external leg is, as a rule, the product of twofactors. Let us start with the simpler one. For the scalar field ϕ (representinga particle with zero spin) this factor is the simplest possible, it equals to 1. Forother fields (representing particles with higher spins) there is a nontrivial firstfactor for each external leg. This factor is different for particles and antiparticles.It also distinguishes between ingoing and outgoing particles (i.e. between theinitial and the final state). The factor depends on the particle momentum andspin, but we are not going to discuss this dependence in any detail here.

As to the massless vector field Aµ (e.g. for the photon, where antiparticle =particle) this factor is

ingoing particle εµoutgoing particle ε∗µ

For the spinor field (e.g. for the electron and positron, which are distinguishedin diagrams by the orientation of the arrow) the factor is

ingoing particle arrow towards the diagram uingoing antiparticle arrow out of the diagram v

outgoing particle arrow out of the diagram uoutgoing antiparticle arrow towards the diagram v

These rules are universal, independent of the specific form of the Lagrangian.

Examples for electrons and photons may illuminate the general rules. Wewill draw diagrams from the left to the right, i.e. ingoing particles (initial state)are on the left and outgoing particles (final state) on the right9.

process typical diagram first external legs factors

e−γ → e−γ u, εµ, u, ε∗ν

e+γ → e+γ v, εµ, v, ε∗ν

e+e− → e+e− v, u, v, u

e−e− → e−e− u, u, u, u

9Note that some authors, including Peskin-Schroeder, draw the Feynman diagrams otherway round, namely from the bottom to the top.

1.1. CONCLUSIONS 7

Now to the second factor corresponding to an external leg. It has a prettysimple appearance, namely it equals to

√Z, where Z is a constant (the so-called

wave-function renormalization constant) dependent on the field correspondingto the given leg. The definition and calculation of Z are, however, anything butsimple.

Fortunately, the dominant part of vast majority of cross-sections and de-cay rates calculated by means of Feynman diagrams is given by the so-calledtree diagrams (diagrams containing no closed loops), and at the tree level theZ constant is always equal to 1. So while staying at the tree level, one canforget about Z completely. And since our first aim is to master the tree levelcalculations, we can ignore the whole Z-affair until the discussion of loops andrenormalization. The following sketch of the Z definition is presented only forthe sake of completeness.

Unlike all other Feynman rules, the Z constant is defined not directly viathe Lagrangian, but rather via an infinite sum of Feynman diagrams10. Thesaid sum, called the full or dressed propagator, contains all diagrams with twoexternal legs corresponding to the field under consideration. These two externallegs are treated in a specific way — the corresponding factor is the propagatorrather then the external leg factor (were it not for this specific treatment, thedefinition of external leg factor would be implicit at best, or tautological atworst). The dressed propagator is a function of the external leg momentum(both legs have the same momentum due to the vertex momentum δ-functions)and, as a rule, has a pole in the p2-variable. The residuum at this pole is thewanted Z.

This definition, as it stands, applies only to the scalar field. For higherspins the dressed propagator is a matrix and the Z constant is defined via theeigenvalues of this matrix. So one can have, in principle, several different Zconstants corresponding to one field. For the electron-positron field, however,there turns out to be only one such constant and the same is true for the photonfield.

In addition to this, there is yet another very important ingredient in theexternal leg treatment. The external leg factor stands not only for the simple(bare) external leg, but rather for the dressed external leg (with all loop correc-tions). In other words, when calculating a scattering amplitude, one should notinclude diagrams with loop corrections on external legs. These diagrams are, ina sense, taken into account via the

√Z factors11.

Too complicated? Never mind. Simply forget everything about Z, it will besufficient to recall it only much later, when dealing with renormalization.

Remark: As we have indicated, in some circumstances the external leg factormay be even more complicated than the product of two terms (one of them being√Z). This happens when there are non-vanishing sums of all diagrams with two

external legs corresponding to different fields. This is only rarely the case andalways indicates that our choice of fields was not the most appropriate one. Theremedy for this trouble is quite ordinary: after a suitable re-definition (just asimple linear combination) of the fields the trouble simply drops out.

10For the defininition of Feynman diagrams see the next subsection.11After being forced to calculate the loop corrections to a simple line in order to obtain Z,

one does not need to calculate them again when calculating the scattering amplitude. Thereis at least some justice in this world.


1.1.2 Feynman diagrams

diagrams for a given process contributing at a given order

A process defines external legs, both ingoing and outgoing. A Feynman diagramcorresponding to this process is any connected diagram (graph) with this set ofexternal legs interconnected by the internal lines (propagators) allowed by thetheory, via the vertices allowed by the theory.

There is usually an infinite number of such diagrams. Still, only a finitenumber contribute at a given order (of the perturbation theory in fact, but thisdoes not concern us here). The order may be defined in at least three differentways, namely as a) the number of vertices, b) the power of the coupling constantor c) the number of (independent) loops. If there is only one type of vertex inthe theory, these three definitions are equivalent12. If one has more types ofvertices, but all characterized by the same coupling constant13, then the firstdefinition is not used and the other two are not equivalent.

As an example, let us consider a scattering AB → 12, described by eitherϕ4- or ϕ3-theory. At the leading order (lowest nonzero order, tree level) one has

ϕ4-theory: ϕ3-theory: Note that the second and the third diagrams for the ϕ3-theory are not equiva-lent, they contain different vertices (intersections of different lines). At the nextto the leading order (1-loop level) in the ϕ4-theory one has The 1-loop diagrams for the ϕ3-theory are left for the reader as an exercise, aswell as are the 2-loops diagrams. Other useful exercises are some QED processes,such as e−γ → e−γ, e−e+ → e−e+, e−e+ → e−e+γ, etc.

12Proof: The equivalence of the first two definitions is evident (every vertex contributesby the same coupling constant). As to the equivalence of the third definition, let us denotethe number of vertices, internal lines, external lines and independent loops by V , I, E and Lrespectively. The famous Euler’s Theorem states V = I −L+1. This is to be combined withnV = 2I +E where n is the number of legs of the vertex of the theory. The latter equation isnothing but a simple observation that we can count the number of lines by counting verticesand multiplying their number by n, but we are double-counting internal lines in this way.When combined, the equations give (n− 2) V = 2L+E−2, i.e. for a given E the dependenceof V on L is linear.

13A good example is the so-called scalar electrodynamics (spinless particles and photons)defined by the Lagrangian L [ϕ] = (Dµϕ)

∗Dµϕ−m2ϕ∗ϕ, where Dµ = ∂µ + iqAµ.

1.1. CONCLUSIONS 9

the factor corresponding to a given diagram

The factor corresponding to a diagram is the product14 of factors correspondingto all external lines, internal lines and vertices of the diagram, multiplied by15

• an extra factor (2π)4for each vertex

• an extra factor∫

d4k(2π)4

for each propagator (with the four-momentum k)

• an extra so-called combinatorial factor, to be discussed later

• some extra factors of (−1) related to fermionic lines16

Examples17: = −ig (2π)4 δ4 (pA + pB − p1 − p2)

= −g2 (2π)8∫

d4k(2π)4

ik2−m2 δ

4 (pA + pB − k) δ4 (k − p1 − p2)

= −i g2

(pA+pB)2−m2 (2π)4δ4 (pA + pB − p1 − p2) = −g2 (2π)8 1

2

∫d4k(2π)4

d4k′

(2π)4i

k2−m2i

k′2−m2 δ4 (pA + pB − k − k′)×

↑combinatorial factor × δ4 (k + k′ − p1 − p2)

= 12g

2∫d4k 1

k2−m21

(pA+pB−k)2−m2 δ4 (pA + pB − p1 − p2)

= −q2 (2π)8∫

d4k(2π)4

uBγµδ4 (pA + pB − k)

i(γλkλ+m)k2−m2 ×

×γνδ4 (k − p1 − p2)u2ε∗1,µεA,ν

= −iq2 uBγµ(γλ(pA+pB)λ+m)γνu2

(pA+pB)2−m2 ε∗1,µεA,ν (2π)4 δ4 (pA + pB − p1 − p2)

14If individual factors are simple numbers, one does not care about their ordering. In somecases, however, these factors are matrices, and then the proper ordering is necessary. Thebasic rule here is that for every line with an arrow, the factors are to be ordered ”against thearrow”, i.e. starting from the end of the arrow and going in the reverse direction.

15The factors (2π)4 and∫

d4k(2π)4

for vertices and propagators could be included in the

definition of vertices and propagators, but it is a common habit to include them in this way.16The factor (−1) for every closed fermionic loop, and the relative minus sign for the di-

agrams, which can be obtained from each other by an interchange of two fermionic lines.Diagrams related to each other by the omission or addition of boson lines have the same sign.

17All momenta are understood to flow from the left to the right. One can, of course, chooseanother orientation of momenta and change the signs in δ-functions correspondingly.


combinatorial factors

Beyond the tree level, a diagram may require the so-called combinatorial fac-tor18, which is usually the most cumbersome factor to evaluate. Therefore, itseems reasonable to start with some simple rules of thumb:

• If two vertices are connected by n different internal lines, the correspondingcombinatorial factor is 1/n! 1

3! 1

2!2!

• If a line starts and ends in the same vertex, it contributes by 1/2 to thecombinatorial factor 1

2 1

2

1

2!

• If N permutations of n vertices do not change the diagram, the correspond-ing combinatorial factor is 1/N (note that if not all n! permutations leave thediagram untouched, then N 6= n!) 1

2

1

3! 1

231

3!

• The above list is not exhaustive, e.g. it does not allow to find the correctcombinatorial factor in the following case 1

8

A systematic, but less illustrative, prescription goes something like this: As-sign a label to each end of every line in the diagram. The labeled diagram ischaracterized by sets of labels belonging to the common vertex, and by pairsof labels connected by a line. Permutation of labels would typically lead to adiagram labeled in a different way. Some permutations, however, lead to theidentically labeled diagram. Find out the number N of all such permutations(leaving the labeled diagram unchanged). The combinatorial factor of the dia-gram is 1/N .

Needless to say, this prescription is difficult to follow practically. Fortunately,in simple cases (and one seldom needs to go much further) it can be reducedeasily to the above rules of thumb. Actually, this prescription is not very popular,since it is not very economical. Probably the most economical way of generatingthe Feynman diagrams with the correct combinatorial factors is provided bythe functional integral formulation of QFT. For a curious reader, a reasonablyeconomical and purely diagrammatic algorithm is presented in 1.1.5.

18Why such a name: the diagrams represent specific terms of perturbative expansion, thenumber of terms corresponding to a given diagram is given by some combinatorics.

1.1. CONCLUSIONS 11

1.1.3 Scattering amplitude

The definition of the scattering amplitude Mfi is quite simple:

the sum of Feynman diagrams = iMfi (2π)4δ(4) (Pf − Pi)

where Pi and Pf are the overall initial and final momentum respectively. By thesum of the diagrams, the sum of the corresponding factors is meant, of course.

Examples (to be checked by the reader):

• ϕ4-theory, AB → 12 scattering

tree-level Mfi = −g

1-loop-level Mfi = − 12g

2 [I (pA + pB) + I (pA − p1) + I (pA − p2)] + . . .

I (p) = i∫

d4k(2π)4

1k2−m2

1(p−k)2−m2

where the ellipsis stands for the four diagrams with ”bubbles” on externallegs19, which will turn out to be irrelevant, no matter how unclear it mayseem now

• ϕ3-theory, AB → 12 scattering

tree-level Mfi = − g2

(pA+pB)2−m2 − g2

(pA−p1)2−m2 − g2

(pA−p2)2−m2

1-loop-level the result is intricate and not that illuminatingbut the reader is encouraged to work out some loop diagrams

• ϕ2Φ-theory20, A→ 12 decay

tree-level Mfi = −g

1-loop-level Mfi = −g3J (p1, p2) + . . .

J (p1, p2) = i∫

d4k(2π)4

1k2−M2

1(p1+k)2−m2

1(p2−k)2−m2

where the ellipsis stands again for the (three, irrelevant) diagrams with”bubbles” on external legs.

19Each such diagram contributes by the factor

− g2

p2i −m2

∫

d4k

(2π)4i

k2 −m2

which is suspicious both because of vanishing denominator (p2i = m2) and divergent integral.20A theory for two different fields ϕ and Φ, defined by the Lagrangian

L [ϕ,Φ] =1

2∂µϕ∂

µϕ− 1

2m2ϕ2 +

1

2∂µΦ∂

µΦ− 1

2M2Φ2 − g

2ϕ2Φ


1.1.4 cross-sections and decay rates

The cross-section for a scattering AB → 12 . . . n is given by

dσ = (2π)4δ4 (Pf − Pi)

1

4√(pA.pB)

2 −m2Am

2B

|Mfi|2n∏

i=1

d3pi

(2π)32Ei

while the analogous quantity for a decay A→ 12 . . . n is

dΓ = (2π)4 δ4 (Pf − Pi)1

2EA|Mfi|2

n∏

i=1

d3pi

(2π)3 2Ei

Because of the δ-function present in these formulae, one can relatively easilyperform four integrations on the RHS. For the quite important case of n = 2,i.e. for AB → 12 and A→ 12, the result after such integrations is in the CMS21

dσCMS =1

64π2

|~p1||~pA|

1

(pA + pB)2 |Mfi|2 dΩ1

dΓCMS =1

32π2

|~p1|m2

A

|Mfi|2 dΩ1

Examples:

• ϕ4-theory, AB → 12 scattering, tree level

dσCMS =1

64π2

1

sg2dΩ s = (pA + pB)

2

In this case the differential cross-section does not depend on angles, so onecan immediately write down the total cross-section σCMS = g2/16πs.

• ϕ3-theory, AB → 12 scattering, tree level

dσCMS =g4

64π2s

(1

s−m2+

1

t−m2+

1

u−m2

)2

dΩt = (pA − p1)

2

u = (pA − p2)2

where s, t, u are the frequently used so-called Mandelstam variables.

Exercises:

• AB → 12 scattering at the tree level, within ”the ϕ4-theory with deriva-tives”, i.e. the theory of scalar fields with the non-quadratic part of theLagrangian being Lint = − g

4ϕ2∂µϕ∂

µϕ.

• Φ → ϕϕ decay rate, ϕϕ→ ϕϕ, ϕΦ → ϕΦ and ϕϕ→ ΦΦ cross-sections atthe tree level, for the ϕ2Φ-theory defined in the footnote on the page 11.

21Once the result is known in the CMS, one can rewrite it into any other system, but thisis not trivial, since experimentally natural quantities like angles are not Lorentz covariant. Itis therefore useful to have explicit formulae for commonly used systems, e.g. for the so-calledtarget system (the rest frame of the particle B)

dσTS =1

64π2

|~p1||~pA|

1

mB

1

(EA +mB) |~p1| − EA |~pA| cos ϑ∣

∣Mfi

∣

∣

2dΩ1

where all quantities (~p,E, ϑ,Ω) are understood in the target frame.

1.1. CONCLUSIONS 13

rough estimations of real life processes (optional reading)

Calculations with scalar fields (spinless particles) are usually significantly easierthan those of real life processes in the Standard Model. To get rough (but stillilluminating) estimations for the realistic processes, however, it is frequentlysufficient to ignore almost all the complications of the higher spin kinematics.To arrive at such rough estimates, the Feynman rules are supplemented by thefollowing rules of thumb:

• All propagators are treated as the scalar propagator i(p2−m2)−1, i.e. thenumerator of a higher spin propagator, which in fact corresponds to thesum over spin projections, is simply ignored. More precisely, it is replacedby a constant, which for the scalar and vector fields is simply 1, and forthe spinor field it is M (this is necessary for dimensional reasons) whereM is usually the mass of the heaviest external particle involved.

• Spins of external legs are treated in the same brutal way — they areignored: this amounts to ignoring the εµ factors for external photons

completely, and to replacing spinor external legs factors u, v etc. by√M .

• Any matrix structures in vertices, like γµ in the QED vertex, are ignored22.

Examples (just for illustration):

• W -boson decay23 W+ → e+νe, µ+νµ, τ+ντ , ud, cs, tb

the SM vertex − i g2√2γµ 1−γ5

2

the amplitude Mfi = −i g2√2εµueγ

µ 1−γ5

2 vν ≈ g22√2MW

the decay rate dΓCMS = 132π2

|~pe|M2

W

g22

8 M2WdΩ ΓCMS =

g22MW

256π

• Z-boson decay24 Z0 → e+e−, νeνe, uu, dd, . . .

the SM vertex − i g22 cosϑW

γµ(v − aγ5

)

the amplitude Mfi ≈ g24 cosϑW

MZ

the decay rate ΓCMS =g22MZ

256 cos2 ϑW

22In fact, if the beast called γ5 enters the game, there is a good reason to treat 12

(

1± γ5)

rather than(

1± γ5)

as 1, but we are going to ignore such subtleties.

23For negligible electron mass |~pe| = Ee = MW2

. The correct tree level result isg22MW

48π,

so we have missed the factor of 16. A decay to another pair (µ+νµ, . . .) is analogous, for

quarks one has to multiply by 3 because of three different colors. For the top quark, theapproximation of negligible mass is not legal.

24We have used vf ≈ af ≈ 12. When compared to the correct result, we have again missed

the factor of 16and some terms proportional to sin2ϑW and sin4ϑW.


• νµ-capture25 νµµ

− → νee−

the diagram the amplitude Mfi ≈ g2

2

8

m2µ

k2−M2W

≈ g22m

2µ

8M2W

the cross-section σCMS =g42

1024π|~pe||~pµ|

m2µ

M4W

• µ-decay26 µ− → νµνee−

the diagram the amplitude Mfi ≈ g2

2

8

m2µ

k2−M2W

≈ g22m

2µ

8M2W

the decay rate dΓ = 1(2π)5

12Eµ

g42m

4µ

64M4Wδ4 (Pf − Pi)

d3pe

2Ee

d3pνe

2Eνe

d3pνµ

2Eνµ

Γ = 1211

1(2π)3

g42m

5µ

M4W

25k2 ≈ m2µ can be neglected when compared to M2

W26The diagram is the same as in the previous case, but the muon-neutrino leg is now

understood to be in the final state.Since there are three particles in the final state, we cannot use the ready-made formulae

for the two-particle final state. The relevant steps when going from dΓ to Γ are: the identity

δ4(

Pf − Pi

) d3pνe2Eνe

d3pνµ2Eνµ

= 18dΩνµ , the standard trick d3pe = p2edpedΩe = peEedEedΩe and

finally pe = Ee for negligible me, leading to∫ maxmin

EedEe =∫mµ/20 EedEe = m2

µ/8. The

correct result is 83times our result.

1.1. CONCLUSIONS 15

1.1.5 A redundant paragraph

This subsection adds nothing new to what has already been said, it just presentsanother (perhaps more systematic) way of obtaining the diagrams for a givenprocess, contributing at a given order, together with appropriate combinato-rial factors. It is based on the so-called Dyson-Schwinger equation, which forcombined ϕ3- and ϕ4-theories reads = + . . . + +

1

2! +1

3!The shaded blob represents sum of all diagrams (including disconnected ones)and the ellipsis stands for the diagrams analogous to the first one on the RHS,with the external leg on the left side of the diagram connected directly with oneof the external legs represented by the dots on the right. The factors 1/n! takecare of the equivalent lines, for non-equivalent lines there is no such factor.

To reveal the combinatorial factors for diagrams corresponding to a givenprocess, the DS equation is used in an iterative way: one starts with the sum ofall diagrams with the corresponding number of external legs, then one takes anyleg and applies the Dyson-Schwinger equation to it, then the same is repeatedwith some other leg etc., until one reaches

(the structure one is interested in)× + . . .

where the ellipsis stands for diagrams with disconnected external legs + higherorders. The combinatorial factors are now directly read out from ”the structureone is interested in” as the factors multiplying the diagrams.

Let us illustrate the whole procedure by the diagram with two externallegs within the ϕ4-theory (in which case the second last term in the above DSequation is missing). The starting point is the DS equation for 2 external legs = × +

1

3!Now the DS equation is applied to some other leg, say to the 2nd external leg = × +

3

3! +1

3!

1

3!If we are going to work only up to the 1st order (in the number of vertices), thenthe last term is already of higher order, and the second term is again processedby the DS equation, to finally give =

+1

2

× + . . .

The factor in front of the second diagram in the brackets is indeed the correctcombinatorial factor for this diagram. (As an exercise the reader may try to goone order higher.)


As another example let us consider the AB → 12 scattering within theϕ4-theory. Again, the starting point is the DS equation =

1

3! + . . .

where the ellipsis stands for terms with disconnected external legs. The DSequation is now applied to some other leg, say the external leg B =

3

3! +1

3!

1

3! + . . .

The RHS enjoys yet another DS equation, now say on the external leg 1, to give

= +1

2

1

3! +1

3!

3

3!+1

3!

3

3!+ . . .

The first two (last two) terms on the RHS come from the first (second) diagramon the RHS above, terms with more than two vertices are included into ellipsis.The first diagram is now treated with the help of the previous result for the2-leg diagrams, the other three are treated all in the same way, which we willdemonstrate only for the second diagram: we use the DS equation once more,now say for the external leg 2

1

2

1

3! =2

2

1

3! +1

2

3

3! + . . .

=

2

2

3

3! +2

2

3

3!

× + . . .

Putting the pieces together, one finally obtains the one-loop diagrams with thecorrect combinatorial factors

12 1

2 12

12 1

2 12 1

2

1.2. MANY-BODY QUANTUM MECHANICS 17

1.2 Many-Body Quantum Mechanics

The main characters of QFT are quantum fields or perhaps creation and annihi-lation operators (since the quantum fields are some specific linear combinationsof the creation and annihilation operators). In most of the available textbookson QFT, the creation and annihilation operators are introduced in the processof the so-called canonical quantization27. This, however, is not the most naturalway. In opinion of the present author, it may be even bewildering, as it maydistort the student’s picture of relative importance of basic ingredients of QFT(e.g. by overemphasizing the role of the second quantization). The aim of thissecond introduction is to present a more natural definition of the creation andannihilation operators, and to demonstrate their main virtues.

1.2.1 Fock space, creation and annihilation operators

Fock space

1-particle system

the states constitute a Hilbert space H1 with an orthonormal basis |i〉, i ∈ N

2-particle system28

the states constitute the Hilbert space H2 or H2B or H2

F, with the basis |i, j〉non-identical particles H2 = H1 ⊗H1 |i, j〉 = |i〉 ⊗ |j〉identical bosons H2

B ⊂ H1 ⊗H1 |i, j〉 = 1√2(|i〉 ⊗ |j〉+ |j〉 ⊗ |i〉)

identical fermions H2F ⊂ H1 ⊗H1 |i, j〉 = 1√

2(|i〉 ⊗ |j〉 − |j〉 ⊗ |i〉)

n-particle system (identical particles)

the Hilbert space is either HnB or Hn

F ⊂ H1 ⊗ . . .⊗H1

︸︷︷︸n

, with the basis

|i, j, . . . , k〉 = 1√n!

∑

permutations

(±1)p |i〉 ⊗ |j〉 ⊗ . . .⊗ |k〉︸︷︷︸n

where p is the parity of the permutation, the upperlower sign applies to bosons

fermions

0-particle system

1-dimensional Hilbert space H0 with the basis vector |0〉 (no particles, vacuum)

Fock spacedirect sum of the bosonic or fermionic n-particle spaces

HB =

∞⊕

n=0

HnB HF =

∞⊕

n=0

HnF

27There are exceptions. In the Weinberg’s book the creation and annihilation operators areintroduced exactly in the spirit we are going to adopt in this section. The same philosophy isto be found in some books on many-particle quantum mechanics. On the other hand, someQFT textbooks avoid the creation and annihilation operators completely, sticking exclusivelyto the path integral formalism.

28This is the keystone of the whole structure. Once it is really understood, the rest followssmoothly. To achieve a solid grasp of the point, the reader may wish to consult the couple ofremarks following the definition of the Fock space.


Remark: Let us recall that for two linear spaces U (basis ei, dimension m) andV (basis fj, dimension n), the direct sum and product are linear spaces U ⊕ V(basis generated by both ei and fj, dimension m+n) and U⊗V (basis generatedby ordered pairs (ei, fj), dimension m.n).

Remark: The fact that the Hilbert space of a system of two non-identicalparticles is the direct product of the 1-particle Hilbert spaces may come as notcompletely obvious. If so, it is perhaps a good idea to start from the question whatexactly the 2-particle system is (provided that we know already what the 1-particlesystem is). The answer within the quantum physics is not that straightforwardas in the classical physics, simply because we cannot count the quantum particlesdirectly, e.g. by pointing the index finger and saying one, two. Still, the answeris not too complicated even in the quantum physics. It is natural to think of aquantum system as being 2-particle iffa) it contains states with sharp quantum numbers (i.e. eigenvalues of a completesystem of mutually commuting operators) of both 1-particle systems, and thisholds for all combinations of values of these quantum numbersb) such states constitute a complete set of statesThis, if considered carefully, is just the definition of the direct product.

Remark: A triviality which, if not explicitly recognized, can mix up one’s mind:H1 ∩H2 = ∅, i.e. the 2-particle Hilbert space contains no vectors correspondingto states with just one particle, and vice versa.

Remark: The fact that multiparticle states of identical particles are representedby either completely symmetric or completely antisymmetric vectors should befamiliar from the basic QM course. The former case is called bosonic, the latterfermionic. In all formulae we will try, in accord with the common habit, totreat these two possibilities simultaneously, using symbols like ± and ∓, wherethe upper and lower signs apply to bosons and fermions respectively.

Remark: As to the basis vectors, our notation is not the only possible one. An-other widely used convention (the so-called occupation number representation)denotes the basis vectors as |n1, n2, . . .〉, where ni is the number of particles inthe i-th 1-particle state. So e.g. |2, 2, 2, 4, 4〉 ⇔ |0, 3, 0, 2, 0, 0, 0, . . .〉, where theLHS is in our original notation while the RHS is in the occupation number rep-resentation. The main drawback of the original notation is that it is not unique,e.g. |1, 2, 3〉 and ± |1, 3, 2〉 denotes the same vector. One should be thereforecareful when summing over all basis states. The main drawback of the occupa-tion number representation is typographical: one cannot write any basis vectorwithout the use of ellipsis, and even this may sometimes become unbearable (trye.g. to write |49, 87, 642〉 in the occupation number representation).

Remark: The basis vectors |i, j, . . . , k〉 or |n1, n2, . . .〉 are not all normalized tounity (they are, but only if all i, j, . . . , k are mutually different, i.e. if none ofni exceeds 1). If some of the i, j, . . . , k are equal, i.e. if at least one ni > 1, thenthe norm of the fermionic state is automatically zero (this is the Pauli exclusionprinciple), while the norm of the bosonic state is

√n1! n2! . . .. Prove this.

Remark: A triviality which, if not explicitly recognized, can mix up one’s mind:the vacuum |0〉 is a unit vector which has nothing to do with the zero vector 0.


creation and annihilation operators

Let |i〉 (i = 1, 2, . . .) be an orthonormal basis of a 1-particle Hilbert space, and|0〉, |i〉, |i, j〉, |i, j, k〉, . . . (i ≤ j ≤ k ≤ . . .) an orthogonal basis of the Fockspace. The creation and annihilation operators are defined as follows

creation operator a+iis a linear operator, which maps the n-particle basis vector to the (n+1)-particlevector by adding one particle in the i-th state (the particle is added at the firstposition in the resulting vector; for bosons this rule does not matter, for fermionsit determines the sign)

a+i |0〉 = |i〉 a+i |j〉 = |i, j〉 a+i |j, k, . . .〉 = |i, j, k, . . .〉

annihilation operator aiis a linear operator, which maps the n-particle basis vector to the (n−1)-particlevector by removing one particle in the i-th state. The particle is removedfrom the first position of the original vector, and if it is not there, the originalvector must be reshuffled (for bosons this rule does not matter, for fermionsit determines the sign). If the original vector contains more than one particlein the i-th state, the whole procedure is performed with each of them and theresults are summed up. If the original vector does not contain a particle in thei-th state, the result is the zero vector.

ai |0〉 = 0 ai |j〉 = δij |0〉ai |j, k, l . . .〉 = δij |k, l, . . .〉 ± δik |j, l, . . .〉+ δil |j, k, . . .〉 ± . . .

Both creation and annihilation operators are linear and they are defined on thebasis vectors. Consequently they are defined for any vector.

Remark: In the occupation number representation, the definitions read

bosons a+i |n1, . . . , ni, . . .〉 = |n1, . . . , ni + 1, . . .〉ai |n1, . . . , ni, . . .〉 = ni |n1, . . . , ni − 1, . . .〉

fermions a+i |n1, . . . , ni = 0, . . .〉 = (−1)pi |n1, . . . , ni = 1, . . .〉

a+i |n1, . . . , ni = 1, . . .〉 = 0ai |n1, . . . , ni = 0, . . .〉 = 0ai |n1, . . . , ni = 1, . . .〉 = (−1)pi |n1, . . . , ni = 0, . . .〉

pi =∑i−1

k=1 nk

Creation and annihilation operators are very useful, because• they enable the most natural description of processes in which the number ofparticles is not conserved, i.e. in which particles are created and/or destroyed• any linear operator can be expressed in terms of the creation and annihilationoperators, namely as a sum of products of these operators• there is a standard and routine method of how to calculate matrix elementsof operators expressed in terms of the creation and annihilation operators.

In view of how frequent the processes of particle creation and annihilation are(decays and inelastic scatterings in the atomic, nuclear, subnuclear and solidstate physics), the first point is evidently very important. And in view of howoften the QM calculations are just the calculations of various matrix elementsof linear operators, the other two points are clearly also very important.


key attributes of the creation and annihilation operators

Perhaps the three most important are29

• a+i = a†i i.e. a+i and ai are Hermitian conjugated

• a+i ai (no summation) is the operator of number of particles in the i-th state

•[ai, a

+j

]∓ = δij [ai, aj]∓ =

[a+i , a

+j

]∓ = 0 where [x, y]∓ = xy∓yx

The proof is an easy exercise, recommended to anybody who wants to becomequickly accustomed to elementary manipulations with the a+i , ai operators.

30

The above relations comprise almost everything we would need as to thecreation and annihilation operators. Still, there are some additional useful re-lations, like the relation between different sets of the creation and annihilationoperators. Let |α〉 (α ∈ N) be an orthonormal basis of the 1-particle Hilbertspace, different from the original basis |i〉. Starting from this new basis, onecan define the new set of creation and annihilation operators a+α and aα. Therelation between these operators and the original ones reads

a+α =∑

i

〈i|α〉 a+i aα =∑

i

〈α|i〉 ai

This follows directly from the relation |α〉 = |i〉〈i|α〉 (with the Einstein summa-tion convention understood) and from the definition of the creation and annihi-lation operators (check it).

29Note that a+i and ai operators could be (and often are) introduced in the reversed order.In that case, the (anti)commutation relations are postulated and the Fock space is constructedafterwards for a+i and ai to have something to live in. It is perhaps just a matter of taste,but the present author strongly prefers the ”more natural logic” of this section. Later in theselectures, however, we will encounter also the reversed logic of the second quantization.

30A sketch of the proof :Hermitian conjugation (for bosons ni, n′

i ∈ N , for fermions ni, n′i ∈ 0, 1)

⟨

. . . n′i . . .

∣

∣ ai |. . . ni . . .〉 =⟨

. . . n′i . . . | . . . ni − 1 . . .

⟩

ni =⟨

. . . n′i . . . | . . . ni − 1 . . .

⟩

ni δn′

i,ni−1

q

〈. . . ni . . .| a+i∣

∣. . . n′i . . .

⟩

=⟨

. . . ni . . . | . . . n′i + 1 . . .

⟩

=⟨

. . . ni . . . | . . . n′i + 1 . . .

⟩

δni,n′

i+1

particle number operatorbosons a+i ai |. . . ni . . .〉 = a+i ni |. . . ni − 1 . . .〉 = nia

+i |. . . ni − 1 . . .〉 = ni |. . . ni . . .〉

fermions a+i ai |. . . 0 . . .〉 = 0

a+i ai |. . . 1 . . .〉 = a+i (−1)pi |. . . 0 . . .〉 = (−1)2pi |. . . 1 . . .〉commutation relation (bosons)[

ai, a+i

]

|. . . ni . . .〉 = ai |. . . ni + 1 . . .〉 − nia+i |. . . ni − 1 . . .〉

= (ni + 1) |. . . ni . . .〉 − ni |. . . ni . . .〉 = |. . . ni . . .〉[

ai, a+j

]

|. . . ni . . . nj . . .〉 = ai |. . . ni . . . nj + 1 . . .〉 − a+j |. . . ni − 1 . . . nj . . .〉= |. . . ni − 1 . . . nj + 1 . . .〉 − |. . . ni − 1 . . . nj + 1 . . .〉 = 0

anticommutation relation (fermions)

ai, a+i

|. . . 1 . . .〉 = 0 + (−1)pi a+i |. . . 0 . . .〉 = (−1)2pi |. . . 1 . . .〉 = |. . . 1 . . .〉

ai, a+i

|. . . 0 . . .〉 = (−1)pi ai |. . . 1 . . .〉+ 0 = (−1)2pi |. . . 0 . . .〉 = |. . . 0 . . .〉

ai, a+j

|. . . 0 . . . 0 . . .〉 = (−1)pj ai |. . . 0 . . . 1 . . .〉+ 0 = 0

ai, a+j

|. . . 0 . . . 1 . . .〉 = 0

ai, a+j

|. . . 1 . . . 0 . . .〉 = (−1)pi+pj |. . . 0 . . . 1 . . .〉 + (−1)pi+pj−1 |. . . 0 . . . 1 . . .〉 = 0

ai, a+j

|. . . 1 . . . 1 . . .〉 = 0 + (−1)pi a+j |. . . 0 . . . 1 . . .〉 = 0

The other (anti)commutation relations are treated in the same way.


Remark: The basis |i, j, . . .〉, or |n1, n2, . . .〉, arising from a particular basis|i〉 of the 1-particle Hilbert space is perhaps the most natural, but not the onlyreasonable, basis in the Fock space. Actually, any complete set of physically rel-evant commuting operators defines some relevant basis (from this point of view,|n1, n2, . . .〉 is the basis of eigenvectors of the occupation number operators).If the eigenvalues of a complete system of commuting operators are discrete andbounded from below, then one can label both eigenvalues and eigenvectors bynatural numbers. In such a case, the basis defined by the considered systemof operators looks like |N1, N2, . . .〉, and for a basis of this type we can definethe so-called raising and lowering operators, just as we have defined the creation

and annihilation operators: A+i (Ai) raises (lowers) the quantum number Ni by

one.Of a special interest are the cases when the Hamiltonian can be written as a sumof two terms, one of which has eigenvectors |N1, N2, . . .〉 and the second one canbe understood as a small perturbation. If so, the system formally looks like analmost ideal gas made of a new type of particles (created by the A+

i operatorsfrom the state in which all Ni vanish). These formal particles are not to bemistaken for the original particles, which the system is built from.It may come as a kind of surprise that such formal particles do appear frequentlyin many-body systems. They are called elementary excitations, and they comein great variety (phonons, plasmons, magnons, etc.). Their relation to the orig-inal particles is more or less known as the result of either detailed calculations,or an educated guess, or some combination of the two. The description of thesystem is, as a rule, much simpler in terms of the elementary excitations thanin terms of the original particles. This explains the wide use of the elementaryexcitations language by both theorists and experimentalists.

Remark: The reader is perhaps familiar with the operators a+ and a, satisfyingthe above commutation relations, namely from the discussion of the LHO (lin-ear harmonic oscillator) in a basic QM course31. What is the relation betweenthe LHO a+ and a operators and the ones discussed here?The famous LHO a+ and a operators can be viewed as the creation and an-nihilation operators of the elementary excitations for the extreme example ofmany-body system presented by one LHO. A system with one particle that canbe in any of infinite number of states, is formally equivalent to the ideal gas ofarbitrary number of formal particles, all of which can be, however, in just onestate.Do the LHO operators a+ and a play any role in QFT? Yes, they do, but onlyan auxiliary role and only in one particular development of QFT, namely in thecanonical quantization of classical fields. But since this is still perhaps the mostcommon development of the theory, the role of the LHO is easy to be overesti-mated. Anyway, as for the present section (with the exception of the optionalsubsection 1.2.4), the reader may well forget about the LHO.

31Recall a+ = x√

mω/2~ − ip/√2~mω , a = x

√

mω/2~ + ip/√2~mω. The canonical

commutation relation [x, p] = i~ implies for a+ and a the commutation relation of the creationand annihilation operators. Moreover, the eigenvalues of the operator N = a+a are naturalnumbers (this follows from the commutation relation).

Note that the definition of a+ and a (together with the above implications) applies to any1-particle system, not only to the LHO. What makes the LHO special in this respect is theHamiltonian, which is very simple in terms of a+, a. These operators are useful also for thesystems ”close to LHO”, where the difference can be treated as a small perturbation.


1.2.2 Important operators expressed in terms of a+i , ai

As already announced, any linear operator in the Fock space can be written asa polynomial (perhaps infinite) in creation and annihilation operators. Howeverinteresting this general statement may sound, the particular examples are evenmore interesting and very important in practice. We will therefore start withthe examples and return to the general statement only later on.

Hamiltonian of a system of non-interacting particles

Let us consider non-interacting particles (ideal gas) in an external classical fieldwith a potential energy U (x). The most suitable choice of basis in the 1-particle Hilbert space is the set of eigenstates of the 1-particle Hamiltonianp2/2m+ U (x), i.e. the states |i〉 satisfying

(1

2mp2 + U (x)

)|i〉 = Ei |i〉

By this choice, the standard basis of the whole Fock space is determined, namely|0〉, |i〉, |i, j〉, |i, j, k〉, etc. And since the particles do not interact, each of thesebasis states has a sharp value of energy, namely 0, Ei, Ei + Ej , Ei + Ej + Ek,etc., respectively. The Hamiltonian of the system with any number of particlesis the linear operator with these eigenvectors and these eigenvalues. It is veryeasy to guess such an operator, namely H0 =

∑i Eini, where ni is the operator

of number of particles in the i-th state. And since we know how to express ni

in terms of the creation and annihilation operators, we are done

H0 =∑

i

Ei a+i ai

• If, for any reason, we would need to express H0 in terms of another set ofcreation and annihilation operators a+α =

∑i 〈i|α〉 a+i and aα =

∑i 〈α|i〉 ai, it is

straightforward to do so: H0 =∑

α,β Eαβa+αaβ where Eαβ =

∑iEi 〈α|i〉〈i|β〉.

• If one has a continuous quantum number q, rather than the discrete i, thenthe sum

∑i is replaced by the integral

∫dq: H0 =

∫dq E (q) a+q aq. Another

change is that any Kronecker δij is replaced by the Dirac δ (q − q′).

Free particles. 1-particle states labeled by the momentum ~p, with E (~p) = p2

2m

H0 =

∫d3p

p2

2ma+~p a~p

Periodic external field (the 0-th approximation for electrons in solids). Bloch

theorem: 1-particle states are labeled by the level n, the quasi-momentum ~k,and the spin σ. The energy εn(~k) depends on details of the periodic field

H0 =∑

n,σ

∫d3k εn(~k) a

+

n,~k,σan,~k,σ

Spherically symmetric external field (the 0-th approximation for electrons inatoms and nucleons in nuclei). 1-particle states are labeled by the quantumnumbers n, l,m, σ. The energy En,l depends on details of the field

H0 =∑

n,l,m,σ

En,l a+n,l,m,σan,l,m,σ


Remark: The ideal gas approximation is very popular for electrons in atoms,molecules and solids. At the first sight, however, it looks like a rather poorapproximation. The dominant (Coulomb) interaction between electrons is enor-mous at atomic scale and cannot be neglected in any decent approach.But there is no mystery involved, the ideal gas approximation used for electronsdoes not neglect the Coulomb interaction. The point is that the external field forthe electron ideal gas contains not only the Coulomb field of positively chargednuclei, but also some kind of mean field of all negatively charged electrons. Thismean field is usually given by the Hartree-Fock approximation. The corner-stoneof this approximation is a restriction on electron states taken into consideration:only the direct products of single electron states are accounted for. In this re-stricted set of states one looks for what in some sense is the best approximationto the stationary states. This leads to the specific integro-differential equationfor the 1-electron states and corresponding energies, which is then solved itera-tively32. The creation and annihilation operators for these Hartree-Fock statesand the corresponding energies then enter the electron ideal gas Hamiltonian.

Remark: The ground state of a fermionic system in the Hartree-Fock approxi-mation (the ideal gas approximation with 1-particle states and energies given bythe Hartree-Fock equation) is quite simple: all the 1-particle states with energiesbelow some boundary energy, the so-called Fermi energy εF , are occupied, whileall the states with energies above εF are free. The Fermi energy depends, ofcourse, on the number of particles in the system.In solids, the 1-particle Hartree-Fock states are characterized by (n,~k, σ) (level,quasi-momentum, spin). The 1-particle n-th level Hartree-Fock energy is, as a

rule, an ascending function of k2 in any direction of ~k. In any direction, there-fore, there exists the level n and the vector ~kF (ϕ, ϑ) for which εn(~kF ) = εF . The

endpoints of vectors ~kF (ϕ, ϑ) form a surface, called the Fermi surface. In themany-body ground state, the 1-particle states beneath (above) the Fermi surfaceare occupied (free).It turns out that for a great variety of phenomena in solids, only the low excitedstates of the electron system are involved. They differ from the ground state byhaving a few 1-particle states above the Fermi surface occupied. The particularform of the Fermi surface therefore determines many macroscopic properties ofthe material under consideration. For this reason the knowledge of the Fermisurface is very important in the solid state physics.

Remark: The ideal gas of fermions is frequently treated by means of a famousformal trick known as the electron-hole formalism. The ground state of the Nfermion ideal gas is called the Fermi vacuum, and denoted by |0F 〉. For i ≤ None defines new operators b+i = ai and bi = a+i . The original a+i and ai operatorsare taken into account only for i > N .Both a- and b-operators satisfy the commutation relations, and both bi (i ≤ N)and ai (i > N) annihilate the Fermi vacuum (indeed, bi |0F 〉 = 0 because of anti-symmetry of fermion states, i.e. because of the Pauli exclusive principle). So,formally we have two types of particles, the holes and the new electrons, createdfrom the Fermi vacuum by b+i and a+i respectively. The Hamiltonian readsH0 =

∑i≤N Eibib

+i +∑

i>N Eia+i ai =

∑i≤N Ei−

∑i≤N Eib

+i bi+

∑i>N Eia

+i ai.

The popular interpretation of the minus sign: the holes have negative energy.

32For details consult any reasonable textbook on QM or solid state physics.


Hamiltonian of a system of particles with the pair interaction

Perhaps the most important interaction to be added to the previous case of theideal gas is the pair interaction, i.e. an interaction characterized by a potentialenergy of pairs of particles (most of applications in the solid state, atomic andnuclear physics involve such an interaction). In this case, the most suitablechoice of basis in the 1-particle Hilbert space is the x-representation |~x〉, sincethe 2-particle states |~x, ~y〉 have a sharp value of the pair potential energy V (~x, ~y).

Due to the fact that we are dealing with the pair interaction, the 3-particlestate |~x1, ~x2, ~x3〉 does also have the sharp value of the potential energy, namelyV (~x1, ~x2) + V (~x1, ~x3) + V (~x2, ~x3), and the same holds for other multiparticlestates (this, in fact, is the definition of the pair interaction).

What is the potential energy of the state with n(~xi) particles at the posi-tion ~xi, where i = 1, 2, . . .? The number of pairs contributing by V (~xi, ~xj) is12n~xi

n~xjfor i 6= j, by which we understand also ~xi 6= ~xj (the 1

2 is there toavoid double-counting). For i = j there is a subtlety involved. One has toinclude the potential energy of a particle with all other particles sharing thesame position, but not with itself (a particle with itself does not constitute apair). The number of pairs contributing by V (~xi, ~xi) is therefore

12n~xi

(n~xi− 1).

This makes the total potential energy in the state under consideration equal to12

∑i,j V (~xi, ~xj)n~xi

n~xj− 1

2

∑i V (~xi, ~xi)n~xi

.Using the same logic as in the case of the ideal gas, it is now easy to write

down the operator of the total potential energy in terms of operators n~x = a+~x a~x.Using the commutation relations for the creation and annihilation operators theresulting expression can be simplified to the form33

Hpair =1

2

∫d3x d3y V (~x, ~y) a+~x a

+~y a~ya~x

Note the order of the creation and annihilation operators, which is mandatory.It embodies the above mentioned subtlety.

As we have seen before, the x-representation is usually not the most suitablefor the ideal gas Hamiltonian. To have the complete Hamiltonian of a systemwith the pair interaction presented in a single representation, it is useful torewrite the potential energy operator Hpair in the other representation.

Free particles. All one needs is a~x =∫d3p 〈~x|~p〉 a~p =

∫d3p 1

(2π~)3/2ei~p.~x/~a~p

Hpair =1

2

∫d3p1d

3p2d3p3d

3p4 V (~p1, ~p2, ~p3, ~p4) a+~p1a+~p2

a~p3a~p4

V (~p1, ~p2, ~p3, ~p4) =

∫d3x

(2π~)3

d3y

(2π~)3V (~x, ~y) exp

i (~p4 − ~p1) .~x

~exp

i (~p3 − ~p2) .~y

~

Periodic external field. The plane waves 〈~x|~p〉 are to be replaced by the Bloch

functions⟨~x|n,~k

⟩= un(~k)e

i~k.~x.

Spherically symmetric external field. The plane waves 〈~x|~p〉 are to be replacedby the product of a solution of the radial Schrodinger equation and a sphericalharmonics 〈~x|n, l,m〉 = Rnl (x)Ylm (ϕ, ϑ).

33 U = 12

∫

d3x d3y V (~x, ~y) a+~xa~xa

+~ya~y − 1

2

∫

d3x V (~x, ~x) a+~xa~x

= 12

∫

d3x d3y V (~x, ~y) a+~x

(

δ (x− y)± a+~ya~x

)

a~y − 12

∫

d3x V (~x, ~x) a+~xa~x

= ± 12

∫

d3x d3y V (~x, ~y) a+~xa+~ya~xa~y = 1

2

∫

d3x d3y V (~x, ~x) a+~xa+~ya~ya~x


Remark: Even if it is not necessary for our purposes, it is hard to resistthe temptation to discuss perhaps the most important pair interaction in thenon-relativistic quantum physics, namely the Coulomb interaction. This is ofgeneral interest not only because of the vast amount of applications, but alsodue to the fact that the standard way of dealing with the Coulomb potential inthe p-representation involves a mathematical inconsistency. The way in whichthis inconsistency is treated is in a sense generic, and therefore quite instructive.

In the x-representation VCoulomb (~x, ~y) = e2

4π1

|~x−~y| , i.e. in the p-representation

(which is relevant in the case with no external field)

VCoulomb (~p1, ~p2, ~p3, ~p4) =e2

4π

∫d3x

(2π)3

d3y

(2π)3

1

|~x− ~y|ei(~p4−~p1).~xei(~p3−~p2).~y

(for the sake of brevity, we use the Heaviside-Lorentz convention in electrody-namics and ~ = 1 units in QM). This integral, however, is badly divergent. Theintegrand simply does not drop out fast enough for |~x| → ∞ and |~y| → ∞.Instead of giving up the use of the p-representation for the Coulomb potentialenergy, it is a common habit to use a dirty trick. It starts by considering the

Yukawa (or Debey) potential energy VDebey (~x, ~y) =e2

4π1

|~x−~y|e−µ|~x−~y|, for which

the p-representation is well defined and can be evaluated readily34

VDebey (~p1, ~p2, ~p3, ~p4) =e2

4π

1

2π2

1

µ2 + (~p4 − ~p1)2 δ (~p1 + ~p2 − ~p3 − ~p4)

Now comes the dirty part. It is based on two simple (almost trivial) observations:the first is that VCoulomb (~x, ~y) = limµ→0 VDebey (~x, ~y), and the second is that thelimit limµ→0 VDebey (~p1, ~p2, ~p3, ~p4) is well defined. From this, a brave heart can

easily conclude that VCoulomb (~p1, ~p2, ~p3, ~p4) =e2

8π31

(~p4−~p1)2 δ (~p1 + ~p2 − ~p3 − ~p4).

And, believe it or not, this is indeed what is commonly used as the Coulombpotential energy in the p-representation.Needless to say, from the mathematical point of view, this is an awful heresy(called illegal change of order of a limit and an integral). How does it comeabout that physicists are working happily with something so unlawful?The most popular answer is that the Debey is nothing else but a screened Coulomb,and that in most systems this is more realistic than the pure Coulomb. This isa reasonable answer, with a slight off-taste of a cheat (the limit µ → 0 convictsus that we are nevertheless interested in the pure Coulomb).Perhaps a bit more fair answer is this one: For µ small enough, one cannot sayexperimentally the difference between the Debey and Coulomb. And the commonplausible belief is that measurable outputs should not depend on immeasurableinputs (if this was not typically true, the whole science would hardly be possi-ble). If the mathematics nevertheless reveals inconsistencies for some values ofa immeasurable parameter, one should feel free to choose another value, whichallows for mathematically sound treatment.

34For ~r = ~x−~y, VYukawa (~p1, ~p2, ~p3, ~p4) =e2

4π

∫

d3r(2π)3

d3y(2π)3

1re−µrei(~p4−~p1).~rei(~p3−~p2+~p4−~p1).~y

= e2

4πδ (~p1 + ~p2 − ~p3 − ~p4)

∫

d3r(2π)3

e−µr

rei~q.~r , where ~q = ~p4 − ~p1. The remaining integral in

the spherical coordinates: 1(2π)2

∫∞0 dr r e−µr

∫ 1−1 d cosϑ e

iqr cosϑ = 12π2

1q

∫∞0 dr e−µr sin qr.

For the integral I =∫∞0dr e−µr sin qr one obtains by double partial integration the relation

I = qµ2 − q2

µ2 I, and putting everything together, one comes to the quoted result.


Hamiltonian of a system of unstable particles

Let us consider a system of particles of three types A, B and C, one of whichdecays into other two, say C → AB. The decay can be understood as the anni-hilation of the decaying particle and the simultaneous creation of the products.The corresponding interaction Hamiltonian, i.e. the part of the time-evolutiongenerator responsible for this kind of change, is almost self-evident. It shouldcontain the combination a+i b

+j ck, where the lowercase letters for creation and

annihilation operators correspond to the uppercase letters denoting the type ofparticle, and subscripts specify the states of the particles involved.

The decay is usually allowed for various combinations of the quantum num-bers i, j, k, so the interaction Hamiltonian should assume the form of the sumof all legal alternatives. This is usually written as the sum of all alternatives,each of them multiplied by some factor, which vanishes for the forbidden com-binations of quantum numbers:

∑i,j,k gijka

+i b

+j ck.

There is still one problem with this candidate for the interaction Hamilto-nian: it is not Hermitian. But this is quite easy to take care of, one just addsthe Hermitian conjugate operator g∗ijkc

+k bjai. So the Hermiticity of the Hamil-

tonian requires, for any decay, the existence of the reverse process AB → C.All in all

Hdecay =∑

i,j,k

gijka+i b

+j ck + g∗ijkc

+k bjai

Generalizations (decays with 3 or more products, or even some more bizarreprocesses, with more than one particle annihilated) are straightforward.

Remark: The factor gijk is usually called the coupling constant, although itdepends on the quantum numbers i, j, k. The reason is that most of decays arelocal and translation invariant (they do not vary with changes in position). Inthe x-representation the locality means that g~x,~y,~z = g~xδ(~x − ~y)δ (~x− ~z) andtranslational invariance requires that g does not depend even on ~x

Hdecay =

∫d3x ga+~x b

+~x c~x + g∗c+~x b~xa~x

Remark: When dealing with various types of particles, one needs a Fock spacewhich contains the Fock spaces of all particle species under consideration. Theobvious first choice is the direct product of the corresponding Fock spaces, e.g.H = HA⊗HB ⊗HC. In such a case any creation/annihilation operator for oneparticle type commutes with any c/a operator for any other particle type.Sometimes, however, it is formally advantageous to modify the commutationrelations among different particle species. Although the bosonic c/a operators arealways chosen to commute with all the other c/a operators, for fermions it maybe preferable to have the c/a operators for different fermions anti-commutingwith each other. The point is that sometimes different particle species may beviewed just as different states of the same particle (due to isospin, eightfold way,etc. symmetries). If so, it is clearly favorable to have the (anti)commutationrules which do not need a radical modification with every change of viewpoint.The question is, of course, if we can choose the (anti)commutation relations atour will. The answer is affirmative. It just requires the appropriate choice ofthe anti-symmetrized subspace of the direct product of the Fock spaces.


any linear operator expressed in terms of a+, a

The only purpose of this paragraph is to satisfy a curious reader (if there isany). It will be of no practical importance to us.

First of all, it is very easy to convince oneself that any linear operator canbe expressed in terms of creation operators a+i , annihilation operators ai and

the vacuum projection operator |0〉〈0|. Indeed, if A is a linear operator, then

A =∑

i,j,...m,n,...

Aij...,kl...a+i a

+j . . . |0〉〈0|akal . . .

where Aij...,kl... =〈i,j,...|A|k,l,...〉

〈i,j,...|i,j,...〉〈k,l,...|k,l,...〉 . Proof: both LHS and RHS have the

same matrix elements for all combinations of basis vectors (check this).

The only question therefore is how to get rid of |0〉〈0|. This is done byinduction. First, one expresses the A operator only within the 0-particle sub-space H0of the Fock space, where it is nothing else but the multiplication bythe constant

A0 = A0,0 ≡ 〈0| A |0〉Then, one expresses the A1 = A − A0 operator within the 0- and 1-particlesubspace H0 ⊕H1. Here one gets (check it)

A1 = Ai,ja+i aj + Ai,0a

+i + A0,jaj

where Aij = 〈i| A− A0 |j〉, Ai,0 = 〈i| A− A0 |0〉 and A0,j = 〈0| A− A0 |j〉. If onerestricts oneself to H0 ⊕H1, then A = A0 + A1 (why?). So we have succeededin writing the operator A in terms of a+i , ai, even if only in the subspace of theFock space. This subspace is now expanded to H0 ⊕H1 ⊕H2, etc.

It may be instructive now to work out the operator A2 = A− A0− A1withinH0 ⊕H1 ⊕H2 in terms of a+i , ai (try it). We will, however, proceed directly to

the general case of An = A−∑n−1m=0 Am within

⊕nm=0 Hm

An =∑

allowed

combinations

Aij...,kl...a+i a

+j . . . akal . . .

Aij...,kl... =〈i, j, . . .| A−∑n−1

m=0 Am |k, l, . . .〉〈i, j, . . . |i, j, . . .〉〈k, l, . . . |k, l . . .〉

and the ”allowed combinations” are either ij . . .︸︷︷︸n

, kl . . .︸︷︷︸m≤n

or ij . . .︸︷︷︸m≤n

, kl . . .︸︷︷︸n

.

If restricted to⊕n

m=0 Hm, then A =∑n

m=0 Am, i.e. we have A expressed interms of a+i , ai. To get an expression valid not only in subspaces, one takes

A =

∞∑

m=0

Am


1.2.3 Calculation of matrix elements — the main trick

One of the most valuable benefits of the use of the creation and annihilationoperator formalism is the completely automatous way of matrix elements cal-culation. The starting point is twofold:• any ket (bra) vector can be written as a superposition of the basis vectors,which in turn can be obtained by a+i (ai) operators acting on |0〉• any linear operator can be written as a linear combination of products of thea+i and ai operatorsConsequently, any matrix element of a linear operator is equal to some linearcombination of the vacuum expectation values (VEVs) of the products of thecreation and annihilation operators.

Some of the VEVs are very easy to calculate, namely those in which thelast (first) operator in the product is the annihilation (creation) one. Indeed,due to ai |0〉 = 0 and 〈0| a+i = 0, any such VEV vanishes. Other VEVs areeasily brought to the previous form, one just has to use the (anti)commutationrelations

[ai, a

+j

]∓ = δij to push the creation (annihilation) operators to the

right (left). By repeated use of the (anti)commutation relations, the originalVEV is brought to the sum of scalar products 〈0|0〉 multiplied by pure numbers,and the VEVs vanishing because of 〈0|a+i = 0 or ai |0〉 = 0. An example isperhaps more instructive than a general exposition.

Example: Let us consider a decay-like Hamiltonian for just one type of particlesHdecay =

∑i,j,k gijk(a

+i a

+j ak + a+k ajai) (e.g. phonons, as well as gluons, enjoy

this kind of interaction). Note that the coupling ”constant” is real gijk = g∗ijk.And let us say we want to calculate 〈l|Hdecay |m,n〉. First, one writes

〈l|Hdecay |m,n〉 =∑

i,j,k

gijk(〈0|ala+i a+j aka+ma+n |0〉+ 〈0|ala+k ajaia+ma+n |0〉

)

Then one starts to reshuffle the operators. Take, e.g., the first two and useala

+i = δli ± a+i al (or with i replaced by k in the second term), to obtain


i,j,k

gijk(δli 〈0| a+j aka+ma+n |0〉 ± 〈0|a+i ala+j aka+ma+n |0〉

+δlk 〈0|ajaia+ma+n |0〉 ± 〈0| a+k alajaia+ma+n |0〉)

Three of the four terms have a+ next to 〈0|, and consequently they vanish. Inthe remaining term (the third one) one continues with reshuffling


i,j,k

gijkδlk(δim 〈0| aja+n |0〉 ± 〈0| aja+maia+n |0〉

)

=∑

i,j,k

gijkδlk(δimδjn ± δjm 〈0|aia+n |0〉+ 0

)

=∑

i,j,k

gijkδlk (δimδjn ± δjmδin) = gmnl ± gnml

The result could be seen directly from the beginning. The point was not to obtainthis particular result, but rather to illustrate the general procedure. It should beclear from the example, that however complicated the operator and the states(between which it is sandwiched) are, the calculation proceeds in the same way:one just reshuffles the creation and annihilation operators.


The example has demonstrated an important common feature of this typeof calculations: after all the rubbish vanishes, what remains are just productsof deltas (Kronecker or Dirac for discrete or continuous cases respectively),each delta originating from some pair of a a+. This enables a short-cut in thewhole procedure, namely to take into account only those terms in which all aand a+operators can be organized (without leftovers) in the pairs a a+(in thisorder!), and to assign the appropriate δ to every such pair.

This is usually done within the ”clip” notation, like e.g. . . . ai . . . a+j . . .,

which tells us that ai is paired with a+j . The factor corresponding to thisparticular clip is therefore δij . In the case of fermions one has to keep track ofsigns. The reader may try to convince him/herself that the rule is: every pairof clips biting into each other, generates the minus sign for fermions.

Example: The same example as above, now using the clip short-cut.


i,j,k

gijk(〈0|ala+i a+j aka+ma+n |0〉+ 〈0| ala+k ajaia+ma+n |0〉

)

The first term cannot be organized (without leftovers) in pairs of aa+, so it doesnot contribute. As to the rest, one has

〈0|ala+k ajaia+ma+n |0〉 = 〈0|ala+k ajaia+ma+n |0〉+ 〈0| ala+k aj a+mai a+n |0〉

= δlkδjnδim ± δlkδjmδin

leading immediately to the same result as before (gmnl ± gnml).

The common habit is to make the short-cut even shorter. Instead of writingthe clips above or under the corresponding operators, one draws just the clipsand then assigns the corresponding factor to the whole picture. In this comics-like formalism, the matrix element from the example would become

l k l k

j n ± j m

i m i n

The picture may suggest some relation to the Feynman diagrams, and this isindeed the case. The propagators in the Feynman diagrams originate from thesame clip trick, although not between the creation and annihilation operatorsthemselves, but rather between some specific combinations thereof. Therefore,the factor corresponding to the Feynman propagator is not just the simpleKronecker or Dirac delta, but rather some more complicated expression.


We can also understand, even if only roughly, the origin of vertices. Thekeyword is locality. Local interactions contain products of operators in thesame position, e.g. a+~x b

+~x c~x, which in our discrete case would correspond to

Hdecay =∑

i g(a+i a

+i ai + a+i aiai). The above picture is then changed to ±

Finally, we may take yet one step further and indicate where the diagramscome from. In the QM the exponential of an operator plays frequently animportant role. The exponential of the interaction Hamiltonian appears in oneversion of the perturbation theory, the one which became very popular in QFT.The power expansion of the exponential of the local decay Hamiltonian, togetherwith the above example concerning this Hamiltonian, should give the basic tasteof the origin of the Feynman diagrams35.

This rough idea of how the Feynman diagrams arise from the formalism ofthe creation and annihilation operators in the Fock space is not where we haveto stop. We can go much further, at this point we are fully prepared to developthe Feynman diagrams for the non-relativistic many-body QM. Nevertheless,we are not going to.

The reason is that our main goal in these lectures is to develop the relativisticquantum theory. And although the Feynman diagrams in the relativistic andnon-relativistic versions have a lot in common, and although their derivationsare quite similar, and although the applications of the non-relativistic versionare important, interesting and instructive, we simply cannot afford to spend toomuch time on the introductions.

It is not our intention, however, to abandon the non-relativistic case com-pletely. We will return to it, at least briefly, once the Feynman diagrams ma-chinery is fully developed in the relativistic case. Then we will be able to graspthe non-relativistic case rather quickly, just because of strong similarities36.

35The reader is encouraged to work out 〈l| expHdecay |m,n〉 up to the second order ing, using all the three (equivalent) methods discussed above (operator reshuffling, clip nota-tion, diagrams). The key part is to calculate 1

2〈l|∑i g(a

+i a

+i ai + a+i aiai)

∑

j g(a+j a

+j aj +

a+j ajaj) |m,n〉.36This is definitely not the best way for anybody interested primarily in the non-relativistic

many-body quantum theory. For such a reader it would be preferable to develop the diagramtechnique already here. Such a development would include a couple of fine points which, whenunderstood well in the non-relativistic case, could be subsequently passed through quickly inthe relativistic case. So it would make a perfect sense not to stop here.

Anyway, it seems to be an unnecessary luxury to discuss everything in detail in both versions(relativistic and non-relativistic). The point is that the details are subtle, difficult and quitea few. Perhaps only one version should be treated in all details, the other one can be thenglossed over. And in these lectures, the focus is on the relativistic version.


1.2.4 The bird’s-eye view of the solid state physics

Even if we are not going to entertain ourselves with calculations within thecreation and annihilation operators formalism of the non-relativistic many-bodyQM, we may still want to spend some time on formulation of a specific example.The reason is that the example may turn out to be quite instructive. This,however, is not guaranteed. Moreover, the content of this section is not requisitefor the rest of the text. The reader is therefore encouraged to skip the sectionin the first (and perhaps also in any subsequent) reading.

The basic approximations of the solid state physics

A solid state physics deals with a macroscopic number of nuclei and electrons,interacting electromagnetically. The dominant interaction is the Coulomb one,responsible for the vast majority of solid state phenomena covering almost ev-erything except of magnetic properties. For one type of nucleus with the pro-ton number Z (generalization is straightforward) the Hamiltonian in the wave-

function formalism37 is H = −∑i∆i

2M −∑j∆j

2m +U(~R)+V (~r)+W (~R,~r), where

U(~R) = 18π

∑i6=j

Z2e2

|~Ri−~Rj| , V (~r) = 18π

∑i6=j

e2

|~ri−~rj| , W (~R,~r) = − 18π

∑i,j

Ze2

|~Ri−~rj|and (obviously) ~R = (~R1, . . .), ~r = (~r1, . . .). If desired, more than Coulomb canbe accounted for by changing U , V and W appropriately.

Although looking quite innocent, this Hamiltonian is by far too difficult tocalculate physical properties of solids directly from it. Actually, no one has eversucceeded in proving even the basic experimental fact, namely that nuclei insolids build a lattice. Therefore, any attempt to grasp the solid state physicstheoretically, starts from the series of clever approximations.

The first one is the Born-Oppenheimer adiabatic approximation which en-ables us to treat electrons and nuclei separately. The second one is the Hartree-Fock approximation, which enables us to reduce the unmanageable many-electronproblem to a manageable single-electron problem. Finally, the third of thecelebrated approximations is the harmonic approximation, which enables theexplicit (approximate) solution of the nucleus problem.

These approximations are used in many areas of quantum physics, let usmention the QM of molecules as a prominent example. There is, however, animportant difference between the use of the approximations for molecules andfor solids. The difference is in the number of particles involved (few nucleiand something like ten times more electrons in molecules, Avogadro numberin solids). So while in the QM of molecules, one indeed solves the equationsresulting from the individual approximations, in case of solids one (usually)does not. Here the approximations are used mainly to setup the conceptualframework for both theoretical analysis and experimental data interpretation.

In neither of the three approximations the Fock space formalism is of anygreat help (although for the harmonic approximation, the outcome is often for-mulated in this formalism). But when moving beyond the three approximations,this formalism is by far the most natural and convenient one.

37If required, it is quite straightforward (even if not that much rewarding) to rewritethe Hamiltonian in terms of the creation and annihilation operators (denoted as uppercase

and lowercase for nuclei and electrons respectively) H =∫

d3p(

p2

2MA+

~pA~p + p2

2ma+~pa~p

)

+

e2

8π

∫

d3x d3y 1|~x−~y|

(

Z2A+~xA+

~yA~yA~x − ZA+

~xa+~ya~yA~x + a+

~xa+~ya~ya~x

)

.


Born-Oppenheimer approximation treats nuclei and electrons on differentfooting. The intuitive argument is this: nuclei are much heavier, therefore theywould typically move much slower. So one can perhaps gain a good insight bysolving first the electron problem for nuclei at fixed positions (these positions,

collectively denoted as ~R, are understood as parameters of the electron problem)

He(~R)ψn(~r) = εn(~R)ψn(~r)

He(~R) = −∑

i

∆i

2m+ V +W

and only afterwards to solve the nucleus problem with the electron energy εn(~R)understood as a part of the potential energy on the nuclei

HNΨm(~R) = EmΨm(~R)

HN = −∑

i

∆i

2M+ U + εn(~R)

The natural question as to which electron energy level (which n) is to be used inHN is answered in a natural way: one uses some kind of mean value, typicallyover the canonical ensemble, i.e. the εn(~R) in the definition of the HN is to be

replaced by ε(~R) =∑

n εn(~R) exp−εn(~R)/kT .

The formal derivation of the above equations is not important for us, butwe can nevertheless sketch it briefly. The eigenfunctions Φm(~R,~r) of the fullHamiltonian H are expanded in terms of the complete system (in the variable ~r)

of functions ψn(~r): Φm(~R,~r) = cmn(~R)ψn(~r). The coefficients of this expansion

are, of course, ~R-dependent. Plugging into HΦm = EmΦm one obtains theequation for cmn(~R) which is, apart of an extra term, identical to the above

equation for Ψm(~R). The equation for cmn(~R) is solved iteratively, the zeroth

iteration ignores the extra term completely. As to the weighted average ε(~R)the derivation is a bit more complicated, since it has to involve the statisticalphysics from the beginning, but the idea remains unchanged.

The Born-Oppenheimer adiabatic approximation is nothing else but the ze-roth iteration of this systematic procedure. Once the zeroth iteration is solved,one can evaluate the neglected term and use this value in the first iteration.But usually one contents oneself by checking if the value is small enough, whichshould indicate that already the zeroth iteration was good enough. In the solidstate physics one usually does not go beyond the zeroth approximation, noteven check whether the neglected term comes out small enough, simply becausethis is too difficult.

In the QM of molecules, the Born-Oppenheimer approximation stands be-hind our qualitative understanding of the chemical binding, which is undoubt-edly one of the greatest and the most far-reaching achievements of QM. Needlessto say, this qualitative understanding is supported by impressive quantitativesuccesses of quantum chemistry, which are based on the same underlying ap-proximation. In the solid state physics, the role of this approximation is moremodest. It serves only as a formal tool allowing for the separation of the electronand nucleus problems. Nevertheless, it is very important since it sets the basicframework and paves the way for the other two celebrated approximations.


Hartree-Fock approximation replaces the many-electron problem by therelated single-electron problem. The intuitive argument is this: even if there isno guarantee that one can describe the system reasonably in terms of states ofindividual electrons, an attempt is definitely worth a try. This means to restrictoneself to electron wave-functions of the form ψ(~r) =

∏i ϕi(~ri). The effective

Hamiltonian for an individual electron should look something like

HHartree = −∆i

2m+W (~R,~ri) +

∑

j 6=i

e2

8π

∫d3rj

ϕ∗j (~rj)ϕj(~rj)

|~ri − ~rj |

where the last term stands for the potential energy of the electron in the elec-trostatic field of the rest of the electrons.

The Schrodinger-like equation Hϕi(~ri) = εiϕi(~ri) is a non-linear equationwhich is solved iteratively. For a system of n electrons one has to find thefirst n eigenvalues ε, corresponding to mutually orthogonal eigenfunctions ϕi,so that the multi-electron wave-function ψ does not contain any two electronsin the same state (otherwise it would violate the Pauli exclusion principle). Theresulting ψ, i.e. the limit of the iteration procedure, is called the self-consistentHartree wave-function.

In the Hartree approximation, however, the Pauli principle is not accountedfor sufficiently. The multi-electron wave-function is just the product of single-electron wave-functions, not the anti-symmetrized product, as it should be. Thisis fixed in the Hartree-Fock approximation where the equation for ϕi(~r) becomes

HHartreeϕi(~r)−∑

j 6=i

e2

8π

∫d3r′

ϕ∗j (~r

′)ϕi(~r′)

|~r − ~r′| δsisjϕj(~r) = εiϕi(~r)

where s stands for spin of the electron. As to the new term on the LHS, calledthe exchange term, it is definitely not easy to understand intuitively. In thiscase, a formal derivation is perhaps more elucidatory.

A sketch of the formal derivation of the Hartree and Hartree-Fock equationslooks like this: one takes the product or the anti-symmetrized product (oftencalled the Slater determinant) in the Hartree and Hartree-Fock approximationsrespectively. Then one use this (anti-symmetrized product) as an ansatz for thevariational method of finding (approximately) the ground state of the system,i.e. one has to minimize the matrix element

∫ψ∗Heψ with constraints

∫ϕ∗iϕi =

1. The output of this procedure are the above equations (with εi entering asthe Lagrange multiplicators).

The main disadvantage of the formal derivation based on the variationalmethod is that there is no systematic way of going beyond this approximation.An alternative derivation, allowing for the clear identification of what has beenomitted, is therefore most welcome. Fortunately, there is such a derivation. Itis based on the Feynman diagram technique. When applied to the system ofelectrons with the Coulomb interaction, the diagram technique enables one toidentify a subset of diagrams which, when summed together, provides preciselythe Hartree-Fock approximation. This, however, is not of our concern here.

In the QM of molecules, the Hartree-Fock approximation is of the highestimportance both qualitatively (it sets the language) and quantitatively (it entersdetailed calculations). In the solid state physics, the main role of the approxi-mation is qualitative. Together with the Bloch theorem, it leads to one of thecardinal notions of this branch of physics, namely to the Fermi surface.


Bloch theorem reveals the generic form of eigenfunctions of the Hamilto-nian which is symmetric under the lattice translations. The eigenfunctions arelabeled by two quantum numbers n and ~k, the former (called the zone number)being discrete and the latter (called the quasi-momentum) being continuous and

discrete for infinite and finite lattices respectively (for large finite lattice, the ~kis so-called quasi-continuous). The theorem states that the eigenfunctions are

of the form ϕn,~k(~r) = ei~k.~run,~k(~r), where the function un,~k(~r) is periodic (with

the periodicity of the lattice)38. In the solid state physics the theorem is sup-plemented by the following more or less plausible assumptions:• The theorem holds not only for the Schrodinger equation, but even for theHartree and Hartree-Fock equations with periodic external potentials.• The energy εn(~k) is for any n a (quasi-)continuous function of ~k.

• The relation εn(~k) = const defines a surface in the ~k-space.The very important picture of the zone structure of the electron states in

solids results from these assumptions. So does the notion of the Fermi sur-face, one of the most crucial concepts of the solid state physic. In the groundstate of the system of non-interacting fermions, all the one-fermion levels withenergies below and above some boundary energy are occupied and empty re-spectively. The boundary energy is called the Fermi energy εF and the surfacecorresponding to εn(~k) = εF is called the Fermi surface.

The reason for the enormous importance of the Fermi surface is that allthe low excited states of the electron system lie in the vicinity of the Fermisurface. The geometry of the Fermi surface therefore determines a great deal ofthe physical properties of the given solid.

Calculation of the Fermi surface from the first principles is only possible withsome (usually quite radical) simplifying assumptions. Surprisingly enough, yetthe brutally looking simplifications use to give qualitatively and even quantita-tively satisfactory description of many phenomena. However, the most reliabledeterminations of the Fermi surface are provided not by the theory, but ratherby the experiment. There are several methods of experimental determinationof the Fermi surface. The mutual consistency of the results obtained by variousmethods serves as a welcome cross-check of the whole scheme.

38The proof is quite simple. The symmetric Hamiltonian commutes with any translationT~R

where ~R is a lattice vector. Moreover, any two translations commutes with each other.Let us consider common eigenfunctions ϕ(~r) of the set of mutually commuting operatorspresented by the Hamiltonian and the lattice translations. We denote by λi the eigenvaluesof the translations along the three primitive vectors of the lattice , i.e.T~ai

ϕ(~r) = λiϕ(~r). Thecomposition rule for the translations (T ~A+~B

= T ~AT~B

) implies T~Rϕ(~r) = λn1

1 λn22 λn3

3 ϕ(~r) for

~R =∑3

i=1 ni~ai. On the other hand, T~Rϕ(~r) = ϕ(~r + ~R). This requires |λi| = 1, since

otherwise the function ϕ(~r) would be unbounded or disobeying the Born-Karman periodicboundary conditions for the infinite and finite lattice respectively. Denoting λi = eikiai (no

summation) one has T~Rϕ(~r) = ei

~k.~Rϕ(~r). At this point one is practically finished, as it is now

straightforward to demonstrate that u~k(~r) = e−i~k.~rϕ(~r) is periodic. Indeed T~Re−i~k.~rϕ(~r) =

e−i~k.(~r+~R)ϕ(~r + ~R) = e−i~k.~re−i~k.~RT~Rϕ(~r) = e−i~k.~rϕ(~r).

For an infinite lattice, ~k can assume any value (~k is a continuous). For finite lattice theallowed values are restricted by the boundary conditions to discrete possible values, neverthe-less for large lattices the ~k-variable turns out to be quasi-continuous (for any allowed value,

the closest allowed value is ”very close”). For a given ~k, the Hamiltonian can have severaleigenfunctions and eigenvalues and so one needs another quantum number n in addition tothe quantum number ~k .


Harmonic approximation describes the system of nuclei or any other sys-tem in the vicinity of its stable equilibrium. It is one of the most fruitfulapproximations in the whole physics.

For the sake of notational convenience, we will consider the one-dimensionalcase. Moreover from now on, we will denote by R the equilibrium position ofthe particular nucleus, rather than the actual position, which will be denotedby R+ ρ. The Hamiltonian of the system of nuclei HN = −∑n

∆n

2M +U(R+ ρ),where U(R+ρ) = U(R+ρ)+ε(R+ρ). The potential energy U(R+ρ) is expandedaround the equilibrium U(R + ρ) = U(R) +∑n Unρn +

∑m,n Umnρmρn + . . .

where Un = ∂∂ρn

U(R + ρ)|ρ=0 and Umn = ∂2

∂ρm∂ρnU(R + ρ)|ρ=0. The first term

in the expansion is simply a constant, which can be dropped out safely. Thesecond term vanishes because of derivatives Ui vanishing at the minimum ofpotential energy. When neglecting the terms beyond the third one, we obtainthe harmonic approximation

HN = −∑

n

∆n

2M+∑

m,n

Umnρmρn

where ∆n = ∂2

∂ρ2nand the matrix Umn of the second derivatives is symmetric and

positive (in the stable equilibrium the potential energy reaches its minimum).

Now comes the celebrated move: the symmetric matrix Umn can be diag-onalized, i.e. there exists an orthogonal matrix D such that K ≡ DUD−1

is diagonal (Kmn = Knδmn no summation). Utilizing the linear combinationsξm =

∑n Dmnρn, one may rewrite the Hamiltonian to the extremely convenient

form

HN = −∑

n

∆n

2M+

1

2Knξ

2n

where now ∆n = ∂2

∂ξ2nis understood39. This is the famous miracle of systems in

the vicinity of their stable equilibriums: any such system is well approximatedby the system of coupled harmonic oscillators which, in turn, is equivalent tothe system of decoupled harmonic oscillators.

Stationary states of the system of the independent harmonic oscillators arecharacterized by the sequence (n1, n2, . . .) where ni defines the energy level ofthe i-th oscillator Energies of individual oscillators are ~ωn(Nn + 1/2) whereωn =

√Kn/M . Energy of the system is

∑i ~ωn(Nn + 1/2). This brings us

to yet another equivalent description of the considered system, namely to the(formal) ideal gas description.

Let us imagine a system of free particles, each of them having energy eigen-states labeled by n with eigenvalues ~ωn. If there are Nn particles in the n-thstate, the energy of the system will be

∑n ~ωnNn. This is equivalent (up to a

constant) to the Hamiltonian of independent oscillators. It is common habit todescribe a system of independent harmonic oscillators in terms of the equivalentsystem of free formal particles. These formal particles are called phonons.

39This is indeed possible since ∂∂ξi

=∑

j Dij∂

∂ρjand ∂2

∂ξi∂ξk=

∑

j,l Dij∂

∂ρjDkl

∂∂ρl

. So if

one sets k = i and then sums over i utilizing symmetry and orthogonality of D (leading to∑

i DijDil =∑

i DjiDil = δjl) one obtains∑

i∂2

∂ξ2i=

∑

j∂

∂ρ2jas claimed.


Phonons are would-be particles widely used for the formal description of realparticles, namely independent harmonic oscillators (see the previous paragraph).Phonon is not a kind of particle. Strictly speaking, it is just a word.

Some people may insist on the use of the word phonon only for systemswith some translational symmetry, rather than in connection with any systemof independent harmonic oscillators. This, however, is just the matter of taste.Anyway, transitionally invariant lattices are of the utmost importance in solidstate physics and, interestingly enough, in the case of a transitionally symmetriclattice (of stable equilibrium positions of individual nuclei) the diagonalizingmatrix is explicitly known. Indeed, the Fourier transformation

ξm =∑

n

eikmRnρn

i.e. Dmn = eikmRn does the job40. For a finite lattice with N sites and periodicboundary conditions41, one has km = 2π

amN where a is the lattice spacing. Since

there is one to one correspondence between m and km, one may label individualindependent oscillators by km rather than by m. In this notation, the energiesof individual oscillators become ω(km). For an infinite lattice, k may assumeany value from the interval of the length 2π

a , say (−πa ,

πa ), and the corresponding

energies form a function ω(k).Three-dimensional case is treated in the same way, even if with more cum-

bersome notation. Phonons are characterized by the continuous and quasi-continuous ~k for infinite and large finite lattices respectively, and moreover bythe polarization σ and by the function ω(~k, σ).

As for the determination of the function ω(~k, σ), very much the same as for

the Fermi surface can be said. Namely: Calculation of U(~R + ~ρ) and conse-

quently of ω(~k, σ) from the first principles is only possible with some (usuallyquite radical) simplifying assumptions. Surprisingly enough, yet the brutallylooking simplifications use to give qualitatively and even quantitatively satis-factory description of many phenomena. However, the most reliable determina-tions of ω(~k, σ) are provided not by the theory, but rather by the experiment.

There are several methods of experimental determination of ω(~k, σ). The mu-tual consistency of the results obtained by various methods serves as a welcomecross-check of the whole scheme.

40The proof goes very much like the one for the Bloch theorem. First, one may want torecall that the eigenvectors of the matrix U can be used for the straightforward constructionof the diagonalizing matrix D (the matrix with columns being the eigenvectors of the matrixU is the diagonalizing one, as one can verify easily). Then one utilizes the invariance of Uwith respect to the lattice translations Tna (Tnaρj = ρj+n) which implies [U , Tna] = 0. As tothe common eigenvectors (of U and all Tna), if one has ∀j Taρj = λρj then ∀j Tnaρj = λnρj(since Tna = (Ta)n) and this implies ρj+n = λnρj . Denoting them-th eigenvalue λm = eikma

one obtains ρn = c.eikmna = c.eikmRn .It is perhaps worth mentioning that the above explanation of the origin of the Fourier trans-

formation in these circumstances seems to be somehow unpopular. Most authors prefer tointroduce it other way round. They discuss the system of classical rather than quantum har-monic oscillators, in which case the Fourier transformation comes as a natural tool of solvingthe system of coupled ordinary differential equations. The resulting system of independentclassical oscillators is canonically quantized afterwards. Nevertheless, there is no need for thisside-step to the classical world. The problem is formulated at the quantum level from thevery beginning and it can be solved naturally entirely within the quantum framework, as wehave demonstrated.

41The postulate of the periodic boundary conditions is just a convenient technical devicerather than a physically sound requirement.


The basic approximations and the a+, a operators

Both the Hartree-Fock and the harmonic approximations lead to the systemsof ideal gases. Their Hamiltonians can be therefore rewritten in terms of thecreation and annihilation operators easily. For the electron system one has

He =∑

n,σ

∫d3k εn(~k) a

+

n,~k,σan,~k,σ

Using the electron-hole formalism (see p. 23) this is commonly rewritten as

He,h = −∑

n,σ

∫d3k εhn(

~k) b+n,~k,σ

bn,~k,σ +∑

n,σ

∫d3k εen(

~k) a+n,~k,σ

an,~k,σ + const

where εen(~k) and εhn(

~k) vanish beneath and above the Fermi surface respectively

(they equal to εn(~k) otherwise). For the nucleus system one has (cf. p. 21)

HN =∑

σ

∫d3k ~ω(~k, σ) A+

~k,σA~k,σ

Note that one can write the Hamiltonian of the system of electrons and nucleiwith Coulomb interactions in terms of the creation and annihilation operatorsof electrons and nuclei (we have learned how to do this in previous sections).The outcome of the basic approximation differs from this straightforward use ofthe creation and annihilation operators in many respects.

• Both the electron-hole and the phonon systems do not have a fixed num-ber of particles. They contain, e.g., no particles in the ground states ofthe electron-hole and the harmonic approximations respectively. The lowexcited states are characterized by the presence of some ”above Fermi”electrons and holes and/or few phonons. The number of the genuine elec-trons and nuclei, on the other hand, is fixed.

• There are no interactions between the Hartree-Fock electrons and holesand the phonons. The interactions between the genuine electrons andnuclei, on the other hand, are absolutely crucial.

• There are no unknown parameters or functions in the Hamiltonian interms of creation and annihilation operators of the genuine electrons andnuclei. The Hamiltonian of both ideal gases of the Hartree-Fock electronsand holes and of the phonons do contain such unknown functions, namelythe energies of individual one-particle states. These usually need to bedetermined phenomenologically.


Beyond the basic approximations

The formalism of creation and annihilation operators is useful even beyond thebasic approximations. Let us briefly mention some examples.

Beyond the Hartree-Fock approximation there is always some (educated-guessed, model-dependent) potential energy between Hartree-Fock electrons andholes. It is straightforward to write it in terms of the electron-hole creation andannihilation operators. Application: excitons (a bound states of an electron anda hole).

Beyond the harmonic approximation there are higher (than quadratic) termsin the Taylor expansion of the potential (they were neglected in the harmonicapproximation). To write them in terms of the phonon creation and annihilationoperators, one has to write the position operator, i.e. the operator of the devi-ation ~ρ of nuclei from the equilibrium position in terms of the phonon variable~ξ and then to write the ~ξ-operator in terms of A+

~k,σA~k,σ (which is just a notori-

ous LHO relation). The result is the phonon-phonon interaction. Application:thermal expansion and thermal conductivity.

Beyond the adiabatic approximation there is some (educated-guessed, model-dependent) potential energy between Hartree-Fock electrons and nuclei. Thisis written in terms of electron and nucleus creation and annihilation operators.The potential energy is Taylor expanded in nuclei position deviations ~ρ andthe first term (others are neglected) is re-expressed in terms of phonon opera-tors. The resulting intearction is the electron-phonon interaction. Application:Cooper pairs in superconductors (a bound states of two electrons).

1.3. RELATIVITY AND QUANTUM THEORY 39

1.3 Relativity and Quantum Theory

Actually, as to the quantum fields, the keyword is relativity. Even if QFT isuseful also in the nonrelativistic context (see the previous section), the verynotion of the quantum field originated from an endeavor to fit together relativ-ity and quantum theory. This is a nontrivial task: to formulate a relativisticquantum theory is significantly more complicated than it is in a nonrelativisticcase. The reason is that specification of a Hamiltonian, the crucial operator ofany quantum theory, is much more restricted in relativistic theories.

To understand the source of difficulties, it is sufficient to realize that to havea relativistic quantum theory means to have a quantum theory with measurablepredictions which remain unchanged by the relativistic transformations. Therelativistic transformations at the classical level are the space-time transforma-tions conserving the interval ds = ηµνdx

µdxν , i.e. boosts, rotations, translationsand space-time inversions (the list is exhaustive). They constitute a group.

Now to the quantum level: whenever macroscopic measuring and/or prepar-ing devices enjoy a relativistic transformation, the Hilbert (Fock) space shouldtransform according to a corresponding linear42 transformation. It is also almostself-evident that the quantum transformation corresponding to a composition ofclassical transformations should be equal to the composition of quantum trans-formations corresponding to the individual classical transformations. So at thequantum level the relativistic transformations should constitute a representationof the group of classical transformations.

The point now is that the Hamiltonian, as the time-translations generator,is among the generators of the Poincare group (boosts, rotations, translations).Consequently, unlike in a nonrelativistic QM, in a relativistic quantum theoryone cannot specify the Hamiltonian alone, one has rather to specify it within acomplete representation of the Poincare algebra. This is the starting point ofany effort to get a relativistic quantum theory, even if it is not always statedexplicitly. The outcome of such efforts are quantum fields. Depending on thephilosophy adopted, they use to emerge in at least two different ways. We willcall them particle-focused and field-focused.

The particle-focused approach, represented mostly by the Weinberg’s book,is very much in the spirit of the previous section. One starts from the Fockspace, which is systematically built up from the 1-particle Hilbert space. Cre-ation and annihilation operators are then defined as very natural objects, namelyas maps from the n-particle subspace into (n± 1)-particle ones, and only after-wards quantum fields are built from these operators in a bit sophisticated way(keywords being cluster decomposition principle and relativistic invariance).

The field-focused approach (represented by Peskin–Schroeder, and sharedby the majority of textbooks on the subject) starts from the quantization of aclassical field, introducing the creation and annihilation operators in this wayas quantum incarnations of the normal mode expansion coefficients, and finallyproviding Fock space as the world for these operators to live in. The logic behindthis formal development is nothing else but construction of the Poincare groupgenerators. So, again, the corner-stone is the relativistic invariance.

42Linearity of transformations at the quantum level is necessary to preserve the super-position principle. The symmetry transformations should not change measurable things,which would not be the case if the superposition |ψ〉 = c1 |ψ1〉 + c2 |ψ2〉 would transformto T |ψ〉 6= c1T |ψ1〉+ c2T |ψ2〉.


1.3.1 Lorentz and Poincare groups

This is by no means a systematic exposition to the Lorentz and Poincare groupsand their representations. It is rather a summary of important relations, someof which should be familiar (at some level of rigor) from the previous courses.

the groups

The classical relativistic transformations constitute a group, the correspondingtransformations at the quantum level constitute a representation of this group.The (active) group transformations are

xµ → Λµνx

ν + aµ

where Λµν are combined rotations, boosts and space-time inversions, while aµ

describe translations.The rotations around (and the boosts along) the space axes are

R1 (ϑ) =

1 0 0 00 1 0 00 0 cosϑ − sinϑ0 0 sinϑ cosϑ

B1 (β) =

chβ − shβ 0 0− shβ chβ 0 0

0 0 1 00 0 0 1

R2 (ϑ) =

1 0 0 00 cosϑ 0 sinϑ0 0 1 00 − sinϑ 0 cosϑ

B2 (β) =

chβ 0 − shβ 00 1 0 0

− shβ 0 chβ 00 0 0 1

R3 (ϑ) =

1 0 0 00 cosϑ − sinϑ 00 sinϑ cosϑ 00 0 0 1

B3 (β) =

chβ 0 0 − shβ0 1 0 00 0 1 0

− shβ 0 0 chβ

where ϑ is the rotation angle and tanhβ = v/c. They constitute the Lorentzgroup. It is a non-compact (because of β ∈ (−∞,∞)) Lie group.

The translations along the space-time axes are

T0 (α) =

α000

T1 (α) =

0α00

T2 (α) =

00α0

T3 (α) =

000α

Together with the boosts and rotations they constitute the Poincare group. Itis a non-compact (on top of the non-compactness of the Lorentz subgroup onehas α ∈ (−∞,∞)) Lie group.

The space-time inversions are four different diagonal matrices. The timeinversion is given by I0 = diag (−1, 1, 1, 1), the three space inversions are givenby I1 = diag (1,−1, 1, 1) I2 = diag (1, 1,−1, 1) I3 = diag (1, 1, 1,−1).


the Lie algebra

The standard technique of finding the representations of a Lie group is to findthe representations of the corresponding Lie algebra (the commutator algebraof the generators). The standard choice of generators corresponds to the above10 types of transformations: infinitesimal rotations Ri (ε) = 1 − iεJi + O(ε2),boosts Bi (ε) = 1− iεKi +O(ε2) and translations Tµ (ε) = −iεPµ +O(ε2)

J1 = i

0 0 0 00 0 0 00 0 0 −10 0 1 0

K1 = i

0 −1 0 0−1 0 0 00 0 0 00 0 0 0

J2 = i

0 0 0 00 0 0 10 0 0 00 −1 0 0

K2 = i

0 0 −1 00 0 0 0−1 0 0 00 0 0 0

J3 = i

0 0 0 00 0 −1 00 1 0 00 0 0 0

K3 = i

0 0 0 −10 0 0 00 0 0 0−1 0 0 0

P0 = i

1000

P1 = i

0100

P2 = i

0010

P3 = i

0001

Calculation

of the commutators is straightforward43 (even if not very exciting)

[Ji, Jj ] = iεijkJk [Ji, P0] = 0

[Ki,Kj ] = −iεijkJk [Ji, Pj ] = iεijkPk

[Ji,Kj] = iεijkKk [Ki, P0] = iPi

[Pµ, Pν ] = 0 [Ki, Pj ] = iP0δij

Remark: The generators of the Lorentz group are often treated in a morecompact way. One writes an infinitesimal transformation as Λµ

ν = δµν + ωµν ,

where ωµν is antisymmetric (as can be shown directly form the definition of the

Lorentz transformation). Now comes the trick: one writes ωµν = − i

2ωρσ (Mρσ)

µν

where ρσ are summation indices, µν are indices of the Lorentz transformationand Mρσ are chosen so that ωρσ are precisely what they look like, i.e. ηρσ +ωρσ

are the Lorentz transformation matrices corresponding to the map of co-vectorsto vectors. One can show, after some gymnastics, that

(Mρσ)µν = i

(δµρ ησν − δµσηρν

)Jk =

1

2εijkMij Ki =Mi0

43Commutators between Ji (or Ki) and Pµ follow from Λµν (xν + aν) −

(

Λµνx

ν + aµ)

=Λµ

νaν − aµ = (Λ− 1)µν a

ν , i.e. one obtains the commutator under consideration directly byacting of the corresponding generator Ji or Ki on the (formal) vector Pµ.


the scalar representation of the Poincare group

Investigations of the Poincare group representations are of vital importanceto any serious attempt to discuss relativistic quantum theory. It is, however,not our task right now. For quite some time we will need only the simplestrepresentation, the so-called scalar one. All complications introduced by higherrepresentations are postponed to the summer term.

Let us consider a Hilbert space in which some representation of the Poincaregroup is defined. Perhaps the most convenient basis of such a space is theone defined by the eigenvectors of the translation generators Pµ, commutingwith each other. The eigenvectors are usually denoted as |p, σ〉, where p =(p0, p1, p2, p3) stands for eigenvalues of P . In this notation one has

Pµ |p, σ〉 = pµ |p, σ〉

and σ stands for any other quantum numbers. The representation of space-timetranslations are obtained by exponentiation of generators. If U (Λ, a) is theelement of the representation, corresponding to the Poincare transformationx→ Λx+ a, then U (1, a) = e−iaP . And since |p, σ〉 is an eigenstate of Pµ, oneobtains

U(1, a) |p, σ〉 = e−ipa |p, σ〉And how does the representation of the Lorentz subgroup look like? Since the

pµ is a fourvector, one may be tempted to try U (Λ, 0) |p, σ〉 ?= |Λp, σ〉. This

really works, but only in the simplest, the so-called scalar representation, inwhich no σ is involved. It is straightforward to check that in such a case therelation

U (Λ, a) |p〉 = e−i(Λp)a |Λp〉defines indeed a representation of the Poincare group.

Let us remark that once the spin is involved, it enters the parameter σ andthe transformation is a bit more complicated. It goes like this: U (Λ, a) |p, σ〉 =∑

σ′ e−i(Λp)aCσσ′ |Λp, σ′〉 and the coefficients Cσσ′ do define the particular rep-resentation of the Lorentz group. These complications, however, do not concernus now. For our present purposes the simplest scalar representation will besufficient.

Now it looks like if we had reached our goal — we have a Hilbert spacewith a representation of the Poincare group acting on it. A short inspectionreveals, however, that this is just the rather trivial case of the free particle. Tosee this, it is sufficient to realize that the operator P 2 = PµP

µ commutes withall the generators of the Poincare group, i.e. it is a Casimir operator of thisgroup. If we denote the eigenvalue of this operator by the symbol m2 then theirreducible representations of the Poincare group can be classified by the valueof the m2. The relation between the energy and the 3-momentum of the state|p〉 is E2 − ~p2 = m2, i.e. for each value of m2 we really do have the Hilbertspace of states of a free relativistic particle. (The reader is encouraged to clarifyhim/herself how should the Hilbert space of the states of free relativistic particlelook like. He/she should come to conclusion, that it has to be equal to what wehave encountered just now.)

Nevertheless, this rather trivial representation is a very important one — itbecomes the starting point of what we call the particle-focused approach to thequantum field theory. We shall comment on this briefly in the next paragraph.


The Hilbert space spanned over the eigenvectors |p, σ〉 of the translationgenerators Pµ is not the only possible choice of a playground for the relativisticquantum theory. Another quite natural Hilbert space is provided by the func-tions ϕ (x). A very simple representation of the Poincare group is obtained bythe mapping

ϕ (x) → ϕ (Λx+ a)

This means that with any Poincare transformation x→ Λx+a one simply pullsback the functions in accord with the transformation. It is almost obvious thatthis, indeed, is a representation (if not, check it in a formal way). Actually,this representation is equivalent to the scalar representation discussed above, aswe shall see shortly. (Let us remark that more complicated representations canbe defined on n-component functions, where the components are mixed by thetransformation in a specific way.)

It is straightforward to work out the generators in this representation (andwe shall need them later on). The changes of the space-time position xµ andthe function ϕ (x) by the infinitesimal Poincare transformation are δxµ andδxµ∂µϕ (x) respectively, from where one can directly read out the Poincaregenerators in this representation

Jiϕ (x) = (Jix)µ∂µϕ (x)

Kiϕ (x) = (Kix)µ∂µϕ (x)

Pµϕ (x) = i∂µϕ (x)

Using the explicit knowledge of the generators Ji and Ki one obtains

Jiϕ (x) =i

2εijk

(δµj ηkν − δµk ηjν

)xν∂µϕ (x) = −iεijkxj∂kϕ (x)

Kiϕ (x) = i (δµi η0ν − δµ0 ηiν)xν∂µϕ (x) = ix0∂iϕ (x)− ixi∂0ϕ (x)

or even more briefly ~J ϕ (x) = −i~x×∇ϕ (x) and ~Kϕ (x) = it∇ϕ (x)− i~xϕ (x).

At this point the reader may be tempted to interpret ϕ (x) as a wave-functionof the ordinary quantum mechanics. There is, however, an important differencebetween what have now and the standard quantum mechanics. In the usual for-mulation of the quantum mechanics in terms of wave-functions, the Hamiltonianis specified as a differential operator (with space rather than space-time deriva-tives) acting on the wave-function ϕ (x). Our representation of the Poincarealgebra did not provide any such Hamiltonian, it just states that the Hamilto-nian is the generator of the time translations.

However, if one is really keen to interpret ϕ (x) as the wave-function, oneis allowed to do so44. Then one may try to specify the Hamiltonian for thisirreducible representation by demanding p2 = m2 for any eigenfunction e−ipx.

44In that case, it is more conveniet to define the transformation as push-forward, i.e. bythe mapping ϕ (x) → ϕ

(

Λ−1 (x− a))

. With this definition one obtains Pµ = −i∂µ with the

exponentials e−ipx being the eigenstates of these generators. These eigenstates could/shouldplay the role of the |p〉 states. And, indeed,

e−ipx → e−ipΛ−1(x−a) = e−iΛ−1Λp.Λ−1(x−a) = ei(Λp)(x−a) = e−ia(Λp)eix(Λp)

which is exactly the transformation of the |p〉 state.


In this way, one is lead to some specific differential equation for ϕ (x), e.g. tothe equation

−i∂tϕ (x) =√m2 − ∂i∂i ϕ (x)

Because of the square root, however, this is not very convenient equation towork with. First of all, it is not straightforward to check, if this operator obeysall the commutation relations of the Poincare algebra. Second, after the Taylorexpansion of the square root one gets infinite number of derivatives, whichcorresponds to a non-local theory (which is usually quite non-trivial to be putin accord with special relativity). Another ugly feature of the proposed equationis that it treated the time and space derivatives in very different manner, which isat least strange in a would-be relativistic theory. The awkwardness of the squareroot becomes even more apparent once the interaction with electromagnetic fieldis considered, but we are not going to penetrate in such details here.

For all these reasons it is a common habit to abandon the above equationand rather to consider the closely related so-called Klein–Gordon equation

(∂µ∂

µ +m2)ϕ (x) = 0

as a kind of a relativistic version of the Schrodinger equation (even if the orderof the Schrodinger and Klein–Gordon equations are different).

Note, however, that the Klein–Gordon equation is only related, but notequivalent to the equation with the square root. One of the consequences of thisnon-equivalence is that the solutions of the Klein-Gordon equation may haveboth positive and negative energies. This does not pose an immediate problem,since the negative energy solutions can be simply ignored, but it becomes reallypuzzling, once the electromagnetic interactions are switched on.

Another unpleasant feature is that one cannot interpret |ϕ (x)|2 as a prob-ability density, because this quantity is not conserved. For the Schrodingerequation one was able to derive the continuity equation for the density |ϕ (x)|2and the corresponding current, but for the Klein–Gordon equation the quan-tity |ϕ (x)|2 does not obey the continuity equation any more. One can, however,perform with the Klein–Gordon equation a simple massage analogous to the oneknown from the treatment of the Schrodinger equation, to get another continuityequation with the density ϕ∗∂0ϕ − ϕ∂0ϕ

∗. But this density has its own draw-back — it can be negative. It cannot play, therefore the role of the probabilitydensity.

All this was well known to the pioneers of the quantum theory and eventu-ally led to rejection of wave-function interpretation of ϕ (x) in the Klein–Gordonequation. Strangely enough, the field ϕ (x) equation remained one of the cor-nerstones of the quantum field theory. The reason is that it was not the functionand the equation which were rejected, but rather only their wave-function in-terpretation.

The function ϕ (x) satisfying the Klein–Gordon is very important — it be-comes the starting point of what we call the field-focused approach to the quan-tum field theory. In this approach the function ϕ (x) is treated as a classical field(transforming according to the considered representation of the Poincare group)and starting from it one develops step by step the corresponding quantum the-ory. The whole procedure is discussed in quite some detail in the followingchapters.


1.3.2 The logic of the particle-focused approach to QFT

The relativistic quantum theory describes, above all, the physics of elementaryparticles. Therefore the particle-focused approach looks like the most natural.Nevertheless, it is by far not the most common, for reasons which are mainlyhistorical. Now we have to confess (embarrassed) that in these lectures we aregoing to follow the less natural, but more wide-spread field-focused approach.45

The particle-focused approach is only very briefly sketched in this paragraph.Not everything should and could be understood here, it is sufficient just to catchthe flavor. If too dense and difficult (as it is) the paragraph should be skipped.

One starts with an irreducible representation of the Poincare group on some1-particle Hilbert space. The usual basis vectors in the Hilbert space are of theform |p, σ〉, where p is the (overall) momentum of the state and all the othercharacteristics are included in σ. For a multiparticle state, the σ should containa continuous spectrum of momenta of particular particles. This provides us witha natural definition of 1-particle states as the ones with discrete σ. In this caseit turns out that values of σ correspond to spin (helicity) projections.

Irreducible representations are characterized by eigenvalues of two Casimiroperators (operators commuting with all generators), one of them being m2, theeigenvalue of the Casimir operator P 2, and the second one having to do withthe spin. The states in the Hilbert space are therefore characterized by eigen-values of 3-momentum, i.e. the notation |~p, σ〉 is more appropriate than |p, σ〉(nevertheless, when dealing with Lorentz transformations, the |p, σ〉 notation isvery convenient). The |~p, σ〉 states are still eigenstates of the Hamiltonian, with

the eigenvalues E~p =√~p2 +m2.

Once a representation of the Poincare group on a 1-particle Hilbert spaceis known, one can systematically build up the corresponding Fock space fromdirect products of the Hilbert ones. The motivation for such a construction isthat this would be a natural framework for processes with nonconserved numbersof particles, and such processes are witnessed in the nature. This Fock spacebenefits from having a natural representation of Poincare group, namely the onedefined by the direct products of the representations of the original 1-particleHilbert space. The Hamiltonian constructed in this way, as well as all the othergenerators, correspond to a system of noninteracting particles. In terms ofcreation and annihilation operators, which are defined as very natural operators

in the Fock space the free Hamiltonian has a simple form H0 =∫

d3p(2π)3

E~pa+~p a~p.

45The explanation for this is a bit funny.As to what we call here the particle-centered approach, the textbook is the Weinberg’s one.

We strongly recommend it to the reader, even if it would mean that he/she will quit thesenotes. The present author feels that he has nothing to add to the Weinberg’s presentation.

But even if the approach of the Weinberg’s book is perhaps more natural than any other,it is certainly not a good idea to ignore the traditional development, which we call here thefield-centered approach. If for nothing else, then simply because it is traditional and thereforeit became a part of the standard background of the majority of particle physicists.

Now as to the textbooks following the traditional approach, quite a few are available. Butperhaps in all of them there are points (and unfortunately not just one or two) which are notexplained clearly enough, and are therefore not easy to grasp. The aim of the present notes isto provide the standard material with perhaps a bit more emphasis put on some points whichare often only glossed over. The hope is, that this would enable reader to penetrate into thesubject in combination of a reasonable depth with a relative painlessness.

Nevertheless, beyond any doubt, this hope is not to be fulfilled. The reader will surely finda plenty of disappointing parts in the text.


The next step, and this is the hard one, is to find another Hamiltonian whichwould describe, in a relativistic way, a system of interacting particles. One doesnot start with a specific choice, but rather with a perturbation theory for ageneric Hamiltonian H = H0 + Hint. The perturbation theory is neatly for-mulated in the interaction picture, where |ψI (t)〉 = U(t, 0)|ψI (0)〉, with U(t, 0)satisfying i∂tU(t, 0) = Hint,I (t)U(t, 0) with the initial condition U(0, 0) = 1.The perturbative solution of this equation leads to the sum of integrals of theform

∫ t

t0dt1 . . . dtn T Hint,I (t1) . . .Hint,I (tn), where T orders the factors with

respect to decreasing time. For a relativistic theory, these integrals should betterbe Lorentz invariant, otherwise the scalar products of the time-evolved stateswould be frame dependent. This presents nontrivial restrictions on the interac-tion Hamiltonian Hint. First of all, the space-time variables should be treatedon the same footing, which would suggest an interaction Hamiltonian of theform Hint =

∫d3x Hint and Hint should be a Lorentz scalar. Furthermore, the

T -ordering should not change the value of the product when going from frameto frame, which would suggest [Hint (x) ,Hint (y)] = 0 for (x− y)

2 ≤ 0 (for time-like intervals, the ordering of the Hamiltonians in the T -product is the same inall the reference frames, for space-like intervals the ordering is frame-dependent,but becomes irrelevant for Hamiltonians commuting with each other).

All these requirements do not have a flavor of rigorous statements, theyare rather simple observations about how could (would) a relativistic quantumtheory look like. It comes as a kind of surprise, that the notion of quantumfields is a straightforward outcome of these considerations. Without going intodetails, let us sketch the logic of the derivation:1. As any linear operator, the Hamiltonian can be written as a sum of productsof the creation and annihilation operators. The language of the a+~p and a~poperators is technicaly advantageous, e.g. in the formulation of the so-calledcluster decomposition principle, stating that experiments which are sufficientlyseparated in space, have unrelated results.2. Poincare transformations of a+~p and a~p (inherited from the transforma-

tions of states) are given by ~p-dependent matrices, and so the products ofsuch operators (with different momenta) have in general complicated trans-formation properties. One can, however, combine the a+~p and a~p operatorsinto simply transforming quantities called the creation and annihilation fieldsϕ+l (x) =

∑σ

∫d3p ul(x, ~p, σ)a

+~p and ϕ−

l (x) =∑

σ

∫d3p vl(x, ~p, σ)a~p, which are

much more suitable for a construction of relativistic quantum theories.3. The required simple transformation properties of ϕ±

l are the ones independentof any x or ~p, namely ϕ±

l (x) → ∑l′ Dll′(Λ

−1)ϕ±l′ (x)(Λx + a), where the D

matrices furnish a representation of the Lorentz group. The coefficients ul and vlare calculable for any such representation, e.g. for the trivial one Dll′(Λ

−1) = 1one gets u(x, ~p, σ) = 1

(2π)3√

2E~p

eipx and v(x, ~p, σ) = 1

(2π)3√

2E~p

e−ipx.

4. One can easily construct a scalar Hint(x) from the creation and annihilationfields. The vanishing commutator of two such Hint(x) for time-like intervals,however, is not automatically guaranteed. But if Hint(x) is constructed from aspecific linear combination of the creation and annihilation fields, namely formthe fields ϕl(x) = ϕ+

l (x)+ϕ−l (x), then the commutator is really zero for time-like

intervals. This is the way how the quantum fields are introduced in the Wein-berg’s approach — as (perhaps the only) the natural objects for construction ofinteraction Hamiltonians leading to relativistic quantum theories.


1.3.3 The logic of the field-focused approach to QFT

The basic idea of the field-focused approach to quantum fields is to take a clas-sical relativistic field theory and to quantize it canonically (the exact meaningof this statement is to be explained in the next chapter). This makes a perfectsense in case of the electromagnetic field, since the primary task of the canonicalquantization is to provide a quantum theory with a given classical limit. If thefield is classically well known, but one suspects that there is some underlyingquantum theory, then the canonical quantization is a handy tool.

This tool, however, is used also for quantum fields for which there is no suchthing as the corresponding classical fields, at least not one observed normally inthe nature (the electron-positron field is perhaps the prominent example). Thismay sound even more surprising after one realizes that there is a well knownclassical counterpart to the quantum electron, namely the classical electron.So if one is really keen on the canonical quantization, it seems very naturalto quantize the (relativistic) classical mechanics of the electron particle, ratherthan a classical field theory of non-existing classical electron field. But still,what is quantized is indeed the classical field. What is the rationale for this?

First, let us indicate why one avoids the quantization of relativistic particles.Actually even for free particles this would be technically more demanding thanit is for free fields. But this is not the main reason in favor of field quantization.The point is that we are not interested in free (particles or field) theory, butrather in a theory with interaction. And while it is straightforward to generalizea relativistic classical free field theory to a relativistic classical field theory withinteraction (and then to quantize it), it is quite non-trivial to do so for particles.

Second, it should be perhaps emphasized that the nickname ”second quan-tization”, which is sometimes used for the canonical quantization of fields, pro-vides absolutely no clue as to any real reasons for the procedure. On the con-trary, the nickname could be very misleading. It suggests that what is quantizedis not a classical field, but rather a wave-function, which may be regarded to bethe result of (the first) quantization. This point of view just obscures the wholeproblem and is of no relevance at all (except of, perhaps, the historical one).

So why are the fields quantized? The reason is this: In the non-relativisticquantum theories the dynamics is defined by the Hamiltonian. Important pointis that any decent Hamiltonian will do the job. In the relativistic quantumtheories, on the other hand, the Hamiltonian, as the time-translations generator,comes in the unity of ten generators of the Poincare group. Not every decentHamiltonian defines a relativistic dynamics. The reason is that for a givenHamiltonian, one cannot always supply the nine friends to furnish the Poincarealgebra. As a matter of fact, it is in general quite difficult, if not impossible,to find such nine friends. Usually the most natural way is not to start withthe Hamiltonian and then to try to find the corresponding nine generators,but to define the theory from the beginning by presenting the whole set often generators46. This is definitely much easier to say than to really provide.And here comes the field quantization, a clever trick facilitating simultaneousconstruction of all ten Poincare generators.

46From this it should be clear that even the formulation, not to speak about solution, ofthe relativistic quantum theory is about 10 times more difficult than that of non-relativisticquantum theory. The situation is similar to the one in general relativity with 10 componentsof the metric tensor as opposed to one potential describing the Newtonian gravity.


The starting point is a relativistic classical field theory. This means, first ofall, that the Poincare transformations of the field are well defined (as an examplewe may take the function ϕ (x) discussed on p. 43, which is now treated as a clas-sical field transforming according to the scalar representation47 of the Poincaregroup). Then only theories which are symmetric under these transformationsare considered. Now one could expect that, after the canonical quantization,the Poincare transformations of classical fields become somehow the desiredPoincare transformations of the Hilbert space of states. The reality, however, isa bit more sophisticated. Here we are going to sketch it only very briefly, detailsare to be found in the next chapter

At the classical level, to each symmetry there is a conserved charge (Noether’stheorem). When formulated in the Hamiltonian formalism, the Poisson bracketsof these charges obey the same algebra, as do the Poincare generators. Aftercanonical quantization, the Noether charges become operators (in the Hilbertspace of states), the Poisson brackets become commutators, and the Poissonbracket algebra becomes the Poincare algebra itself (or, strictly speaking, somerepresentation of the Poincare algebra). Consequently, the Noether chargesbecome, in the process of the canonical quantization, the generators of the sym-metry at the quantum level.

Precisely this is going to be the logic behind the field quantization adopted inthese lecture notes: field quantization is a procedure leading in a systematic wayto a quantum theory with a consistent package of the ten Poincare generators.

Let us emphasize once more that another important aspect of canonicalquantization, namely that it leads to a quantum theory with a given classicallimit, is not utilized here. We ignore this aspect on purpose. In spite of theimmense role it has played historically and in spite of the undisputed importanceof this aspect in the case of the electromagnetic field, for other fields it is illusoryand may lead to undue misconceptions.

To summarize: Enlightened by the third introduction (Relativity and Quan-tum Theory) we are now going to penetrate a bit into the technique of thecanonical quantization of relativistic classical fields. The result will be a rela-tivistic quantum theory in terms of creation and annihilation operators familiarfrom the second introduction (Many-Body Quantum Mechanics). Clever ver-sion of the perturbation theory formulated within the obtained theories will thenlead us to the Feynman rules discussed in the first introduction (Conclusions).

Remark: For the sake of completeness let us mention yet another approachto the quantum field theory — the one which can be naturally called the pathintegral-focused approach. We will have much to say about it in the chapter ??

47Why representation, why not any (possibly non-linear) realization? We have answered thisquestion (the keyword was the superposition principle) supposing realization in the Hilbertspace of quantum states. Now ϕ (x) does not correspond to the quantum state, so it islegitimate to raise the question again.

The answer (pretty unclear at the moment) is that non-linear transformations of classicalfields would lead, after quantization, to transformations not conserving the number of particles,which is usually ”unphysical” in a sense that one could discriminate between two inertialsystems by counting particles.

Chapter 2

Free Scalar Quantum Field

In this chapter the simplest QFT, namely the theory of the free scalar field, isdeveloped along the lines described at the end of the previous chapter. The key-word is the canonical quantization (of the corresponding classical field theory).

2.1 Elements of Classical Field Theory

2.1.1 Lagrangian Field Theory

mass points nonrelativistic fields relativistic fields

qa (t) a = 1, 2, . . . ϕ (~x, t) ~x ∈ R3 ϕ (x) x ∈ R (3, 1)

S =∫dt L (q, q) S =

∫d3x dt L (ϕ,∇ϕ, ϕ) S =

∫d4x L (ϕ, ∂µϕ)

ddt

∂L∂qa

− ∂L∂qa

= 0 −→ ∂µ∂L

∂(∂µϕ) − ∂L∂ϕ = 0

The third column is just a straightforward generalization of the first one, withthe time variable replaced by the corresponding space-time analog. The purposeof the second column is to make such a formal generalization easier to digest1.For more than one field ϕa (x) a = 1, . . . , n one hasL (ϕ1, ∂µϕ1, . . . , ϕn, ∂µϕn)and there are n Lagrange-Euler equations

∂µ∂L

∂(∂µϕa)− ∂L∂ϕa

= 0

1The dynamical variable is renamed (q → ϕ) and the discrete index is replaced by thecontinuous one (qa → ϕx = ϕ (x)). The kinetic energy T , in the Lagrangian L = T − U ,is written as the integral (

∑

a T (qa) →∫

d3x T (ϕ (x))). The potential energy is in generalthe double integral (

∑

a,b U (qa, qb) →∫

d3x d3y U (ϕ (x) , ϕ (y))), but for the continuous

limit of the nearest neighbor interactions (e.g. for an elastic continuum) the potential energyis the function of ϕ (x) and its gradient, with the double integral reduced to the single one(∫

d3x d3y U (ϕ (x) , ϕ (y)) →∫

d3x u (ϕ (x) ,∇ϕ (x)))

49

50 CHAPTER 2. FREE SCALAR QUANTUM FIELD

The fundamental quantity in the Lagrangian theory is the variation of theLagrangian density with a variation of fields

δL = L (ϕ+ εδϕ, ∂µϕ+ ε∂µδϕ) − L (ϕ, ∂µϕ)

= ε

[∂L (ϕ, ∂µϕ)

∂ϕδϕ+

∂L (ϕ, ∂µϕ)

∂(∂µϕ)∂µδϕ

]+O(ε2)

= ε

[δϕ

(∂L∂ϕ

− ∂µ∂L

∂(∂µϕ)

)+ ∂µ

(∂L

∂(∂µϕ)δϕ

)]+O(ε2)

It enters the variation of the action

δS =

∫δL d4x

= ε

∫ [δϕ

(∂L∂ϕ

− ∂µ∂L

∂(∂µϕ)

)+ ∂µ

(∂L

∂(∂µϕ)δϕ

)]d4x+O(ε2)

which in turn defines the equations of motion, i.e. the Lagrange–Euler equationsfor extremal action S (for δϕ vanishing at space infinity always and for initialand final time everywhere)

δS = 0 ⇒∫δϕ

(∂L∂ϕ

− ∂µ∂L

∂(∂µϕ)

)d4x+

∫∂L

∂(∂µϕ)δϕ d3Σ = 0

The second term vanishes for δϕ under consideration, the first one vanishes forany allowed δϕ iff ∂L

∂ϕ − ∂µ∂L

∂(∂µϕ) = 0.

Example: Free real Klein-Gordon field L [ϕ] = 12∂µϕ∂

µϕ− 12m

2ϕ2

∂µ∂µϕ+m2ϕ = 0

This Lagrange-Euler equation is the so-called Klein-Gordon equation. It enteredphysics as a relativistic generalization of the Schrodinger equation (~p = −i∇,E = i∂t,

(p2 −m2

)ϕ = 0, recall ~ = c = 1). Here, however, it is the equation

of motion for some classical field ϕ.

Example: Interacting Klein-Gordon field L [ϕ] = 12∂µϕ∂

µϕ− 12m

2ϕ2− 14!gϕ

4

∂µ∂µϕ+m2ϕ+

1

3!gϕ3 = 0

Nontrivial interactions lead, as a rule, to nonlinear equations of motion.

Example: Free complex Klein-Gordon field L [ϕ] = ∂µϕ∗∂µϕ−m2ϕ∗ϕ

The fields ϕ∗ and ϕ are treated as independent.2

(∂µ∂

µ +m2)ϕ = 0

(∂µ∂

µ +m2)ϕ∗ = 0

The first (second) equation is the Lagrange-Euler equation for ϕ∗(ϕ) respectively.

2It seems more natural to write ϕ = ϕ1 + iϕ2, where ϕi are real fields and treat thesetwo real fields as independent variables. However, one can equally well take their linearcombinations ϕ′

i = cijϕj as independent variables, and if complex cij are allowed, then ϕ∗

and ϕ can be viewed as a specific choice of such linear combinations.

2.1. ELEMENTS OF CLASSICAL FIELD THEORY 51

Noether’s Theorem

Symmetries imply conservation laws. A symmetry is an (infinitesimal, local)field transformation

ϕ (x) → ϕ (x) + εδϕ (x)

leaving unchanged either the Lagrangian density L, or (in a weaker version)the Lagrangian L =

∫L d3x, or at least (in the weakest version) the action

S =∫L d4x. Conservation laws are either local ∂µj

µ = 0 or global ∂tQ = 0,and they hold only for fields satisfying the Lagrange-Euler equations.

symmetry conservation law current or charge

δL = 0 ∂µjµ = 0 jµ = ∂L

∂(∂µϕ)δϕ

δL = ε∂µJ µ (x) ∂µjµ = 0 jµ = ∂L

∂(∂µϕ)δϕ− J µ

δL = 0 ∂tQ = 0 Q =∫d3x ∂L

∂(∂0ϕ)δϕ

δL = ε∂tQ (t) ∂tQ = 0 Q =∫d3x ∂L

∂(∂0ϕ)δϕ−Q

The first two lines (the strongest version of the Noether’s theorem) follow di-rectly from δL given above (supposing the untransformed field ϕ obeys theLagrange-Euler equations). The next two lines follow from the δL integrated

through the space∫∂µ

(∂L

∂(∂µϕ)δϕ)d3x = 0, which can be written as

∂0

∫d3x

∂L∂(∂0ϕ)

δϕ =

∫d3x ∂i

(∂L

∂(∂iϕ)δϕ

)=

∫dSi

∂L∂(∂iϕ)

δϕ = 0

for the fields obeying the Lagrange-Euler equations and vanishing in the spatialinfinity. The conserved quantity has a form of the spatial integral of some”density”, but this is not necessary a time-component of a conserved current.

Remark: For more than one field in the Lagrangian and for the symmetrytransformation ϕa (x) → ϕa (x) + εδaϕ (x), the conserved current is given by

jµ =∑

a

∂L∂(∂µϕa)

δϕa

Proof: δL = ε∑

a

[δϕa

(∂L∂ϕa

− ∂µ∂L

∂(∂µϕa)

)+ ∂µ

(∂L

∂(∂µϕa)δϕa

)]+O(ε2).

On the other hand, if more symmetries are involved, i.e. if the Lagrangiandensity is symmetric under different transformations ϕ (x) → ϕ (x) + εδkϕ (x),then there is one conserved current for every such transformation

jµk =∑

a

∂L∂(∂µϕ)

δkϕ

Proof: δL = ε∑[

δϕ(

∂L∂ϕ − ∂µ

∂L∂(∂µϕ)

)+ ∂µ

(∂L

∂(∂µϕ)δkϕ)]

+O(ε2).


Example: Phase change — field transformations ϕ → ϕe−iα, ϕ∗ → ϕ∗eiα.Infinitesimal form ϕ→ ϕ− iεϕ, ϕ∗ → ϕ∗+ iεϕ∗, i.e. δϕ = −iϕ and δϕ∗ = iϕ∗.Lagrangian density L [ϕ] = ∂µϕ

∗∂µϕ−m2ϕ∗ϕ− 14g (ϕ

∗ϕ)2. Symmetry δL = 0

jµ =∂L

∂(∂µϕ)δϕ+

∂L∂(∂µϕ∗)

δϕ∗ = −iϕ∂µϕ∗ + iϕ∗∂µϕ

Q =

∫d3x j0 (x) = i

∫d3x

(ϕ∗∂0ϕ− ϕ∂0ϕ∗)

Once the interaction with the electromagnetic field is turned on, this happens tobe the electromagnetic current of the Klein-Gordon field.

Example: Internal symmetries — field transformations ϕi → Tijϕj (i, j =1, . . . , N), where T ∈ G and G is some Lie group of linear transformations.Infinitesimal form ϕi → ϕi − iεk (tk)ij ϕj, i.e. δkϕi = −i (tk)ij ϕj. Lagrangian

density L [ϕ1, . . . , ϕN ] = 12∂µϕi∂

µϕi − 12m

2ϕ2i − 1

4g (ϕiϕi)2. Symmetry δL = 0

jµk =∂L

∂(∂µϕi)δkϕi = −i (∂µϕi) (tk)ij ϕj

Qk =

∫d3x j0 (x) = −i

∫d3x ϕi (tk)ij ϕj

Example: Space-time translations — field transformations ϕ (x) → ϕ (x+ a)(four independent parameters aν will give four independent conservation laws).Infinitesimal transformations ϕ (x) → ϕ (x)+εν∂

νϕ (x), i.e. δνϕ (x) = ∂νϕ (x).The Lagrangian density L [ϕ] = 1

2∂µϕ∂µϕ− 1

2m2ϕ2− 1

4!gϕ4 as a specific example,

but everything holds for any scalar Lagrangian density. Symmetry δL = εν∂νL

(note that a scalar Lagrangian density is transformed as L (x) → L (x+ a),since this holds for any scalar function of x). Technically more suitable form ofthe symmetry3 δL = εν∂µJ µν (x) = εν∂µη

µνL (x).

jµν =∂L

∂(∂µϕ)δνϕ− J µν =

∂L∂(∂µϕ)

∂νϕ− ηµνL

Qν =

∫d3x j0ν (x) =

∫d3x

(∂L

∂(∂0ϕ)∂νϕ− η0νL

)

The conserved quantity is the energy-momentum

Q0 = E =

∫d3x

(∂L∂ϕ

ϕ− L)

Qi = P i =

∫d3x

∂L∂ϕ

∂iϕ

which in our specific example gives

Q0 = E =1

2

∫d3x (ϕ2 + |∇ϕ|2 +m2ϕ2 +

1

12gϕ4)

~Q = ~P =

∫d3x ϕ ∇ϕ

3The index µ is the standard Lorentz index from the continuity equation, the ν specifiesthe transformation, and by coincidence in this case it has the form of a Lorentz index.


Example: Lorentz transformations — field transformations ϕ (x) → ϕ (Λx).Infinitesimal transformations ϕ (x) → ϕ(x) − i

2ωρσ (Mρσ)

µν x

ν∂µϕ(x) (sorry)4,i.e. δρσϕ = −i (Mρσ)

µν x

ν∂µϕ =(δµρ ησν − δµσηρν

)xν∂µϕ = (xρ∂σ − xσ∂ρ)ϕ.

The Lagrangian density L [ϕ] = 12∂µϕ∂

µϕ− 12m

2ϕ2− 14!gϕ

4 again only as a spe-cific example, important is that everything holds for any scalar Lagrangian density.

Symmetry δL = − i2ω

ρσ (Mρσ)λν x

ν∂λL = ωλνxν∂λL, is processed further toδL = ωλν∂λ (xνL) − ωλνηλνL = ωλν∂λ (xνL) = ωλν∂µ (gλµxνL). So one canwrite δL = ωλν∂µJ µλν where5 J µλν = gλµxνL

jµλν =∂L

∂(∂µϕ)

(xν∂λ − xλ∂ν

)ϕ− ηλµxνL

rotations

jµij =∂L

∂(∂µϕ)

(xj∂i − xi∂j

)ϕ− ηiµxjL

Qij = −∫d3x

∂L∂ϕ

(xi∂j − xj∂i

)ϕ

boosts

jµ0i =∂L

∂(∂µϕ)

(xi∂0 − x0∂i

)ϕ− η0µxiL

Q0i = −∫d3x

∂L∂ϕ

(x0∂i − xi∂0

)ϕ+ xiL

In a slightly different notation

rotations ~QR =

∫d3x

∂L∂ϕ

~x×∇ϕ

boosts ~QB = t

∫d3x

∂L∂ϕ

∇ϕ−∫d3x ~x

(∂L∂ϕ

ϕ− L)

and finally in our specific example

rotations ~QR = −∫d3x ϕ ~x×∇ϕ

boosts ~QB = −t∫d3x ϕ ∇ϕ+

∫d3x ~x

(ϕ2 − L

)

These bunches of letters are not very exciting. The only purpose of showing themis to demonstrate how one can obtain conserved charges for all 10 generators ofthe Poincare group. After quantization, these charges will play the role of thegenerators of the group representation in the Fock space.

4For an explanation of this spooky expression see 1.3.1. Six independent parameters ωρσ

correspond to 3 rotations (ωij ) and 3 boosts (ω0i). The changes in ϕ due to these six trans-formations are denoted as δρσϕ with ρ < σ.

5The index µ is the standard Lorentz index from the continuity equation, the pair λνspecifies the transformation, and by coincidence in this case it has the form of Lorentz indices.


2.1.2 Hamiltonian Field Theory

mass points relativistic fields

qa (t) , pa (t) =∂L∂qa

ϕ (x) , π (x) = δLδϕ(x) =

∂L(x)∂ϕ(x)

H =∑

a qapa − L H =∫d3x (ϕ (x) π (x)− L (x))

ddtf = H, f+ ∂

∂tfddtF = H,F+ ∂

∂tF

f, g =∑

a∂f∂pa

∂g∂qa

− ∂g∂pa

∂f∂qa

F,G =∫d3z

(δF

δπ(z)δG

δϕ(z) − δGδπ(z)

δFδϕ(z)

)

The only non-trivial issue is the functional derivative δF/δϕ(x) which is thegeneralization of the partial derivative ∂f/∂xn (note that in the continuous caseϕ plays the role of the variable and x plays the role of index, while in the discretecase x is the variable and n is the index). For functions of n variables one hasδf [~x] = f [~x+ εδ~x] − f [~x] = εδ~x. grad f + O(ε2) =

∑n εδxn.∂f/∂xn + O(ε2).

For functionals6, i.e. for functions with continuous infinite number of variables

δF [ϕ] = F [ϕ+ εδϕ]− F [ϕ] =

∫dx εδϕ(x)

δF [ϕ]

δϕ(x)+O(ε2)

Clearly δFGδϕ(x) =

δFδϕ(x)G+ F δG

δϕ(x) and δf(G[ϕ])δϕ(x) = df [G]

dGδG

δϕ(x) , which are the basic

properties of anything deserving the name derivative.

For our purposes, the most important functionals are going to be of the form∫dy f (ϕ, ∂yϕ), where ∂y ≡ ∂

∂y . In such a case one has7

δ

δϕ (x)

∫dy f (ϕ (y) , ∂yϕ (y)) =

∂f (ϕ (x) , ∂yϕ (x))

∂ϕ (x)− ∂y

∂f (ϕ (x) , ∂yϕ (x))

∂ (∂yϕ (x))

For 3-dimensional integrals in field Lagrangians this reads

δ

δϕ (x)

∫d3y f (ϕ, ϕ,∇ϕ) = ∂f (ϕ, ϕ,∇ϕ)

∂ϕ− ∂i

∂f (ϕ, ϕ,∇ϕ)∂ (∂iϕ)

δ

δϕ (x)

∫d3y f (ϕ, ϕ,∇ϕ) = ∂f (ϕ, ϕ,∇ϕ)

∂ϕ

where RHS are evaluated at the point x.

As illustrations one can take δLδϕ(x) = ∂L(x)

∂ϕ(x) used in the table above, andδδϕ

∫d3x |∇ϕ|2 = −2∇∇ϕ = −2ϕ used in the example below.

6Functional is a mapping from the set of functions to the set of numbers (real or complex).7δ

∫

dy f (ϕ, ∂yϕ) =∫

dy f (ϕ+ εδϕ, ∂yϕ+ ε∂yδϕ) − f (ϕ, ∂yϕ)

= ε∫

dy∂f(ϕ,∂yϕ)

∂ϕδϕ +

∂f(ϕ,∂yϕ)∂(∂yϕ)

∂yδϕ+O(ε2)

= ε∫

dy

(

∂f(ϕ,∂yϕ)∂ϕ

− ∂y∂f(ϕ,∂yϕ)∂(∂yϕ)

)

δϕ+vanishing

surface term+O(ε2)


Example: Klein-Gordon field L [ϕ] = 12∂µϕ∂

µϕ− 12m

2ϕ2(− 1

4!gϕ4)

π (x) =∂L (x)

∂ϕ (x)= ϕ (x) H =

∫d3x H (x)

H (x) = ϕ (x) π (x)− L (x) =1

2π2 +

1

2|∇ϕ|2 + 1

2m2ϕ2 (+

1

4!gϕ4)

ϕ = H,ϕ = π π = H, π = ϕ−m2ϕ (−1

6gϕ3)

Inserting the last relation to the time derivative of the second last one obtainsthe Klein-Gordon equation ϕ−ϕ+m2ϕ = 0

(− 1

6gϕ3)

Poisson brackets of Noether charges

The conserved Noether charges have an important feature, which turns out tobe crucial for our development of QFT: their Poisson brackets obey the Liealgebra of the symmetry group. That is why after the canonical quantization,which transfers functions to operators in a Hilbert space and Poisson bracketsto commutators, the Noether charges become operators obeying the Lie algebraof the symmetry group. As such they are quite natural choice for the generatorsof the group representation in the Hilbert space.

The proof of the above statement is straightforward for internal symmetries.The infinitesimal internal transformations are δkϕi = −i (tk)ij ϕj , where tk arethe generators of the group, satisfying the Lie algebra [ti, tj ] = ifijktk. ThePoisson brackets of the Noether charges are

Qk, Ql =

∫d3x πi(x)δkϕi(x),

∫d3y πm(y)δlϕm(y)

= − (tk)ij (tl)mn

∫d3x πi(x)ϕj(x),

∫d3y πm(y)ϕn(y)

= −∫d3z

∑

a

(tk)ij (tl)mn (δiaϕj(z)πm(z)δna − δmaϕn(z)πi(z)δja)

= −∫d3z π(z) [tl, tk]ϕ(z) =

∫d3z π(z)ifklmtmϕ(z)

= ifklmQm

For the Poincare symmetry the proof is a bit more involved. First of all,the generators now contain derivatives, but this is not a serious problem. Forthe space translations and rotations, e.g., the proof is just a continuous indexvariation of the discrete index proof for the internal symmetries, one just writesP i =

∫d3x d3y π (y) ti(y, x)ϕ (x) and Qij =

∫d3x d3y π (y) tij(y, x)ϕ(x) where

ti(x, y) = δ3(x− y)∂i and tij(x, y) = δ3(x− y)(xj∂i − xi∂j).For the time-translation and boosts the generators are not linear in π and ϕ,

the universal proof for such generators is a bit tricky8. But for any particularsymmetry one may prove the statement just by calculating the Poisson bracketsfor any pair of generators. In the case of the Poincare group this ”exciting”exercise is left to the reader9.

8P. Severa, private communication.9Just kidding, the reader has probably better things to do and he or she is invited to take

the statement for granted even for the Poincare group.


2.2 Canonical Quantization

2.2.1 The procedure

The standard way of obtaining a quantum theory with a given classical limit10:

classical mechanics in any formalism↓

classical mechanics in the Hamiltonian formalism(with Poisson brackets)

replacement of canonical variables by linear operators

replacement of Poisson brackets by commutators f, g → i~

[f , g]

↓explicit construction of a Hilbert space H

explicit construction of the operators↓

quantum mechanics in the Heisenberg picture

Example: A particle in a potential U (x)

L = mx2

2 − U (x)↓

H = p2

2m + U (x)

p, x = 1 dF (x,p)dt = H,F

H = p2

2m + U (x)

[p, x] = −i~ dF (x,p)dt = i

~

[H, F

]

↓

H = L2

xψ (x) = xψ (x) pψ (x) = −i~∂xψ (x)↓

dpdt = i

~

[H, p

]= −∂xU (x) dx

dt = i~

[H, x

]= 1

m∂x

When written in the Schrodinger picture, the Schrodinger equation is obtained.

The above example is not very impressive, since the final result (in theSchrodinger picture) is the usual starting point of any textbook on quantummechanics. More instructive examples are provided by a particle in a generalelectromagnetic field or by the electromagnetic field itself. The latter has playeda key role in the development of the quantum field theory, and is going to bediscussed thoroughly later on (QFT II, summer term). Here we are going toconcentrate on the scalar field quantization. Even this simpler case is sufficientto illustrate some new conceptual problems, not present in the quantization ofordinary mechanical systems with a finite number of degrees of freedom.

10As already mentioned in the first chapter, for our purposes, the classical limit is not theissue. Nevertheless, the technique (of the canonical quantization) is going to be very useful.

2.2. CANONICAL QUANTIZATION 57

Example: The scalar field

L =∫d3x 1

2∂µϕ∂µϕ− 1

2m2ϕ2

(− 1

4!gϕ4)

↓

H =∫d3x 1

2π2 + 1

2 |∇ϕ|2+ 1

2m2ϕ2

(+ 1

4!gϕ4)

π (~x, t) , ϕ (~y, t) = δ3 (~x− ~y) dF [ϕ,π]dt = H,F

H =∫d3x 1

2 π2 + 1

2 |∇ϕ|2+ 1

2m2ϕ2

(+ 1

4!gϕ4)

[π (~x, t) , ϕ (~y, t)] = −i~δ3 (~x− ~y) dF [ϕ,π]dt = i

~

[H, F

]

↓

H =???to be continued

The problem with the example (the reason why it is not finished): H is ingeneral a non-separable Hilbert space. Indeed: for one degree of freedom (DOF)one gets a separable Hilbert space, for finite number of DOF one would expectstill a separable Hilbert space (e.g. the direct product of Hilbert spaces for oneDOF), but for infinite number of DOF there is no reason for the Hilbert spaceto be separable. Even for the simplest case of countable infinite many spins 1/2the cardinality of the set of orthogonal states is 2ℵ0 = ℵ1. For a field, beinga system with continuously many DOF (with infinitely many possible valueseach) the situation is to be expected at least this bad.

The fact that the resulting Hilbert space comes out non-separable is, on theother hand, a serious problem. The point is that the QM works the way it worksdue to the beneficial fact that many useful concepts from linear algebra survive atrip to the countable infinite number of dimensions (i.e. the functional analysisresembles in a sense the linear algebra, even if it is much more sophisticated).For continuously many dimensions this is simply not true any more.

Fortunately, there is a way out, at least for the free fields. The point isthat the free quantum field can be, as we will see shortly, naturally placed intoa separable playground — the Fock space11. This is by no means the onlypossibility, there are other non-equivalent alternatives in separable spaces andyet other alternatives in non-separable ones. However, it would be everythingbut wise to ignore this nice option. So we will, together with the rest of theworld, try to stick to this fortunate encounter and milk it as much as possible.

The Fock space enters the game by changing the perspective a bit and view-ing the scalar field as a system of coupled harmonic oscillators. This is done inthe next section. The other possibilities and their relation to the Fock space areinitially ignored, to be discussed afterwards.

11For interacting fields the Fock space is not so natural and comfortable choice any more, butone usually tries hard to stay within the Fock space, even if it involves quite some benevolenceas to the rigor of mathematics in use. These issues are the subject-matter of the followingchapter.


Scalar Field as Harmonic Oscillators

The linear harmonic oscillator is just a special case of the already discussedexample, namely a particle in the potential U(x). One can therefore quantizethe LHO just as in the above general example (with the particular choice U(x) =mω2x2/2), but this is not the only possibility.

Let us recall that search of the solution of the LHO in the QM (i.e. theeigenvalues and eigenvectors of the Hamiltonian) is simplified considerably byintroduction of the operators a and a+. Analogous quantities can be introducedalready at the classical level12 simply as

a = x

√mω

2+ p

i√2mω

a+ = x

√mω

2− p

i√2mω

The point now is that the canonical quantization can be performed in terms ofthe variables a and a+

L = mx2

2 − mω2x2

2

↓

H = p2

2m + mω2x2

2 = ωa+a or H = ω2 (a+a+ aa+)

a, a+ = i a = −iωa a+ = iωa+

H = ωa+a or H = ω2 (a+a+ aa+)

[a, a+] = 1 a = −iωa a+ = iωa+

↓

H = space spanned by |0〉 , |1〉 , . . .

a |n〉 = |n− 1〉 a+ |n〉 = |n+ 1〉

Note that we have returned back to the convention ~ = c = 1 and refrainedfrom writing the hat above operators. We have considered two (out of manypossible) Hamiltonians equivalent at the classical level, but non-equivalent atthe quantum level (the standard QM choice being H = ω (a+a+ 1/2)). Thebasis |n〉 is orthogonal, but not orthonormal.

12The complex linear combinations of x (t) and p (t) are not as artificial as they may appearat the first sight. It is quite common to write the solution of the classical equation of motionfor LHO in the complex form as x (t) = 1

2(Ae−iωt+Beiωt) and p (t) = − imω

2(Ae−iωt−Beiωt).

Both x (t) and p (t) are in general complex, but if one starts with real quantities, then B = A∗,

and they remain real forever. The a(t) is just a rescaled Ae−iωt: a(t) =√

mω/2Ae−iωt.


The relevance of the LHO in the context of the QFT is given by a ”miracle”furnished by the 3D Fourier expansion

ϕ (~x, t) =

∫d3p

(2π)3 e

i~p.~xϕ (~p, t)

which when applied to the Klein-Gordon equation ∂µ∂µϕ (~x, t)+m2ϕ (~x, t) = 0

leads toϕ (~p, t) + (~p2 +m2)ϕ (~p, t) = 0

for any ~p. Conclusion: the free classical scalar field is equivalent to the (infinite)system of decoupled LHOs13, where ϕ (~p, t) plays the role of the coordinate (notnecessarily real, even if ϕ (~x, t) is real), π = ϕ the role of the momentum and~p the role of the index. Note that m has nothing to do with the mass of theoscillators which all have unit mass and

ω2~p = ~p2 +m2

Quantization of each mode proceeds in the standard way described above.At the classical level we define

a~p (t) = ϕ (~p, t)√

ω~p

2 + π (~p, t) i√2ω~p

A+~p (t) = ϕ (~p, t)

√ω~p

2 − π (~p, t) i√2ω~p

We have used the symbol A+~p instead of the usual a+~p , since we want to reserve

the symbol a+~p for the complex conjugate to a~p. It is essential to realize that

for the ”complex oscillators” ϕ (~p, t) there is no reason for A+~p (t) to be equal to

the complex conjugate a+~p (t) = ϕ∗ (~p, t)√ω~p/2− iπ∗ (~p, t) /

√2ω~p.

For the real classical field, however, the condition ϕ (~x, t) = ϕ∗ (~x, t) impliesϕ (−~p, t) = ϕ∗ (~p, t) (check it) and the same holds also for the conjugate mo-mentum π (~x, t). As a consequence a+~p (t) = A+

−~p (t) and therefore one obtains

ϕ (~x, t) =

∫d3p

(2π)3

1√2ω~p

(a~p (t) e

i~p.~x + a+~p (t) e−i~p.~x)

π (~x, t) =

∫d3p

(2π)3(−i)

√ω~p

2

(a~p (t) e

i~p.~x − a+~p (t) e−i~p.~x)

Now comes the quantization, leading to commutation relations

[a~p (t) , a

+~p′ (t)

]= (2π)

3δ (~p− ~p′) [a~p (t) , a~p′ (t)] =

[a+~p (t) , a+~p′ (t)

]= 0

The reader may want to check that these relations are consistent with anotherset of commutation relations, namely with [ϕ (~x, t) , π (~y, t)] = iδ3 (~x− ~y) and

[ϕ (~x, t) , ϕ (~y, t)] = [π (~x, t) , π (~y, t)] = 0 (hint:∫

d3x(2π)3

e−i~k.~x = δ3(~k)).

13What is behind the miracle: The free field is equivalent to the continuous limit of a systemof linearly coupled oscillators. Any such system can be ”diagonalized”, i.e. rewritten as anequivalent system of decoupled oscillators. For a system with translational invariance, thediagonalization is provided by the Fourier transformation. The keyword is diagonalization,rather than Fourier.


The ”miracle” is not over yet. The free field have turned out to be equivalentto the system of independent oscillators, and this system will now turn out tobe equivalent to still another system, namely to the system of free non-teractingrelativistic particles. Indeed, the free Hamiltonian written in terms of a~p (t) anda+~p (t) becomes14

H =

∫d3x

(1

2π2 +

1

2|∇ϕ|2 + 1

2m2ϕ2

)

=

∫d3p

(2π)3 ω~p

(a+~p (t) a~p (t) +

1

2

[a~p (t) , a

+~p (t)

])

where the last term is an infinite constant (since [a~p (t) , a+~p (t)] = (2π)3δ3(~p−~p)).

This is our first example of the famous (infinite) QFT skeletons in the cupboard.This one is relatively easy to get rid of (to hide it away) simply by subtractingthe appropriate constant from the overall energy, which sounds as a legal step.

Another way leading to the same result is to realize that the canonical quan-tization does not fix the ordering in products of operators. One can obtaindifferent orderings at the quantum level (where the ordering does matter) start-ing from the different orderings at clasical level (where it does not). One maytherefore choose any of the equivalent orderings at the classical level to get thedesired ordering at the quantum level. Then one can postulate that the correctordering is the one leading to the decent Hamiltonian. Anyway, the standardform of the free scalar field Hamiltonian in terms of creation and annihilationoperators is

H =

∫d3p

(2π)3 ω~p a

+~p (t) a~p (t)

This looks pretty familiar. Was it not for the explicit time dependence of thecreation and annihilation operators, this would be the Hamiltonian of the idealgas of free relativistic particles (relativistic because of the relativistic energy

ω~p =√~p2 +m2). The explicit time dependence of the operators, however, is

not an issue — the hamiltonian is in fact time-independent, as we shall seeshortly (the point is that the time dependence of the creation and annihilationoperators turns out to be a+~p (t) = a+~p e

iω~pt and a~p (t) = a~pe−iω~pt respectively).

Still, it is not the proper Hamiltonian yet, since it has nothing to act on.But once we hand over an appropriate Hilbert space, it will indeed become theold friend.

14The result is based on the following algebraic manipulations

∫

d3x π2 = −∫ d3xd3pd3p′

(2π)6

√ω~pω~p′

2

(

a~p (t) − a+−~p

(t))(

a~p′ (t)− a+−~p′

(t))

ei(~p+~p′).~x

= −∫ d3pd3p′

(2π)3

√ω~pω~p′

2

(

a~p (t) − a+−~p

(t))(

a~p′ (t)− a+−~p′

(t))

δ3(~p + ~p′)

= −∫ d3p

(2π)3ω~p

2

(

a~p (t) − a+−~p

(t))(

a−~p (t)− a+~p(t)

)

∫

d3x |∇ϕ|2+m2ϕ2 =∫ d3xd3pd3p′

(2π)6−~p.~p′+m2

2√

ω~pω~p′

(

a~p (t) + a+−~p

(t))(

a~p′ (t) + a+−~p′

(t))

ei(~p+~p′).~x

=∫ d3p

(2π)3ω~p

2

(

a~p (t) + a+−~p

(t))(

a−~p (t) + a+~p(t)

)

where in the last line we have used (~p2 +m2)/ω~p = ω~p.

Putting everything together, one obtains the result.


For the relativistic quantum theory (and this is what we are after) the Hamil-tonian is not the whole story, one rather needs all 10 generators of the Poincaregroup. For the space-translations ~P =

∫d3x π ∇ϕ one obtains15

~P =

∫d3p

(2π)3 ~p a

+~p (t) a~p (t)

while for the rotations and the boosts, i.e. for Qij =∫d3x π

(xj∂i − xi∂j

)ϕ

and Q0i =∫d3x ∂L

∂ϕ

(xi∂0 − x0∂i

)ϕ− xiL the result is16

Qij = i

∫d3p

(2π)3 a

+~p (t)

(pi∂j − pj∂i

)a~p (t)

Q0i = i

∫d3p

(2π)3 ω~pa

+~p (t) ∂ia~p (t)

where ∂i = ∂/∂pi in these formulae.The reason why these operators are regarded as good candidates for the

Poincare group generators is that at the classical level their Poisson bracketsobey the corresponding Lie algebra. After the canonical quantization they aresupposed to obey the algebra as well. This, however, needs a check.

The point is that the canonical quantization does not fix the ordering ofterms in products. One is free to choose any ordering, each leading to someversion of the quantized theory. But in general there is no guarantee that aparticular choice of ordering in the 10 generators will preserve the Lie algebra.Therefore one has to check if his or her choice of ordering did not spoil thealgebra. The alert reader may like to verify the Lie algebra of the Poincaregroup for the generators as given above. The rest of us may just trust theprinted text.

15 ~P =∫ d3p

(2π)3~p2(a~p (t)− a+

−~p(t))(a−~p (t) + a+

~p(t))

=∫ d3p

(2π)3~p2(a~p (t) a−~p (t)− a+−~p (t) a−~p (t) + a~p (t) a+~p (t) − a+−~p (t) a+~p (t))

Now both the first and the last term vanish since they are integrals of odd functions, so~P =

∫ d3p(2π)3

~p2(a+~p (t) a~p (t) + a~p (t) a+~p (t)) =

∫ d3p(2π)3

~p(a+~p (t) a~p (t) + 12[a~p (t) , a+~p (t)])

and the last term again vanishes as a symmetric integral of an odd function.16Qij =

∫ d3xd3pd3p′

2(2π)6

√

ω~p/ω~p′(a~p (t)− a+−~p

(t))(a~p′ (t) + a+−~p′

(t))xjp′iei(~p+~p′).~x − i ↔ j

We start with a~pa+−~p′

= [a~p, a+−~p′

] + a+−~p′

a~p = (2π)3 δ (~p+ ~p′) + a+−~p′

a~p, where a~p stands for

a~p (t) etc. The term with the δ-function vanishes after trivial integration. Then we write

xjp′iei(~p+~p′).~x as −ip′i∂′jei(~p+~p′).~x, after which the d3x integration leads to ∂′jδ3(~p+~p′) andthen using

∫

dk f (k)∂kδ (k) = −∂kf (k) |k=0 one gets

Qij = −i∫ d3p

2(2π)3∂′j

√

ω~p/ω~p′ (a+−~p′

a~p − a+−~pa~p′ + a~pa~p′ − a+

−~pa+−~p′

)p′i|~p′=−~p + i ↔ j

Now using the Leibniz rule and symetric×antisymmetric cancelations one obtains

Qij = −i∫ d3p

2(2π)3pi((∂ja+

~p)a~p − a+

−~p∂ja−~p + a~p∂

ja−~p − a+−~p∂ja+

~p) + i↔ j

At this point one uses per partes integration for the first term, the substitution ~p → −~p forthe second term, and the commutation relations (and ~p→ −~p ) for the last two terms, to get

Qij = i∫ d3p

(2π)3a+~p(pi∂j − pj∂i)a~p − i

∫ d3p4(2π)3

(pi∂j − pj∂i)(2a+~pa~p + a~pa−~p − a+

~pa+−~p

)

In the second term the dpi or dpj integral is trivial, leading to momenta with some componentsinfinite. This term would be therefore important only if states containing particles with infinitemomenta are allowed in the game. Such states, however, are considered unphysical (havinge.g. infinite energy in the free field case). Therefore the last term can be (has to be) ignored.(One can even show, that this term is equal to the surface term in the x-space, which was setto zero in the proof of the Noether’s theorem, so one should set this term to zero as well.)

Boost generators are left as an exercise for the reader.


Side remark on complex fields

For the introductory exposition of the basic ideas and techniques of QFT, thereal scalar field is an appropriate and sufficient tool. At this point, however, itseems natural to say a few words also about complex scalar fields. If nothingelse, the similarities and differences between the real and the complex scalarfields are quite illustrative. The content of this paragraph is not needed forthe understanding of what follows, it is presented here rather for sake of futurereferences.

The Lagrangian density for the free complex scalar fields reads

L [ϕ∗, ϕ] = ∂µϕ∗∂µϕ−m2ϕ∗ϕ

where ϕ = ϕ1 + iϕ2. The complex field ϕ is a (complex) linear combinationof two real fields ϕ1 and ϕ2. One can treat either ϕ1 and ϕ2, or ϕ and ϕ∗ asindependent variables, the particular choice is just the mater of taste. Usuallythe pair ϕ and ϕ∗ is much more convenient.

It is straightforward to check that the Lagrange-Euler equation for ϕ andϕ∗ (as well as for ϕ1 and ϕ2) is the Klein-Gordon equation. Performing nowthe 3D Fourier transformation of both ϕ (~x, t) and ϕ∗ (~x, t), one immediatelyrealizes (just like in the case of the real scalar field) that ϕ (~p, t) and ϕ∗ (~p, t)play the role of the coordinate of a harmonic oscillator

ϕ (~p, t) + (~p2 +m2)ϕ (~p, t) = 0

ϕ∗ (~p, t) + (~p2 +m2)ϕ∗ (~p, t) = 0

while π (~p, t) = ϕ∗ (~p, t) and π∗ (~p, t) = ϕ (~p, t) play the role of the correspondingmomenta. The variable ~p plays the role of the index and the frequency of theoscillator with the index ~p is ω2

~p = ~p2 +m2

Quantization of each mode proceeds again just like in the case of the realfield. For the ϕ field one obtains17

a~p (t) = ϕ (~p, t)√

ω~p

2 + π∗ (~p, t) i√2ω~p

A+~p (t) = ϕ (~p, t)

√ω~p

2 − π∗ (~p, t) i√2ω~p

but now, on the contrary to the real field case, there is no relation between A+~p

and a+~p . It is a common habit to replace the symbol A+~p by the symbol b+−~p = A+

~p

and to write the fields as18

ϕ (~x, t) =

∫d3p

(2π)3

1√2ω~p

(a~p (t) e

i~p.~x + b+~p (t) e−i~p.~x)

ϕ∗ (~x, t) =

∫d3p

(2π)3

1√2ω~p

(a+~p (t) e−i~p.~x + b~p (t) e

i~p.~x)

17For the ϕ∗ field one has the complex conjugated relationsa+~p (t) = ϕ∗ (~p, t)

√

ω~p/2 + iπ (~p, t) /√

2ω~p and A~p (t) = ϕ∗ (~p, t)√

ω~p/2− iπ (~p, t) /√

2ω~p18For the momenta one has

π (~x, t) =∫ d3p

(2π)3(−i)

√

ω~p

2

(

a~p (t) ei~p.~x − b+~p(t) e−i~p.~x

)

π∗ (~x, t) =∫ d3p

(2π)3i√

ω~p

2

(

a+~p(t) e−i~p.~x − b~p (t) ei~p.~x

)


Now comes the quantization, leading to the commutation relations

[a~p (t) , a

+~p′ (t)

]= (2π)

3δ (~p− ~p′)

[b~p (t) , b

+~p′ (t)

]= (2π)

3δ (~p− ~p′)

while all the other commutators vanish.The standard form of the Hamiltonian becomes

H =

∫d3p

(2π)3ω~p

(a+~p (t) a~p (t) + b+~p (t) b~p (t)

)

which looks very much like the Hamiltonian of the ideal gas of two types (a andb) of free relativistic particles. The other generators can be obtained just likein the case of the real field.

The main difference with respect to the real field is that now there aretwo types of particles in the game, with the creation operators a+~p and b+~prespectively. Both types have the same mass and, as a rule, they correspondto a particle and its antiparticle. This becomes even more natural when theinteraction with the electromagnetic field is introduced in the standard way(which we are not going to discuss now). It turns out that the particles createdby a+~p and b+~p have strictly opposite electric charge.

Remark: .


Time dependence of free fields

Even if motivated by the free field case, the operators a~p (t), a+~p (t) can be

introduced equally well in the case of interacting fields. The above (so-calledequal-time) commutation relations would remain unchanged. Nevertheless, inthe case of interacting fields, one is faced with very serious problems which are,fortunately, not present in the free field case.

The crucial difference between the free and interacting field lies in the factthat for the free fields the time dependence of these operators is explicitly known.At the classical level, the independent oscillators enjoy the simple harmonicmotion, with the time dependence e±iω~pt. At the quantum level the same istrue, as one can see immediately by solving the equation of motion

a+~p (t) = i[H, a+~p (t)

]= i

[∫d3p′

(2π)3ω~p′ a+~p′ (t) a~p′ (t) , a+~p (t)

]= iω~p a

+~p (t)

and a~p (t) = −iω~p a~p (t) along the same lines. From now on, we will thereforewrite for the free fields

a+~p (t) = a+~p eiω~pt

a~p (t) = a~pe−iω~pt

where a+~p and a~p are time-independent creation and annihilation operators (they

coincide with a+~p (0) and a~p (0)). This enables us to write the free quantum fieldin a bit nicer way as

ϕ (x) =

∫d3p

(2π)3

1√2ω~p

(a~pe

−ipx + a+~p eipx)

where p0 = ω~p. For interacting fields there is no such simple expression and thisvery fact makes the quantum theory of interacting fields such a complicatedaffair.

Remark: The problem with the interacting fields is not only merely that wedo not know their time dependence explicitly. The problem is much deeper andconcerns the Hilbert space of the QFT. In the next section we are going tobuild the separable Hilbert space for the free fields, the construction is basedon the commutation relations for the operators a~p (t) and a+~p (t). Since thesecommutation relations hold also for the interacting fields, one may consider thisconstruction as being valid for both cases. This, however, is not true.The problem is that once the Hilbert space is specified, one has to check whetherthe Hamiltonian is a well defined operator in this space, i.e. if it defines adecent time evolution. The explicitly known time-dependence of the free fieldsanswers this question for free fields. For the interacting fields the situation ismuch worse. Not only we do not have a proof for a decent time evolution, oncontrary, in some cases we have a proof that the time evolution takes any initialstate away from this Hilbert space. We will come back to these issues in the nextchapter. Until then, let us enjoy the friendly (even if not very exciting) worldof the free fields.


Hilbert Space

The construction of the Hilbert space for any of the infinitely many LHOsrepresenting the free scalar field is straightforward, as described above. Mergingall these Hilbert spaces together is also straightforward, provided there is acommon ground state |0〉. Once such a state is postulated19, the overall Hilbertspace is built as an infinite direct sum of Hilbert spaces of individual LHOs.

Such a space is simply the space spanned by the basis

|0〉|~p〉 =

√2ω~pa

+~p |0〉

|~p, ~p′〉 =√2ω~p′a+~p′ |~p〉

|~p, ~p′, ~p′′〉 =√2ω~pa

+~p |~p′, ~p′′〉

...

where all creation operators are taken at a fixed time, say t = 0, i.e. a+~p ≡ a+~p (0).

The normalization (with notation E~p = ω~p =√~p2 +m2)

〈~p|~p′〉 = 2E~p (2π)3 δ3 (~p− ~p′)

is Lorentz invariant (without√2E~p in the definition of |~p〉 it would not be).

Reason: The integral∫d3p δ3(~p) = 1 is Lorentz invariant, while d3p and δ3(~p)

individually are not. The ratio E/d3p, on the other hand, is invariant20 and sois the d3p δ3(~p) E/d3p = E δ3(~p).

The Hilbert space constructed in this way is nothing else but the Fock spaceintroduced in the first chapter. This leads to another shift in perspective: firstwe have viewed a classical field as a system of classical oscillators, now it turnsout that the corresponding system of quantum oscillators can be viewed as asystem of particles. Perhaps surprising, and very important.

Let us remark that the Fock space is a separable Hilbert space. Originallyour system looked like having continuously infinite dimensional space of states,nevertheless now the number of dimensions seems to be countable. How come?This question is definitely worth discussing, but let us postpone it until the lastsection of this chapter.

At this point we can continue with the scalar field quantization. It was in-terrupted at the point H =???, where one now takes H = the Fock space. Oncethe Hilbert space is given explicitly, the last step is the explicit constructionof the relevant operators. As to the Hamiltonian, we know it in terms of cre-ation and annihilation operators already, and so it happened that it is just theHamiltonian of a system of free noninteracting relativistic particles.

19For infinite number of oscillators (unlike for the finite one) the existence of such a stateis not guaranteed. One is free to assume its existence, but this is an independent assumption,not following from the commutation relations. We will search more into this issue in a while.

20Indeed, let us consider the boost along the x3 axis, with the velocity β. The Lorentztransformation of a 4-momentum p = (E, ~p) is E → γE + γβp3, p1 → p1, p2 → p2 and

p3 → γp3+γβE (where γ =√

1− β2), and the same transformation holds for an infinitesimal4-vector dp. Clearly, d3p is not invariant d3p→ dp1dp2 (γdp3 + γβdE) = d3p (γ + γβdE/dp3).For dE = 0 this would be just a Lorentz contraction, but if both p and p + dp correspondto the same mass m, then E =

√

m2 + ~p2 and dE/dp3 = p3/√

m2 + ~p2 = p3/E. Therefore

d3p→ d3p (γ + γβp3/E) and finally d3pE

→ d3p(γ+γβp3/E)γE+γβp3

= d3pE


An important consequence of the explicit form of the Poincare generatorsare the transformation properties of the basis vectors |~p〉. For this purpose thesuitable notation is the 4-vector one: instead of |~p〉 one writes |p〉 where p0 = ω~p

(dependent variable). The Lorentz transformation takes a simple form

|p〉 Λ→ |Λp〉

This may seem almost self-evident, but it is not. As we will see in a moment, foranother basis (x-representation) where this transformation rule looks equallyself-evident, it simply does not hold. The proof of the transformation rule,i.e. the calculation of how the generators act on the states |p〉, is thereforemandatory. For rotations at, say, t = 0 one has

−iQij |p〉 =∫

d3p′

(2π)3 a

+~p′

(p′i∂′j − p′j∂′i

)a~p′

√2ω~pa

+~p |0〉

=√2ω~p

∫d3p′a+~p′

(p′i∂′j − p′j∂′i

)δ (~p′ − ~p) |0〉

= −√2ω~p

(pi∂j − pj∂i

)a+~p |0〉 = −

(pi∂j − pj∂i

)|p〉

where in the last step we have used (pi∂j − pj∂i)√2ω~p = 0. Now the derivative

of |p〉 in a direction k is defined by |p+ ǫk〉 = |p〉 + ǫkµ∂µ |p〉. For rotations

k = −iJkp (ki = −i(Jk)ijpj = −εijkpj) ⇒∣∣p− iǫJkp

⟩= |p〉− ǫ.εijkpj∂i |p〉, i.e.

(1− iǫQij

)|p〉 =

∣∣(1− iǫJk)p⟩

which is an infinitesimal form of the transformation rule for rotations.For boosts one gets along the same lines

−iQ0i |p〉 =∫

d3p′

(2π)3ω~p′a+~p′∂

′ia~p′

√2ω~pa

+~p |0〉

=√2ω~p

∫d3p′ω~p′a+~p′∂

′iδ (~p′ − ~p) |0〉

= −√2ω~p∂

iω~pa+~p |0〉 = − pi

2ω~p|~p〉 − ω~p∂

i |p〉

and since∣∣p− iεKip

⟩= |p〉 − iǫ

(Ki)jkpk∂j |p〉 = |p〉 − ǫpi∂0 |p〉 − ǫp0∂i |p〉, one

finally obtains (realizing that ∂0 |p〉 = ∂0√2p0a

+~p |0〉 = 1√

2p0a+~p |0〉 = 1

2p0|p〉)

(1− iǫQ0i

)|p〉 =

∣∣(1− iǫKi)p⟩

which is an infinitesimal form of the transformation rule for boosts.As to the translations, the transformation rule is even simpler

|p〉 a→ e−ipa |p〉

as follows directly from the explicit form of the translation generators, whichimplies P |p〉 = p |p〉 (where P 0 = H).

So far everything applied only to t = 0. However, once the explicit timedependence of the creation and annihilation operators in the free field case isfound in the next section, the proof is trivially generalized for any t.


Quasilocalized states

So far, the quantum field have played a role of merely an auxiliary quantity,appearing in the process of the canonical quantization. The creation and annihi-lation operators look more ”physical”, since they create or annihilate physicallywell defined states (of course, only to the extent to which we consider the stateswith the sharp momentum being well defined). Nevertheless the fields will ap-pear again and again, so after a while one becomes so accustomed to them, thatone tends to consider them to be quite natural objects. Here we want to stressthat besides this psychological reason there is also a good physical reason whythe quantum fields really deserve to be considered ”physical”.

Let us consider the Fourier transform of the creation operator

a+ (x) =

∫d3p

(2π)3 e

−i~p.~xa+~p (t) =

∫d3p

(2π)3 e

ipxa+~p

Acting on the vacuum state one obtains a+ (x) |0〉 =∫ d3p

(2π)3eipx 1√

2ω−~p

|~p〉, whichis the superposition (of normalized momentum eigenstates) corresponding to thestate of the particle localized at the point x. This would be a nice object todeal with, was there not for the unpleasant fact that it is not covariant. Thestate localized at the point x is in general not Lorentz transformed21 to thestate localized at the point Λx. Indeed

a+ (x) |0〉 →∫

d3p

(2π)3

1√2ω~p

eipx |Λp〉

and this is in general not equal to a+ (Λx) |0〉. The problem is the non-invarianceof d3p/

√2ω~p. Were it invariant, the substitution p→ Λ−1p would do the job.

Now let us consider ϕ (x) |0〉 =∫

d3p(2π)3

12Ep

eipx |p〉 where Ep = ω~p = p0

ϕ (x) |0〉 →∫

d3p

(2π)3

1

2Epeipx |Λp〉 p→Λ−1p

=

∫d3Λ−1p

(2π)3

1

2EΛ−1pei(Λ

−1p)x ∣∣ΛΛ−1p⟩

=

∫d3p

(2π)3

1

2Epei(Λ

−1p)(Λ−1Λx) |p〉 =∫

d3p

(2π)3

1

2Epeip(Λx) |p〉 = ϕ (Λx) |0〉

so this object is covariant in a well defined sense. On the other hand, the stateϕ (x) |0〉 is well localized, since

〈0| a (x′)ϕ (x) |0〉 = 〈0|∫

d3p

(2π)3

1√2ω~p

eip(x−x′) |0〉

and this integral decreases rapidly for |~x− ~x′| greater than the Compton wave-length of the particle ~/mc, i.e.1/m. (Exercise: convince yourself about this.Hint: use Mathematica or something similar.)

Conclusion: ϕ (x) |0〉 is a reasonable relativistic generalization of a state ofa localized particle. Together with the rest of the world, we will treat ϕ (x) |0〉as a handy compromise between covariance and localizability.

21We are trying to avoid the natural notation a+ (x) |0〉 = |x〉 here, since the symbol |x〉 isreserved for a different quantity in Peskin-Schroeder. Anyway, we want to use it at least inthis footnote, to stress that in this notation |x〉 9 |Λx〉 in spite of what intuition may suggest.This fact emphasizes a need for proof of the transformation |p〉 9 |Λp〉 which is intuitivelyequally ”clear”.


2.2.2 Contemplations and subtleties

Let us summarize our achievements: we have undergone a relatively exhaustivejourney to come to almost obvious results. The (relativistic quantum) theory(of free particles) is formulated in the Fock space, which is something to beexpected from the very beginning. The basis vectors of this space transform inthe natural way. Hamiltonian of the system of free particles is nothing else butthe well known beast, found easily long ago (see Introductions).

Was all this worth the effort, if the outcome is something we could guesswith almost no labor at all? Does one get anything new? One new thing is thatnow we have not only the Hamiltonian, but all 10 Poincare generators — this isthe ”leitmotiv” of our development of the QFT. All generators are expressible interms of a+~p (t) and a~p (t) but, frankly, for the free particles this is also relativelystraightforward to guess.

The real yield of the whole procedure remains unclear until one proceeds tothe interacting fields or particles. The point is that, even if being motivated bythe free field case, the Fourier expansion of fields and quantization in terms ofthe Fourier coefficients turns out to be an efficient tool also for interacting fields.Even in this case the canonical quantization provides the 10 Poincare generatorsin terms of the fields ϕ (~x, t), i.e. in terms of a+~p (t) and a~p (t), which again have

(in a sense) a physical meaning of creation and annihilation operators.Unfortunately, all this does not go smoothly. In spite of our effort to pretend

the opposite, the canonical quantization of systems with infinitely many DOFis much more complex than of those with a finite number of DOF. The onlyreason why we were not faced with this fact hitherto, is that for the free fields thedifficulties are not inevitably manifest. More precisely, there is a representation(one among infinitely many) of canonical commutation relations which looksalmost like if the system has a finite number of DOF. Not surprisingly, this isthe Fock representation — the only one discussed so far. For interacting fields,however, the Fock space is not the trouble-free choice any more. In this caseneither the Fock space, nor any other explicitly known representation, succeedsin avoiding serious difficulties brought in by the infinite number of DOF.

In order to understand, at least to some extent, the problems with the quan-tization of interacting fields, the said difficulties are perhaps worth discussionalready for the free fields. So are the reasons why these difficulties are not soserious in the free field case.

Let us start with recollections of some important properties of systems de-fined by a finite number of canonical commutation relations [pi, qj ] = −iδij and[pi, pj ] = [qi, qj ] = 0, where i, j = 1, . . . , n. One can always introduce operatorsai = qici/2+ipi/ci and a

+i = qici/2−ipi/ci where ci is a constant (for harmonic

oscillators the most convenient choice is ci =√2miωi) satisfying

[ai, a

+j

]= δij

and [ai, aj ] =[a+i , a

+j

]= 0. The following holds:

• a state |0〉 annihilated by all ai operators does exist (∃ |0〉 ∀iai |0〉 = 0)

• the Fock representation of the canonical commutation relations does exist

• all irreducible representations are unitary equivalent to the Fock one

• Hamiltonians and other operators are usually well-defined


For infinite number of DOF, i.e. for the same set of commutation relations,but with i, j = 1, . . . ,∞ the situation is dramatically different:

• existence of |0〉 annihilated by all ai operators is not guaranteed

• the Fock representation, nevertheless, does exist

• there are infinitely many representations non-equivalent to the Fock one

• Hamiltonians and other operators are usually ill-defined in the Fock space

Let us discuss these four point in some detail.As to the existence of |0〉, for one oscillator the proof is notoriously known

from QM courses. It is based on well-known properties of the operatorN = a+a:1¯. N |n〉 = n |n〉 ⇒ Na |n〉 = (n− 1)a |n〉 (a+aa = [a+, a]a+ aa+a = −a+ aN)

2¯. 0 ≤ ‖a |n〉‖2 = 〈n| a+a |n〉 = n 〈n|n〉 implying n ≥ 0, which contradicts 1

¯unless the set of eigenvalues n contains 0. The corresponding eigenstate is |0〉.

For a finite number of independent oscillators the existence of the common|0〉 (∀i ai |0〉 = 0) is proven along the same lines. One considers the set ofcommuting operators Ni = a+i ai and their sum N =

∑i a

+i ai. The proof is

basically the same as for one oscillator.For infinite number of oscillators, however, neither of these two approaches

(nor anything else) really works. The step by step argument proves the state-ment for any finite subset of Ni, but fails to prove it for the whole infinite set.The proof based on the operator N refuses to work once the convergence of theinfinite series is discussed with a proper care.

Instead of studying subtleties of the breakdown of the proofs when passingfrom finite to infinite number of oscillators, we will demonstrate the existence ofthe so-called strange representations of a+i , ai (representations for which thereis no vacuum state |0〉) by an explicit construction (Haag 1955). Let a+i , aibe the creation and annihilation operators in the Fock space with the vacuumstate |0〉. Introduce their linear combinations bi = ai coshα + a+i sinhα andb+i = ai sinhα + a+i coshα. Commutation relations for the b-operators are thesame as for the a-operators (check it). Now let us assume the existence of astate vector |0α〉 satisfying ∀i bi |0α〉 = 0. For such a state one would have

0 = 〈i| bj |0α〉 = 〈i, j|0α〉 coshα+ 〈0|0α〉 δij sinhα

which implies 〈i, i|0α〉 = const (no i dependence). Now for i being an ele-ment of an infinite index set this constant must vanish, because otherwisethe norm of the |0α〉 state comes out infinite (〈0α|0α〉 ≥ ∑∞

i=1 |〈i, i|0α〉|2=∑∞

i=1 const2). And since const = −〈0|0α〉 tanhα the zero value of this constant

implies 〈0|0α〉 = 0. Moreover, vanishing 〈0|0α〉 implies 〈i, j|0α〉 = 0.It is straightforward to show that also 〈i|0α〉 = 0 (0 = 〈0| bi |0α〉 = 〈i|0α〉 coshα)

and finally 〈i, j, . . . |0α〉 = 0 by induction

〈i, j, . . .|︸︷︷︸n

bk |0α〉 = 〈i, j, . . .|︸︷︷︸n+1

0α〉 coshα+ 〈i, j, . . .|︸︷︷︸n−1

0α〉 sinhα

But 〈i, j, . . .| form a basis of the Fock space, so we can conclude that withinthe Fock space there is no vacuum state, i.e. a non-zero normalized vector |0α〉satisfying ∀i bi |0α〉 = 0.


Representations of the canonical commutation relations without the vacuumvector are called the strange representations. The above example22 shows notonly that such representations exist, but that one can obtain (some of) themfrom the Fock representation by very simple algebraic manipulations.

As to the Fock representation, it is always available. One just has to pos-tulate the existence of the vacuum |0〉 and then to build the basis of the Fockspace by repeated action of a+i on |0〉. Let us emphasize that even if we haveproven that existence of such a state does not follow from the commutationrelations in case of infinite many DOF, we are nevertheless free to postulate itsexistence and investigate the consequences. The very construction of the Fockspace guarantees that the canonical commutation relations are fulfilled.

Now to the (non-)equivalence of representations. Let us consider two repre-sentations of canonical commutation relations, i.e. two sets of operators ai, a

+i

and a′i, a′+i in Hilbert spaces H and H

′ correspondingly. The representations

are said to be equivalent if there is an unitary mapping HU→ H

′ satisfyinga′i = UaiU

−1 and a′+i = Ua+i U−1.

It is quite clear that the Fock representation cannot be equivalent to astrange one. Indeed, if the representations are equivalent and the non-primedone is the Fock representation, then defining |0′〉 = U |0〉 one has ∀i a′i |0′〉 =UaiU

−1U |0〉 = Uai |0〉 = 0, i.e. there is a vacuum vector in the primed repre-sentation, which cannot be therefore a strange one.

Perhaps less obvious is the fact that as to the canonical commutation re-lations, any irreducible representation (no invariant subspaces) with the vac-uum vector is equivalent to the Fock representation. The proof is constructive.The considered space H

′ contains a subspace H1 ⊂ H′ spanned by the basis

|0′〉, a′+i |0′〉, a′+i a′+j |0′〉, . . . One defines a linear mapping U from the sub-

space H1 on the Fock space H as follows: U |0′〉 = |0〉, Ua+′i |0′〉 = a+i |0〉,

Ua′+i a′+j |0′〉 = a+i a

+j |0〉, . . . The mapping U is clearly invertible and pre-

serves the scalar product, which implies unitarity (Wigner’s theorem). It isalso straightforward that operators are transformed as Ua′+i U

−1 = a+i andUa′iU

−1 = ai. The only missing piece is to show that H1 = H′ and this follows,

not surprisingly, form the irreducibility assumption23.

22Another instructive example (Haag 1955) is provided directly by the free field. Here thestandard annihilation operators are given by a~p = ϕ (~p, 0)

√

ω~p/2 + iπ (~p, 0) /√

2ω~p, where

ω~p = ~p2 + m2. But one can define another set a′~pin the same way, just with m replaced

by some m′ 6= m. Relations between the two sets are (check it) a′~p= c+a~p + c−a

+−~p

and

a′+~p

= c−a+~p+ c+a−~p, where 2c± =

√

ω′~p/ω~p ±

√

ω~p/ω′~p. The commutation relations become

[

a′~p, a′+

~k

]

= (2π)3 δ(~p−~k)(ω′~p−ω~p)/2ω

′~pand

[

a′~p, a′

~k

]

=[

a′+~p, a′+

~k

]

= 0. The rescaled operators

b~p = r~pa′~p and b+

~p= r~pa

+′~p

where r2~p = 2ω′~p/(ω

′~p − ω~p) constitutes a representation of the

canonical commutation relations.If there is a vacuum vector for a-operators, i.e. if ∃ |0〉 ∀~p a~p |0〉 = 0, then there is no |0′〉

satisfying ∀~p b~p |0′〉 = 0 (the proof is the same as in the example in the main text). In otherwords at least one of the representations under consideration is a strange one.

Yet another example is provided by an extremely simple prescription b~p = a~p +α(~p), where

α(~p) is a complex-valued function. For∫

|α(~p)|2 = ∞ this representation is a strange one (theproof is left to the reader as an exercise)

23As always with this types of proofs, if one is not quite explicit about definition domainsof operators, the ”proof” is a hint at best. For the real, but still not complicated, proof alongthe described lines see Berezin, Metod vtornicnovo kvantovania, p.24.


An immediate corollary of the above considerations is that all irreducible rep-resentations of the canonical commutation relations for finite number of DOFare equivalent (Stone–von Neumann). Indeed, having a finite number of DOFthey are obliged to have a vacuum state, and having a vacuum state they arenecessarily equivalent. As to the (non-)equivalence of various strange represen-tations, we are not going to discuss the subject here. Let us just remark that acomplete classification of strange representations of the canonical commutationrelations is not known yet.

Before going further, we should mention an important example of a reduciblerepresentation with a vacuum state. Let us consider perhaps the most natural(at least at the first sight) representation of a quantized system with infinitemany DOF — the one in which a state is represented by a function ψ(q1, q2, . . .)of infinitely many variables24. The function ψ0(q1, q2, . . .) =

∏∞i=1 ϕ0(qi), where

ϕ0 is a wavefunction of the ground state of LHO, is killed by all annihilationoperators, so it represents the vacuum state. Nevertheless, the Hilbert space H

of such functions cannot be unitary mapped on the Fock space HB, because ofdifferent dimensionalities (as already discussed, H is non-separable, while HB

is separable). The Fock space can be constructed from this vacuum, of course,and it happens to be a subspace of H (invariant with respect to creation andannihilation operators). This Fock space, however, does not cover the whole H.What is missing are states with actually infinite number of particles. The pointis that only states with finite, although arbitrarily large, number of particles areaccessible by repeated action of the creator operators on the vacuum vector25.

This brings us back to the question of how does it come that for infinitelymany oscillators we got a separable, rather then a non-separable, Hilbert space.It should be clear now that this is just a matter of choice — the Fock space isnot the only option, we could have chosen a non-separable Hilbert space (or aseparable strange representation) as well. The main advantage of the Fock spaceis that the relevant mathematics is known. On the other hand, the Fock spacealso seems to be physically acceptable, as far as all physically relevant states donot contain infinitely many particles. It would be therefore everything but wiseto ignore the fact that thanks to the Fock space we can proceed further withouta development of a new and difficult mathematics. So we will, like everybodydoes, try to stick to this fortunate encounter and milk it as much as possible.

Anyway, the choice of the Fock space as the playground for QFT does notclose the discussion. It may turn out that the Hamiltonian and other Poincaregenerators are ill-defined in the Fock space. For the free field, fortunately, thegenerators turn out to be well defined. But the reader should make no mistake,this is an exception rather than a rule.

24Of course, not any such function can represent a state. Recall that for one variable, onlyfunctions from L2 qualify for states. To proceed systematically, one has to define a scalarproduct, which can be done for specific functions of the form ψ(q1, q2, . . .) =

∏∞i=1 ψi(qi)

in a simple way as ψ.ψ′ =∏∞

i=1

∫

dqiψ∗i (qi)ψ

′i(qi). This definition can be extended to the

linear envelope of the ”quadratically integrable specific functions” and the space of normalizedfunctions is to be checked for completeness. But as to the mathematical rigor, this remarkrepresents the utmost edge of our exposition.

25This may come as a kind of surprise, since due to the infinite direct sum ⊕∞n=0H

n in thedefinition of the Fock space, one may expect (incorrectly) that it also contains something likeH

∞. This symbol, however, is just an abuse of notation — it does not correspond to anymany-particle subspace of the Fock space. An analogy may be of some help in clarifying thisissue: the set of natural numbers N does not contain an infinite number ∞, even if it containsevery n where n = 1, . . . ,∞.


The last point from the above lists of characteristic features of systems offinite and infinite DOF concern definitions of operators. This is a subtle pointalready for a finite number of DOF26. For systems with an infinite number ofDOF the situation is usually even worse. The reason is that many ”natural”operators are of the form On where O =

∑∞i=1 cia

+i + c∗i ai. The trouble now is

that for infinite sum one can have∑∞

i=1 |ci|2= ∞, the quantum field ϕ (x) is

a prominent example. Such an operator leads to a state with an infinite normacting on any standard basis vector in the Fock space (convince yourself). Butthis simply means that such operators are not defined within the Fock space.

Nevertheless, the operator O, as well as the quantum field ϕ (x), has a finitematrix elements between any two standard basis vectors. This enables us totreat them as objects having not a well defined meaning as they stand, but onlywithin scalar products — a philosophy similar to that of distributions like theδ-function. Operators which can be defined only in this sense are sometimescalled improper operators.

But the main problem is yet to come. Were all operators appearing in QFTproper or improper ones, the QFT would be perhaps much easier and betterunderstood then it actually is. Unfortunately, for many ”natural” operators eventhe scalar products are infinite. Such objects are neither proper, nor improperoperators, they are simply senseless expressions.

Nevertheless, the free field Hamiltonian H =∫d3p a+~p (t) a~p (t)ω~p/ (2π)

3

is a proper (even if unbounded) operator in the Fock space, since it maps ann-particle basis state to itself, multiplied by a finite number27. The other gen-erators map n-particle basis states to normalized n-particle states, so all theseoperators are well defined. That is why all difficulties discussed in this sectionremain hidden in the free field case. But they will reappear quickly, once theinteracting fields are considered.

26The point is that (unbounded) operators in quantum theory usually enter the game inthe so-called formal way, i.e. without a precise specification of domains. Precise domains, onthe other hand, are of vital importance for such attributes as selfadjointness, which in turnis a necessary condition for a Hamiltonian to define a dynamics, i.e. a unitary time evolution(Stone theorem). Formal Hamiltonians are usually Hermitian (symmetric) on a dense domainin the Hilbert space, and for some (but not for all) such symmetric operators the selfadjointextensions do exist. If so, the Hamiltonian is considered to be well-defined.

For our present purposes the important thing is that the Hamiltonian of the LHO is well-defined in this strict sense. One can even show, using sophisticated techniques of modernmathematical physics, that the Hamiltonian of an anharmonic oscillator H = p2/2m+q2+q4

is well defined (see e.g. Reed-Simon, volume 2, for five proofs of this statement) and thisholds for any finite number of oscillators. On the other hand, some formal Hamiltonians aredoomed to be ill-defined and lead to no dynamics whatsoever.

27The remaining question is if this unbounded Hamiltonian has a selfadjoint extension. Theanswer is affirmative, the proof, however, is to be looked for in books on modern mathematicalphysics rather than in introductory texts on QFT. One may raise an objection that we havedemonstrated selfadjointness of the free field Hamiltonian indirectly by finding the explicitunitary time evolution of states (which follows from the time evolution of the creation andannihilation operators). This, however, was found by formal manipulations, without botheringabout if the manipulated objects are well defined. Needless to say, such an approach can leadto contradictions. Anyway, for the free fields the formal manipulations are fully supported bymore careful analysis.

All this applies, of course, only for one particular ordering of creation and annihilationoperators — not surprisingly the one we have adopted. Other orderings are, strictly speaking,the above mentioned senseless expressions with infinite matrix elements between basis vectors.

Chapter 3

Interacting Quantum Fields

3.1 Naive approach

In the last section of the previous chapter we have discussed several unpleasantfeatures which may appear in a theory of interacting fields (strange representa-tions, ill-defined operators, no dynamics in a sense of unitary time evolution).In the first two sections of the present chapter we are going to ignore all thiscompletely. On top of that, in the first section we will oversimplify the matterseven more than is the common habit.

The reason for this oversimplification is purely didactic. As we will see, onecan get pretty far using a bit simple-minded approach and almost everythingdeveloped in this framework will survive, with necessary modifications, latercritical reexamination. The said modifications are, on the other hand, quitesophisticated and both technically and conceptually demanding. We prefer,therefore, to postpone their discussion until the basic machinery of dealing withinteracting fields is developed in the simplified naive version1.

As the matter of fact, the naive approach is the most natural one. It isbased on the assumption that the free field lagrangian defines what particlesare2, and the interaction lagrangian defines how do these particles interact witheach other. Life, however, turns out to be surprisingly more complex.

So it happens that by switching on the interaction, one in fact redefines whatparticles are. This rather non-trivial and surprising fact has to be taken intoaccount — otherwise one is, sooner or later, faced with serious inconsistenciesin the theory. In the standard approach one indeed develops the theory ofinteracting quantum fields having in mind from the very beginning that ”particlecontent” of the free and interacting theories may differ significantly.

In our naive approach we will ignore all this and move on happily until wewill understand almost completely where the Feynman rules come from. The fewmissing ingredients will be obtained afterwards within the standard approach.

1It should be stressed that even after all known modifications (see section 3.2) the resultingtheory of interacting quantum fields is still not satisfactory in many respects (see section ??).The difference between the oversimplified and the standard approach is not that they areincorrect and correct respectively, but rather that they are incorrect to different degrees.

2According to the naive approacg, one-particle states are those obtained by acting of thecreation operator a+

~pon the vacuum state |0〉, two-particle states are those obtained by acting

of two creation operators on the vacuum state, etc.

73

74 CHAPTER 3. INTERACTING QUANTUM FIELDS

canonical quantization of interacting fields

For interacting fields the quantization proceeds basically along the same linesas for the free fields. Fields and the conjugated momenta are Fourier expanded

ϕ (~x, t) =

∫d3p

(2π)3

1√2ω~p

(a~p (t) e

i~p.~x + a+~p (t) e−i~p.~x)

π (~x, t) = −i∫

d3p

(2π)3

√ω~p

2

(a~p (t) e

i~p.~x − a+~p (t) e−i~p.~x)

and the canonical Poisson brackets for ϕ(~x, t) and π(~x, t) imply the standardPoisson brackets for a~p (t) and a

+~p′ (t) which lead to the commutation relations

[a~p (t) , a+~p′ (t)] = (2π)

3δ(~p− ~p′)

[a~p (t) , a~p′ (t)] = [a+~p (t) , a+~p′ (t)] = 0.

holding for arbitrary time t (which, however, must be the same for both oper-ators in the commutator — that is why they are known as ”equal-time com-mutation relations”). At any fixed time, these commutation relations can berepresented by creation and annihilation operators in the Fock space.

So far, it looks like if there is no serious difference between the free fields andthe interacting ones. In the former case the appearence of the Fock space wasthe last important step which enabled us to complete the canonical quantizationprogram. For the interacting fields, however, one does not have the Fock space,but rather Fock spaces. The point is that for different times t the a+~p (t) and

a~p′ (t) operators are, in principle, represented in different Fock subspaces of the”large” non-separable space. For the free fields all these Fock spaces coincide,they are in fact just one Fock space — we were able to demonstrate this due tothe explicit knowledge of the time evolution of a+~p and a~p′ . But for interacting

fields such a knowledge is not at our disposal anymore3.One of the main differences between the free and interacting fields is that

the time evolution becomes highly nontrivial for the latter. In the Heisenbergpicture, the equations for a~p (t) and a

+~p′ (t) do not lead to a simple harmonic time-

dependence, nor do the equations for the basis states in the Schrodinger picture.These difficulties are evaded (to a certain degree) by a clever approximativescheme, namely by the perturbation theory in the so-called interaction picture.In this section, we will develop the scheme and learn how to use it in thesimplified version. The scheme is valid also in the standard approach, but itsusage is a bit different (as will be discussed thoroughly in the next section).

3Not only one cannot prove for the interacting fields that the Fock spaces for a+~p(t) and

a~p′ (t) at different times coincide, one can even prove the opposite, namely that these Fockspaces are, as a rule, different subspaces of the ”large” non-separable space. This means that,strictly speaking, one cannot avoid the mathematics of the non-separable linear spaces whendealing with interacting fields. We will, nevertheless, try to ignore all the difficulties relatedto non-separability as long as possible. This applies to the present naive approach as well asto the following standard approach. We will return to these difficulties only in the subsequentsection devoted again to ”contemplations and subtleties”.

3.1. NAIVE APPROACH 75

3.1.1 Interaction picture

Our main aim will be the development of some (approximate) techniques of solv-ing the time evolution of interacting fields. It is common to use the interactionpicture of the time evolution in QFT. Operators and states in the interactionpicture are defined as4

AI = eiH0te−iHtAHeiHte−iH0t

|ψI〉 = eiH0te−iHt |ψH〉

where the operators H and H0 are understood in the Schrodinger picture.

The time evolution of operators in the interaction picture is quite simple, itis equal to the time evolution of the free fields. Indeed, both time evolutions(the one of the free fields and the one of the interacting fields in the interactionpicture) are controlled by the same Hamiltonian H0.

Let us emphasize the similarities and the differences between the interactingfields in the Heisenberg and the interaction pictures. In both pictures one hasidentically looking expansions

ϕH (~x, t) =

∫d3p

(2π)31√2ω~p

(a~p,H (t) + a+−~p,H (t)

)ei~p.~x

ϕI (~x, t) =

∫d3p

(2π)3

1√2ω~p

(a~p,I (t) + a+−~p,I (t)

)ei~p.~x

However, the explicit time dependence of the creation and annihilation operatorsin the Heisenberg picture is unknown, while in the interaction picture it is knownexplicitly as a+~p,I (t) = a+~p e

iω~pt and a~p,I (t) = a~pe−iω~pt (see section??). Using

the free field results, one can therefore write immediately

ϕI(x) =

∫d3p

(2π)3

1√2ω~p

(a~pe

−ipx + a+~p eipx)

where a+~p and a~p are the creation and annihilation operators at t = 0 (in any

picture, they all coincide at this moment). The explicit knowledge and thespace-time structure (scalar products of 4-vectors) of the ϕI -fields are going toplay an extremely important role later on.

The time evolution of states in the interaction picture is given by

i∂t |ψI〉 = HI (t) |ψI〉 HI (t) = eiH0t (H −H0) e−iH0t

where H and H0 are understood in the Schrodinger picture. The operatorHI (t)is the interaction Hamiltonian in the interaction picture.

4Relations between the Schrodinger, Heisenberg and interaction pictures:

AH = eiHtASe−iHt |ψH 〉 = eiHt |ψS〉

AI = eiH0tASe−iH0t |ψI 〉 = eiH0t |ψS〉

The operators H and H0 are understood in the Schrodinger picture. Their subscripts areomitted for mainly esthetic reasons (to avoid too much make-up in the formulae). Anyway,directly from the definitions one has HH = HS and H0,I = H0,S , therefore the discussedsubscripts would be usually redundant.


Needless to say, solving the evolution equation for states in the interactionpicture is the difficult point. Nevertheless, we will be able to give the solutionas a perturbative series in terms of ϕI (x). To achieve this, however, we willneed to express all quantities, starting with HI (t), in terms of ϕI (x).

The canonical quantization provides the Hamiltonian as a function of fieldsin the Heisenberg picture. What we will need is HI (t) expressed in termsof ϕI (x). Fortunately, this is straightforward: one just replaces ϕH and πHoperators in the Heisenberg picture by these very operators in the interactionpicture, i.e. by ϕI and πI . Proof: one takes t = 0 in the Heisenberg picture, andin thus obtained Schrodinger picture one simply inserts e−iH0teiH0t between anyfields or conjugate momenta5.

Example: ϕ4-theory L [ϕ] = 12∂µϕ∂

µϕ− 12m

2ϕ2 − 14!gϕ

4

H =

∫d3x(

1

2π2H +

1

2|∇ϕH |2 + 1

2m2ϕ2

H +1

4!gϕ4

H)

and taking t = 0 one gets Hint =∫d3x 1

4!gϕ4S, leading to

HI =

∫d3x

1

4!gϕ4

I

In what follows, the key role is going to be played by the operator U (t, t′),which describes the time evolution of states in the interaction picture

|ψI (t)〉 = U (t, t′) |ψI (t′)〉

Directly from the definition one has

U (t, t′′) = U (t, t′)U (t′, t′′) U−1 (t, t′) = U (t′, t)

where the second relation follows from the first one and the obvious identityU (t, t) = 1. Differentiating with respect to t one obtains i∂tU (t, t′) |ψI (t

′)〉 =HI (t)U (t, t′) |ψI (t

′)〉 for every |ψI (t′)〉 and therefore

i∂tU (t, t′) = HI (t)U (t, t′)

with the initial condition U (t, t) = 1.For t′ = 0 the solution of this equation is readily available6

U (t, 0) = eiH0te−iHt

(H0 and H in the Schrodinger picture). This particular solution shows that (inaddition to providing the time evolution in the interaction picture) the U (t, 0)operator enters the relation between the field operators in the Heisenberg andinteraction pictures7

ϕH (x) = U−1(x0, 0

)ϕI (x)U

(x0, 0

)

5Remark: the simple replacement ϕH → ϕI , πH → πI works even for gradients of fields,one simply has to realize that e−iH0t∇ϕSe

iH0t = ∇(

e−iH0tϕSeiH0t

)

= ∇ϕI , which holdsbecause H0 does not depend on the space coordinates.

6Indeed ∂teiH0te−iHt = eiH0t(iH0 − iH)e−iHt = −ieiH0tHinte−iH0teiH0te−iHt =

= −iHI (t)eiH0te−iHt. Note that the very last equality requires t′ = 0 and therefore one

cannot generalize the relation to any t′. In general U(t, t′) 6= eiH0(t−t′)e−iH(t−t′).7AH = eiHte−iH0tAIe

iH0te−iHt = U−1(t, 0)AIU(t, 0)


perturbation theory

The practically useful, even if only approximate, solution for U (t, t′) is obtainedby rewriting the differential equation to the integral one

U (t, t′) = 1− i

∫ t

t′dt′′HI (t

′′)U (t′′, t′)

which is then solved iteratively

0th iteration U (t, t′) = 1

1st iteration U (t, t′) = 1− i∫ t

t′ dt1HI (t1)

2nd iteration U (t, t′) = 1− i∫ t

t′dt1HI (t1)− i2

∫ t

t′dt1HI (t1)

∫ t1t′dt2HI (t2)

etc.

Using a little artificial trick, the whole scheme can be written in a morecompact form. The trick is to simplify the integration region in the multipleintegrals In =

∫ t

t′ dt1∫ t1t′ dt2 . . .

∫ tn−1

t′ dtn HI (t1)HI (t2) . . . HI (tn). Let us con-sider the n-dimensional hypercube t′ ≤ ti ≤ t and for every permutation ofthe variables ti take a region for which the first variable varies from t′ up tot, the second variable varies from t′ up to the first one, etc. There are n! suchregions (n! permutations) and any point of the hypercube lies either inside ex-actly one of these regions, or at a common border of several regions. Indeed,for any point the ordering of the coordinates (t1, . . . , tn), from the highest tothe lowest one, reveals unambiguously the region (or a border) within which itlies. The integral of the product of Hamiltonians over the whole hypercube isequal to the sum of integrals over the considered regions. In every region oneintegrates the same product of Hamiltonians, but with different ordering of thetimes. The trick now is to force the time ordering to be the same in all regions.This is achieved in a rather artificial way, namely by introducing the so-calledtime-ordered product T A (t1)B (t2) . . ., which is the product with the termsorganized from the left to the right with respect to decreasing time (the lat-est on the very left etc.). Integrals of this T -product over different regions areequal to each other, so we can replace the original integrals by the integrals overhypercubes In = 1

n!

∫ t

t′

∫ t

t′ dt1 . . . dtn T HI (t1) . . .HI (tn) and consequently

U (t, t′) =∞∑

n=0

(−i)nn!

∫ t

t′. . .

∫ t

t′dt1 . . . dtn T HI (t1) . . . HI (tn)

which is usually written in a compact form as

U (t, t′) = Te−i∫ tt′

dt′′ HI(t′′)

where the definition of the RHS is the RHS of the previous equation.Note that if the interaction Hamiltonian is proportional to some constant

(e.g. a coupling constant) then this iterative solution represents the power ex-pansion (perturbation series8) in this constant.

8The usual time-dependent perturbation theory is obtained by inserting the expansion|ψI (t)〉 = an (t) |ϕn〉, where |ϕn〉 are eigenvectors of the free Hamiltonian H0, into the originalequation i∂t |ψI 〉 = HI (t) |ψI 〉. From here one finds a differential equation for an (t), rewritesit as an integral equation and solves it iteratively.


3.1.2 Transition amplitudes

The dynamical content of a quantum theory is encoded in transition amplitudes,i.e. the probability amplitudes for the system to evolve from an initial state |ψi〉at ti to a final state |ψf 〉 at tf . These probability amplitudes are coefficientsof the expansion of the final state (evolved from the given initial state) in aparticular basis.

In the Schrodinger picture the initial and the final states are |ψi,S〉 andUS (tf , ti) |ψi,S〉 respectively, where US (tf , ti) = exp−iH(tf − ti) is the timeevolution operator in the Schrodinger picture. The ”particular basis” is specifiedby the set of vectors |ψf,S〉

transition amplitude = 〈ψf,S |US (tf , ti) |ψi,S〉

It should be perhaps stressed that, in spite of what the notation might suggest,|ψf,S〉 does not define the final state (which is rather defined by |ψi,S〉 and thetime evolution). Actually |ψf,S〉 just defines what component of the final statewe are interested in.

In the Heisenberg picture, the time evolution of states is absent. Neverthe-less, the transition amplitude can be easily written in this picture as

transition amplitude = 〈ψf,H |ψi,H 〉

Indeed, 〈ψf,H |ψi,H〉 = 〈ψf,S | e−iHtf eiHti |ψi,S〉 = 〈ψf,S |US (tf , ti) |ψi,S〉. Po-tentially confusing, especially for a novice, is the fact that formally |ψi,H〉 and|ψf,H〉 coincide with |ψi,S〉 and |ψf,S〉 respectively. The tricky part is that inthe commonly used notation the Heisenberg picture bra and ket vectors labelthe states in different ways. Heisenberg picture ket (bra) vectors, representingthe so-called the in-(out -)states, coincide with the Schrodinger picture state atti (tf ), rather then usual t=0 or t0. Note that these times refers only to theSchrodinger picture states. Indeed, in spite of what the notation may suggest,the Heisenberg picture in- and out -states do not change in time. The in- andout - prefixes have nothing to do with the evolution of states (there is no suchthing in the Heisenberg picture), they are simply labelling conventions (whichhave everything to do with the time evolution of the corresponding states in theSchrodinger picture).Why to bother with the sophisticated notation in the Heisenberg picture, ifit anyway refers to the Schrodinger picture? The reason is, of course, that inthe relativistic QFT it is preferable to use a covariant formalism, in which fieldoperators depend on time and space-position on the same footing. It is sim-ply preferable to deal with operators ϕ (x) rather than ϕ (~x), which makes theHeisenberg picture more appropriate for relativistic QFT.

Finally, the interaction picture shares the intuitive clarity with the Schrodingerpicture and the relativistic covariance with the Heisenberg picture, even if bothonly to certain extent. In the interaction picture

transition amplitude = 〈ψf,I |U (tf , ti) |ψi,I〉

This follows from 〈ψf,S |US (tf , ti) |ψi,S〉 = 〈ψf,I | eiH0tf e−iH(tf−ti)e−iH0ti |ψi,I〉 =〈ψf,I |U(tf , 0)U

−1(ti, 0) |ψi,I〉 and from U(tf , 0)U−1(ti, 0) = U(tf , 0)U(0, ti) =

U(tf , ti).


green functions

For multiparticle systems, a particularly useful set of initial and final statesis given by the states of particles simultaneously created at various positions,i.e. by the localized states. But as we have seen already, in the relativis-tic QFT the more appropriate states are the quasilocalized ones created bythe field operators. The corresponding amplitude is apparently something like〈0|ϕH(~x1, tf )ϕH(~x2, tf ) . . . ϕH(~xn, ti) |0〉. The vacuum state in this amplitude,however, is not exactly what it should be.

The time-independent state |0〉 in the Heisenberg picture corresponds to aparticular time-evolving state |ψ0 (t)〉 in the Schrodinger picture, namely to theone for which |ψ0 (0)〉 = |0〉. This state contains no particles at the time t = 0.But the fields ϕH(~x, ti) should rather act on a different state, namely the onewhich contains no particles at the time ti. In the Schrodinger picture such astate is described by a particular time-evolving state |ψi (t)〉 for which |ψi (ti)〉 =|0〉. How does this state look like in the Heisenberg picture? It coincides, bydefinition, with |ψi (0)〉, which is in turn equal to expiHti |ψi (ti)〉. It is acommon habit to denote this state in the Heisenberg picture as |0〉in and withthis notationn the above cosiderations leads to the following relation

|0〉in = eiHti |0〉In a complete analogy one has to replace the bra-vector 〈0| by

〈0|out = 〈0| e−iHtf

The quantity of interest is therefore given by the product of fields sandwichednot between 〈0| and |0〉, but rather

〈0|out ϕH(~x1, tf )ϕH(~x2, tf ) . . . ϕH(~xn, ti) |0〉inBecause of relativity of simultaneity, however, this quantity looks differently

for other observers, namely the time coordinates x0i are not obliged to coincide.These time coordinates, on the other hand, are not completely arbitrary. To anyobserver the times corresponding to the simultaneous final state in one particularframe, must be all greater than the times corresponding to the simultaneousinitial state in this frame. The more appropriate quantity would be a slightlymore general one, namely the time-ordered T -product of fields9 sandwichedbetween 〈0|out and |0〉in

〈0|out T ϕH(x1)ϕH(x2) . . . ϕH(xn) |0〉inThe dependence on ti and tf is still present in |0〉in and 〈0|out. It is a commonhabit to get rid of this dependence by taking ti = −T and tf = T with T → ∞.The exact reason for this rather arbitrary step remains unclear until the moreserious treatment of the whole machinery becomes available in the next section).

The above matrix element is almost, but not quite, the Green function —one of the most prominent quantities in QFT. The missing ingredient is not tobe discussed within this naive approach, where we shall deal with functions

g (x1, . . . , xn) = limT→∞

〈0| e−iHTT ϕH(x1) . . . ϕH(xn) e−iHT |0〉9For fields commuting at space-like intervals ([ϕH (x), ϕH(y)] = 0 for (x−y)2 < 0) the time

ordering is immaterial for times which coincide in a particular reference frame. For time-likeintervals, on the other hand, the T -product gives the same ordering in all reference frames.


We shall call these functions the green functions (this notion is not commonin literature, but this applies for the whole naive approach presented here).The genuine Green functions G are to be discussed later (we will distinguishbetween analogous quantities in the naive and the standard approaches by usinglowercase letters in the former and uppercase letter in the latter case).

Actual calculations of the green functions are performed, not surprisingly,in the interaction picture. The transition from the Heisenberg picture to theinteraction one is provided by the relations from the page 76. It is useful tostart with

|0〉in = e−iHT |0〉 = e−iHT eiH0T |0〉 = U−1 (−T, 0) |0〉 = U (0,−T ) |0〉〈0|out = 〈0| e−iHT = 〈0| e−iH0T e−iHT = 〈0|U (T, 0)

which holds10 forH0 |0〉 = 0. The next step is to use ϕH (x) = U−1(x0, 0)ϕI (x)U(x0, 0)for every field in the green function, then to write U(T, 0)ϕH (x1)ϕH (x2) . . . as

U(T, 0)U−1(x01, 0)︸︷︷︸U(T,0)U(0,x0

1)

ϕI (x1)U(x01, 0)U−1(x02, 0)︸︷︷︸

U(x01,0)U(0,x0

2)

ϕI (x2)U(x02, 0) . . .

and finally one utilizes U(T, 0)U(0, x01) = U(T, x01), etc. This means that forx01 ≥ . . . ≥ x0n the green function g (x1, . . . , xn) is equal to

limT→∞

〈0|U(T, x01)ϕI (x1)U(x01, x02)ϕI (x2) . . . ϕI (xn)U(x0n,−T ) |0〉

and analogously for other orderings of times.Let us now define a slightly generalized time ordered product as

T U(t, t′)A(t1)B(t2) . . . C(tn) = U(t, t1)A(t1)U(t1, t2)B(t2) . . . C(tn)U(tn, t′)

for t ≥ t1 ≥ t2 ≥ . . . ≥ tn ≥ t′ and for other time orderings the order ofoperators is changed appropriately. With this definition we can finally write

g (x1, . . . , xn) = limT→∞

〈0|T U(T,−T )ϕI (x1) . . . ϕI (xn) |0〉

This form of the green function is what we were after. It has the formof the vacuum expectation value of the products of the field operators in theinteraction picture and it is relatively straightforward to develop the techniquefor calculation of these objects. This technique will lead us directly to theFeynman rules. The rules were introduced in the introductory chapter, butthey were not derived there. Now we are going to really derive them.

Remark: The Feynman rules discussed in the Introductions/Conclusions con-cerned the scattering amplitude Mfi, while here we are dealing with the greenfunctions. This, however, represents no contradiction. The green functions,as well as the genuine Green functions, are auxiliary quantities which are, aswe will see briefly, closely related to the scattering amplitudes. It is thereforequite reasonable first to formulate the Feynman diagrams for the green or Greenfunctions and only afterwards for the scattering amplitudes.

10If H0 |0〉 = E0 |0〉 with E0 6= 0, then the relations hold up-to the irrelevant phase factorexpiE0T.


Wick’s theorem

The perturbative expansion of U(T,−T ) is a series in HI(t), which in turn is afunctional of ϕI (x), so our final expression for the green function gives them asa series of VEVs (vacuum expectation values) of products of ϕI -fields.

As we already know from the Introductions, the most convenient way of cal-culating the VEVs of products of creation and annihilation operators is to rushthe creation and annihilation operators to the left and to the right respectively.We are now going to accommodate this technique to the VEVs of time-orderedproducts of ϕI -fields.

The keyword is the normal product of fields. First one writes ϕI = ϕ+I +ϕ−

I ,where ϕ+

I and ϕ−I are parts of the standard expansion of ϕI(x) containing only

the annihilation and the creation operators respectively11. The normal productof fields, denoted as NϕI (x)ϕI (y) . . . or :ϕI (x)ϕI (y) . . . :, is defined as theproduct in which all ϕ−

I -fields are reshuffled by hand to the left of all ϕ+I -

fields, e.g. NϕI (x)ϕI (y) = ϕ−I (x)ϕ−

I (y) + ϕ−I (x)ϕ+

I (y) + ϕ−I (y)ϕ+

I (x) +ϕ+I (x)ϕ+

I (y). Everybody likes normal products, because their VEVs vanish.

The trick, i.e. the celebrated Wick’s theorem, concerns the relation betweenthe time-ordered and normal products. For two fields one has ϕI (x)ϕI (y) =N ϕI (x)ϕI (y) +

[ϕ+I (x) , ϕ−

I (y)]. It is straightforward to show (do it) that[

ϕ+I (x) , ϕ−

I (y)]= D (x− y) where

D (x− y) =

∫d3p

(2π)3

1

2ω~pe−ip(x−y)

The relation between T ϕIϕI and N ϕIϕI is now straightforward

T ϕI (x)ϕI (y) = N ϕI (x)ϕI (y)+ dF (x− y)

where

dF (ξ) = ϑ(ξ0)D (ξ) + ϑ

(−ξ0

)D (−ξ)

The function dF is almost equal to the so-called Feynman propagator DF (seep.87). Everybody likes dF (x− y), DF (x− y) and similar functions, becausethey are not operators and can be withdrawn out of VEVs.

For three fields one obtains in a similar way12

ϕI (x)ϕI (y)ϕI (z) = N ϕI (x)ϕI (y)ϕI (z)+D (x− y)ϕI (z) +D (x− z)ϕI (y) +D (y − z)ϕI (x)

T ϕI (x)ϕI (y)ϕI (z) = N ϕI (x)ϕI (y)ϕI (z)+ dF (x− y)ϕI (z) + dF (x− z)ϕI (y) + dF (y − z)ϕI (x)

11Strange, but indeed ϕ+I (x) =

∫ d3p

(2π)31√2ω~p

a~pe−ipx and ϕ−

I (x) =∫ d3p

(2π)31√2ω~p

a+~peipx.

The superscript ± is not in honour of the creation and annihilation operators, but rather inhonour of the sign of energy E = ±ω~p.

12One starts with ϕI (x)ϕI (y)ϕI (z) = ϕI (x)ϕI (y)ϕ+I (z)+ϕI (x)ϕI (y)ϕ

−I (z), followed

by ϕI (x)ϕI (y)ϕ−I (z) = ϕI (x) [ϕI (y) , ϕ

−I (z)]+[ϕI (x) , ϕ

−I (z)]ϕI (y)+ϕ

−I (z)ϕI (x)ϕI (y),

so that ϕI (x)ϕI (y)ϕI (z) = ϕI (x)ϕI (y)ϕ+I (z) + ϕI (x)D (y − z) + D (x− z)ϕI (y) +

ϕ−I (z)ϕI (x)ϕI . At this point one utilizes the previous result for two fields, and finally one has

to realize that NϕI (x)ϕI (y)ϕ+I (z) + ϕ−

I (z)NϕI (x)ϕI (y) = NϕI (x)ϕI (y)ϕI (z).


Now we can formulate and prove the Wick’s theorem for n fields

T ϕI (x1) . . . ϕI (xn) = N ϕI (x1) . . . ϕI (xn)+ dF (x1 − x2)N ϕI (x3) . . . ϕI (xn)+ . . .

+ dF (x1 − x2) dF (x3 − x4)N ϕI (x5) . . . ϕI (xn)+ . . .

+ . . .

In each line the ellipsis stands for terms equivalent to the first term, but withthe variables xi permutated in all possible ways. The number of the Feynmanpropagators is increased by one when passing to the next line. The proof isdone by induction, the method is the same as a we have used for three fields.

The most important thing is that except for the very last line, the RHS ofthe Wick’s theorem has vanishing VEV (because of normal products). For nodd even the last line has vanishing VEV, for n even the VEV of the last lineis an explicitly known number. This gives us the quintessence of the Wick’stheorem: for n odd 〈0|T ϕI (x1) . . . ϕI (xn) |0〉 = 0, while for n even

〈0|T ϕI (x1) . . . ϕI (xn) |0〉 = dF (x1 − x2) . . . dF (xn−1 − xn) + permutations

Remark: It is a common habit to economize a notation in the following way.Instead of writing down the products dF (x1 − x2) . . . dF (xn−1 − xn) one writesthe product of fields and connects by a clip the fields giving the particular dF .In this notation

dF (x1 − x2) dF (x3 − x4) = ϕI (x1)ϕI (x2)ϕI (x3)ϕI (x4)

dF (x1 − x3) dF (x2 − x4) = ϕI (x1) ϕI (x3)ϕI (x2) ϕI (x4)

dF (x1 − x4) dF (x2 − x3) = ϕI (x1) ϕI (x4)ϕI (x2)ϕI (x3)

At this point we are practically done. We have expressed the green functionsas a particular series of VEVs of time-ordered products of ϕI -operators and wehave learned how to calculate any such VEV by means of the Wick’s theorem.All one has to do now is to expand U(T,−T ) in the green function up-to a givenorder and then to calculate the corresponding VEVs.

Example: g (x1, x2, x3, x4) in the ϕ4-theorynotation: ϕi := ϕI (xi), ϕx := ϕI (x), dij := dF (xi − xj), dix := dF (xi − x)

g = limT→∞

〈0|T U(T,−T )ϕ1ϕ2ϕ3ϕ4 |0〉 = g(0) + g(1) + . . .

g(0) = 〈0|T ϕ1ϕ2ϕ3ϕ4 |0〉 = d12d34 + d13d24 + d14d23

g(1) = − ig4!

〈0|T∫

d4xϕ4xϕ1ϕ2ϕ3ϕ4

|0〉

= − ig4!

∫d4x 24× d1xd2xd3xd4x + 12× d12dxxd3xd4x + . . .

where we have used U(∞,−∞) = 1− i∫∞−∞ dtHI (t)+ . . . = 1− ig

4!

∫d4xϕ4

x+ . . .


Feynman rules

The previous example was perhaps convincing enough in two respects: first thatin principle the calculations are quite easy (apart from integrations, which mayturn out to be difficult), and second that practically they become almost unman-ageable rather soon (in spite of our effort to simplify the notation). Conclusion:further notational simplifications and tricks are called for urgently.

The most wide-spread trick uses a graphical representation of various termsin green function expansion. Each variable x is represented by a point labeled byx. So in the previous example we would have 4 points labeled by xi i = 1, 2, 3, 4and furthermore a new point x, x′, . . . for each power of HI . Note that theHamiltonian density HI contains several fields, but all at the same point —this is the characteristic feature of local theories. For each dF (y− z) the pointslabeled by y and z are connected by a line. If there are several different fields,there are several different Feynman propagators, and one has to use severaldifferent types of lines.

In this way one assigns a diagram to every term supplied by the team-work of the perturbative expansion of U(T,−T ) and the Wick’s theorem. Suchdiagrams are nothing else but the famous Feynman diagrams. Their structure isevident from the construction. Every diagram has external points, given by theconsidered g-function, and internal points (vertices), given by HI . The numberof internal points is given by the order in the perturbative expansion. Thestructure of the vertices (the number and types of lines entering the vertex) isgiven by the structure of HI , each product of fields represent a vertex, each fieldin the product represent a line entering this vertex.

A diagram, by construction, represents a number. This number is a productof factors corresponding to lines and vertices. The factor corresponding to a line(internal or external) connecting x, y is dF (x− y). The factor corresponding toa vertex in the above example is − ig

4!

∫d4x , while in the full generality it is

−i× what remains of HI after the fields are ”stripped off”×∫d4x

Further simplification concerns combinatorics. Our procedure, as describedso-far, gives a separate diagram for each of the 24 terms − ig

4!

∫d4x d1xd2xd3xd4x

in g(1) in the above example. As should be clear from the example, this factoris purely combinatorial and since it is typical rather than exceptional, it isreasonable to include this 24 into the vertex factor (and to draw one diagraminstead of 24 identical diagrams). This is achieved by doing the appropriatecombinatorics already in the process of ”stripping the fields off”, and it amountsto nothing more than to the multiplication by n! for any field appearing in HI inthe n-th power. An economic way of formalizing this ”stripping off” procedure,with the appropriate combinatorics factors, is to use the derivatives of HI withrespect to the fields.

Having included the typical combinatorial factor into the vertex, we have topay a special attention to those (exceptional) diagrams which do not get thisfactor. The 12 terms − ig

4!

∫d4x d12dxxd3xd4x in g(1) in the example can serve

as an illustration. Twelve identical diagrams are represented by one diagramaccording to our new viewpoint, but this diagram is multiplied by 24, hidden inthe vertex factor, rather then by 12. To correct this, we have to divide by 2 —one example of the infamous explicit combinatorial factors of Feynman rules.


The rules can be summarized briefly as

the Feynman rules for the green functions in the x-representation

line (internal or external) dF (x− y)

vertex (n legs) −i ∂nHI

∂ϕnI

∣∣∣ϕI=0

∫d4x

These are not the Feynman rules from the Introductions yet, but we are on theright track.

The first step towards the rules from the Introductions concerns the relationbetween HI and Lint. For interaction Lagrangians with no derivative terms(like the ϕ4-theory), the definition H =

∫d3x (ϕπ − L) implies immediately

Hint = −Lint. And since Hint in the Heisenberg picture is the same functionof ϕH -fields, as HI is of ϕI -fields (as we have convinced ourselves), one canreplace −∂nHI/∂ϕ

nI by ∂nLint/∂ϕ

n. Finally, for vertices one can replace Lint

by L in the last expression, because the difference is the quadratic part of theLagrangian, and vertices under consideration contain at least three legs. Forinteractions with derivative terms (say Lint ∼ ϕ∂µϕ∂

µϕ) the reasoning is morecomplicated, but the result is the same. We will come back to this issue shortly(see p.85). For now let us proceed, as directly as possible, with the easier case.

Another step is the use of the Fourier expansion13

dF (x− y) =

∫d4p

(2π)4 dF (p) e−ip(x−y)

This enables us to perform the vertex x-integrations explicitly, using the identity∫d4x e−ix(p+p′+...) = (2π)

4δ4 (p+ p′ + . . .), what results in

the Feynman rules for the green functions in the p-representation

internal line∫

d4p(2π)4

dF (p)

external line∫

d4p(2π)4

dF (p) e±ipxi

vertex i ∂nL∂ϕn

∣∣∣ϕ=0

(2π)4δ4 (p+ p′ + . . .)

Let us remark that some authors prefer to make this table simpler-looking,by omitting the factors of (2π)

4as well as the momentum integrations, and

shifting them to the additional rule requiring an extra (2π)4for each vertex

and (2π)−4 ∫ d4p for each line (internal or external). We have adopted such aconvention in the Introductions.

13One may be tempted to use i(p2 − m2)−1 or i(p2 −m2 + iε)−1 as dF (p) (see p. 87),but neither would be correct. Both choices lead to results differing from dF (x− y) by somefunctions of ε, which tend to disappear when ε → 0. Nevertheless the ε-differences are theimportant ones, they determine even the seemingly ε-independent part of the result.

Anyway, apart from the iε subtleties, dF (p) comes out equal to what we have calculatedin the Introductions (from quite different definition of propagator). This may seem like acoincidence, and one may suspect if one gets equal results even beyond the real scalar fieldexample. The answer is affirmative, but we are not going to prove it here in the full generality.The reason is that the general statement is more transparent in another formulation of QFT,namely in the path integral formalism. So we prefer to discuss this issue within this formalism.


derivative couplings

Now to the interaction Lagrangians with derivative terms. The prescriptionfrom the Introductions was quite simple: any ∂µ in the interaction Lagrangianfurnishes the −ipµ factor for the corresponding vertex in the p-representationFeynman rules (pµ being the momentum assigned to the corresponding leg,oriented toward the vertex). To understand the origin of this factor, it is (seem-ingly) sufficient to differentiate the Wick’s theorem, e.g. for two fields

TϕI (x) ∂

′µϕI (x

′)= ∂′µ (N ϕI (x)ϕI (x

′)+ dF (x− x′))

When calculating the green function, the derivative can be withdrawn fromVEV, and once dF (x−x′) is Fourier expanded, it produces the desired factor (thereader is encouraged to make him/her-self clear about momentum orientations).

There is, however, a subtlety involved. The above identity is not straight-forward, even if it follows from the straightforward identity ϕI(x)∂

′µϕI(x

′) =∂′µ(ϕI(x)ϕI(x

′)) = ∂′µ(NϕI(x)ϕI (x′) + D(x − x′)). The point is that when

combining two such identities to get the T -product at the LHS, one obtainsϑ(ξ0)∂′µD(ξ)+ϑ(−ξ0)∂′µD(−ξ) instead of ∂µdF (ξ) on the RHS (with ξ = x−x′).The extra term, i.e. the difference between what is desired and what is obtained,is D(ξ)∂0ϑ(ξ

0) +D(−ξ)∂0ϑ(−ξ0) = (D(ξ) −D(−ξ))δ(ξ0) and this indeed van-ishes, as can be shown easily from the explicit form of D(ξ) (see page 81).

Unfortunately, this is not the whole story. Some extra terms (in the abovesense) are simply die-hard. They do not vanish as such, and one gets rid ofthem only via sophisticated cancellations with yet another extras entering thegame in case of derivative couplings14. Attempting not to oppress the reader,we aim to outline the problem, without penetrating deeply into it.

The troublemaker is the T -product of several differentiated fields. An illus-trative example is provided already by two fields, where one obtains

T ∂µϕI (x) ∂′νϕI (x

′) = ∂µ∂′ν (N ϕI (x)ϕI (x

′)+ dF (x− x′))+δ0µδ0ν∆(x− x′)

with15 ∆(ξ) = −iδ4 (ξ). The same happens in products of more fields and theWick’s theorem is to be modified by the non-vanishing extra term −iδ0µδ0νδ4 (ξ),on top of the doubly differentiated standard propagator. In the ”clip notation”

∂µϕI (x) ∂′νϕI (x

′) = ∂µ∂′ν

standard

ϕI (x)ϕI (x′) +

extra

ϕI (x)ϕI (x′)

where

standard

ϕI (x)ϕI (x′) = dF (x− x′) and

extra

ϕI (x)ϕI (x′) = −iδ0µδ0νδ4 (x− x′)

The rather unpleasant feature of this extra term is its non-covariance, whichseems to ruin the highly appreciated relativistic covariance of the perturbationtheory as developed so-far.

14Similar problems (and similar solutions) haunt also theories of quantum fields with higherspins, i.e. they are not entirely related to derivative couplings.

15First one gets, along the same lines as above, ∆(ξ) = (D(ξ) − D(−ξ))∂0δ(ξ0) +2δ(ξ0)∂0(D(ξ) − D(−ξ)). Due to the identity f(x)δ′(x) = −f ′(x)δ(x) this can be broughtto the form ∆(ξ) = δ(ξ0)∂0(D(ξ) − D(−ξ)) and plugging in the explicit form of D(ξ)

one obtains ∆(ξ) = δ(ξ0)∫ d3p

(2π)31

2ω~p∂0(e−ipξ − eipξ) = δ(ξ0)

∫ d3p

(2π)3−ip02ω~p

(e−ipξ + eipξ) =

−iδ(ξ0)∫ d3p

(2π)3ei~p

~ξ = −iδ4(ξ).


Because of the δ-function, the extra term in the propagator can be tradedfor an extra vertex. To illustrate this, let us consider as an example L [ϕ] =12∂µϕ∂

µϕ− 12m

2ϕ2+ g2ϕ∂µϕ∂

µϕ. A typical term in the perturbative expansion ofa green function containsHint [ϕ (x)]Hint [ϕ (x′)] and clipping the fields togethervia the extra term gives16

−ig2ϕ (x) ∂µϕ (x)

extra

ϕI (x)ϕI (x′)−ig2ϕ (x′) ∂νϕ (x′) = i

g2

4ϕ2 (x) ϕ2 (x)

effectively contracting two original vertices into the new extra one. In this wayone can get rid of the extra non-covariant term in the propagator, at the price ofintroduction of the non-covariant effective vertex. In our example this effectivevertex corresponds to an extra term in the Lagrangian: Lextra = 1

2g2ϕ2ϕ2.

The factor 12 follows from a bit of combinatorics. There are four possibilities

for the extra clipping between the two Hint, endowing the new effective vertexwith the factor of 4. Less obvious is another factor of 1

2 , coming from thefact that interchange of the two contracted original vertices does not changethe diagram. According to the rules for combinatoric factors (see section??)this requires the factor of 1

2 . Once the vertices are contracted, there is no(combinatoric) way to reconstruct this factor, so it has to be included explicitly.

The story is not over yet. There is another source of non-covariant vertices.The point is that once derivative couplings are present, the canonical momentumis not equal to the time derivative of the field any more. As an illustration let usconsider our example again. Here one gets π = ϕ+ gϕϕ, i.e. ϕ = (1 + gϕ)

−1π.

The corresponding Hamiltonian density can be written asH = H0+Hint where17

H0 =1

2π2 +

1

2|∇ϕ|2 + 1

2m2ϕ2

Hint =g

2ϕ |∇ϕ|2 − g

2ϕ (1 + gϕ)−1 π2

H0 corresponds to the Hamiltonian density of the free field, expressed in termsof conjugate quantities, obeying (after quantization) the standard commutationrelation [ϕ (x) , π (y)] = iδ3 (~x− ~y). Using this H0 one can develop the pertur-bation theory in the standard way. Doing so it is convenient, as we have seen,to re-express the canonical momentum in terms of the field variables, leading to

Hint = −1

2gϕ∂µϕ∂

µϕ− 1

2g2ϕ2ϕ2

As announced, this interaction Hamiltonian density contains, on top of theexpected covariant term −Lint, a non-covariant one. But now, the fanfaresbreaks out, and the non-covariant vertices originating from two different sources,cancel each other.18 This miracle is not an exceptional feature of the exampleat hand, it is rather a general virtue of the canonical quantization: at the end ofthe day all non-covariant terms in vertices and propagators tend to disappear.

16Here we pretend that Hint = −Lint, which is not the whole truth in the case at hand.We will correct this in the moment.

17H = ϕπ −L = ϕπ − 12ϕ2 + 1

2|∇ϕ|2 + 1

2m2ϕ2 − g

2ϕ∂µϕ∂µϕ

= 12(1 + gϕ)−1 π2 + 1

2|∇ϕ|2 + 1

2m2ϕ2 + g

2ϕ |∇ϕ|2

= 12π2 + 1

2|∇ϕ|2 + 1

2m2ϕ2 + g

2ϕ |∇ϕ|2 − g

2ϕ (1 + gϕ)−1 π2

18One may worry about what happens to the non-covariant part of Hint contracted (withwhatever) via the non-covariant part of the propagator. Indeed, we have not consider suchcontractions, but as should be clear from what was said so-far, for any such contraction thereis a twin contraction with opposite sign, so all such terms cancels out.


propagator

In the Introductions/Conclusions we have learned that the propagator of thescalar field is equal to i/(p2 −m2). Let us check now, whether this ansatz fordF (p) really leads to the correct expression for dF (ξ), i.e. if

dF (ξ)?=

∫d4p

(2π)4

i

p2 −m2e−ipξ

where dF (ξ) = ϑ(ξ0)D (ξ) + ϑ

(−ξ0

)D (−ξ) and D (ξ) =

∫d3p(2π)3

12ω~p

e−ipξ.

It is very useful to treat the p0-variable in this integral as a complex variable.Writing p2 − m2 = (p0 − ω~p)(p0 + ω~p) (recall that ω~p =

√~p2 +m) one finds

that the integrand has two simple poles in the p0-variable, namely at p0 = ±ω~p

with the residua ±(2ω~p)−1e∓iω~pξ0ei~p.

~ξ. The integrand is, on the other hand,sufficiently small at the lower (upper) semicircle in the p0-plane for ξ0 > 0(ξ0 < 0), so that it does not contribute to the integral for the radius of thesemicircle going to infinity. So it almost looks like if

∫d4p

(2π)4

i

p2 −m2e−ipξ ?

= ±∫

d3p

(2π)3

(e−ipξ

p0 + ω~p|p0=ω~p

+e−ipξ

p0 − ω~p|p0=−ω~p

)

(the sign reflects the orientation of the contour) which would almost give thedesired result after one inserts appropriate ϑ-functions and uses ~p→ −~p substi-tution in the last term.

Now, was that not for the fact that the poles lay on the real axis, one couldperhaps erase the questionmarks safely. But since they do lay there, one canrather erase the equality sign.

It is quite interesting, however, that one can do much better if one shiftsthe poles off the real axis. Let us consider slightly modified ansatz for thepropagator, namely i/(p2 − m2 + iε) with positive ε (see the footnote on thepage 5). The pole in the variable p20 lies at ω2

~p − iε, i.e. the poles in the variablep0 lie at ω~p − iε and −ω~p + iε and so trick with the complex plane now worksperfectly well, leading to (convince yourself that it really does)

∫d4p

(2π)4i

p2 −m2 + iεe−ipξ =

∫d3p

(2π)31

2ω~p

ϑ(ξ0)e−ipξ−εξ0 + ϑ(−ξ0)eipξ+εξ0

At this point one may be tempted to send ε to zero and then to claim theproof of the identity dF (p) = i/(p2 −m2) being finished. This, however, wouldbe very misleading. The limit ε→ 0+ is quite non-trivial and one cannot simplyreplace ε by zero (that is why we were not able to take the integral in the caseof ε = 0).

Within the naive approach one cannot move any further. Nevertheless, theresult is perhaps sufficient to suspect the close relation between the result forthe propagator as found in the Introductions/Conclusions and in the presentchapter. Later on we will see that the iε prescription is precisely what is neededwhen passing from the naive approach to the standard one.


s-matrix

The green functions, discussed so-far, describe the time evolution between theinitial and final states of particles created at certain positions. Most experimen-tal setups in the relativistic particle physics correspond to different settings,namely to the initial and final states of particles created with certain momentain the remote past and the remote future respectively. We will therefore inves-tigate now the so-called s-matrix (almost, but not quite the famous S-matrix)

sfi = limT→∞

〈~p1, . . . , ~pm|U (T,−T ) |~pm+1, . . . , ~pn〉

where f and i are abbreviations for ~p1, . . . , ~pm and ~pm+1, . . . , ~pn respectively. Wehave presented the definition directly in the interaction picture, which is mostsuitable for calculations. Of course, it can be rewritten in any other picture, asdiscussed on p.78.

The difference between the s-matrix and the genuine S-matrix (which isto be discussed within the standard approach) is in the states between whichU (T,−T ) is sandwiched. Within our naive approach we adopt a natural andstraightforward choice, based on the relation19 |~p〉 =√2ω~pa

+~p |0〉, leading to

sfi = limT→∞

〈0|√2ω~p1

a~p1,I (T ) . . . U (T,−T ) . . .√2ω~pn

a+~pn,I(−T ) |0〉

Intuitively this looks quite acceptable, almost inevitable: the multi-particle stateis created, by the corresponding creation operators, from the state with noparticles at all. There is, however a loophole in this reasoning.

The main motivation for the s-matrix was how do the real experiments looklike. The states entering the definition should therefore correspond to sometypical states prepared by accelerators and detected by detectors. The firstobjection which may come to one’s mind is that perhaps we should not use thestates with sharp momenta (plane waves) but rather states with ”well-defined,even if not sharp” momenta and positions (wave-packets). This, however, is notthe problem. One can readily switch from plane-waves to wave-packets and viceversa, so the difference between them is mainly the difference in the languageused, rather than a matter of principle.

The much more serious objection is this one: Let us suppose that we haveat our disposal apparatuses for measurement of momenta. Then we can preparestates more or less close to the states with sharp energy and 3-momentum.The above considered states |~p〉 =

√2ω~pa

+~p |0〉 are such states, but only for

the theory with the free Hamiltonian. Once the interaction is switched on, thesaid |~p〉 states may differ significantly from what is prepared by the availableexperimental devices. Once more and aloud: typical experimentally accessiblestates in the worlds with and without interaction may differ considerably. And,as a rule, they really do.

19The relation is given in a bit sloppy way. In the Schrodinger picture, it is to be un-derstood as |~p,−T 〉S =

√

2ω~pa+~p,S

|0〉, where |0〉 is just a particular state in the Fock

space (no time dependence of |0〉 is involved in this relation). In the interaction picturethe relation reads |~p,−T 〉I =

√

2ω~pa+~p,I

(−T ) |0〉 (this is equivalent to the Schrodinger

picture due to the fact that H0 |0〉 = 0). In the Heisenberg picture, however, one has|~p〉H =

√

2ω~pa+~p,H (−T ) e−iHT |0〉 6=

√

2ω~pa+~p,H (−T ) |0〉 (due to the fact that |0〉 is usually

not an eigenstate of H).


One may, of course, ignore this difference completely. And it is precisely thisignorance, what constitutes the essence of our naive approach. Indeed, the coreof this approach is the work with the s-matrix. defined in terms of explicitlyknown simple states, instead of dealing with the S-matrix, defined in terms ofthe states experimentally accessible in the real world (with interactions). Thelatter are usually not explicitly known, so the naivity simplifies life a lot.

What excuse do we have for such a simplification? Well, if the interac-tion may be viewed as only a small perturbation of the free theory — and wehave adopted this assumption already, namely in the perturbative treatment ofU (T,−T ) operator — then one may hope that the difference between the twosets of states is negligible. To take this hope too seriously would be indeed naive.To ignore it completely would be a bit unwise. If nothing else, the s-matrix isthe zeroth order approximation to the S-matrix, since the unperturbed statesare the zeroth order approximation of the corresponding states in the full the-ory. Moreover, the developments based upon the naive assumption tend to bevery useful, one can get pretty far using this assumption and almost everythingwill survive the more rigorous treatment of the standard approach.

As to the calculation of the s-matrix elements, it follows the calculationof the green functions very closely. One just uses the perturbative expansion

U (T,−T ) = T exp−i∫ T

−T ′dt HI (t) (see p.77) and the Wick’s theorem, which

is to be supplemented by20

ϕI (x) a+~p,I (−T ) = NϕI (x) a

+~p,I (−T )+

1√2ω~p

e−ipxe−iω~pT

a~p,I (T )ϕI (x) = Na~p,I (T )ϕI (x)+1√2ω~p

eipxe−iω~pT

or in the ”clipping notation”

ϕI (x) a+

~p,I (−T ) =1√2ω~p

e−ipxe−iω~pT

a~p,I (T )ϕI(x) =1√2ω~p

eipxe−iω~pT

Consequently, the s-matrix elements are obtained in almost the same way as arethe green functions, i.e. by means of the Feynman rules. The only difference isthe treatment of the external lines: instead of the factor dF (x− y), which waspresent in the case of the green functions, the external legs provide the factorse∓ipxe−iω~pT for the s-matrix elements, where the upper and lower sign in theexponent corresponds to the ingoing and outgoing particle respectively. (Notethat the

√2ω~p in the denominator is canceled by the

√2ω~p in the definition of

sfi.) In the Feynman rules the factor e−iω~pT is usually omitted, since it leads tothe pure phase factor exp−iT∑n

i=1 ω~pn, which is redundant for probability

densities, which we are interested in21.

20Indeed, first of all one has ϕI(x) a+~p,I

(t′) = NϕI (x)a+~p,I

(t′)+[ϕI (x), a+~p,I

(t′)] and then

[ϕI(x), a+~p,I (t

′)] =∫ d3p′

(2π)3ei~p

′.~x√2ω~p′

[a~p′,I(t), a+~p,I (t

′)] =∫ d3p′√

2ω~p′ei(~p′.~x−ω~p′ t+ω~pt

′)δ(~p − ~p′) and

the same gymnastics (with an additional substitution p′ → −p′ in the integral) is performedfor [a~p,I (t

′) , ϕI (x)].21Omission of the phase factor is truly welcome, otherwise we should bother about the ill-


In this way one obtains

the Feynman rules for the s-matrix elements in the x-representation

internal line dF (x− y)

ingoing external line e∓ipx

vertex (n legs) −i δnLδϕn

∣∣∣ϕ=0

∫d4x

The next step is the use of the Fourier expansion of dF (x− y) allowing forexplicit x-integrations (see p.84), resulting in

the Feynman rules for the s-matrix elements in the p-representation

internal line∫

d4p(2π)4

dF (p)

external line 1

vertex i δnLδϕn

∣∣∣ϕ=0

(2π)4δ4 (p+ p′ + . . .)

Omitting the factors of (2π)4 as well as the momentum integrations, and

shifting them to the additional rule requiring an extra (2π)4for each vertex and

(2π)−4 ∫

d4p for each internal line, one obtains

another form of the Feynman rules

for the s-matrix elements in the p-representation

internal line dF (p)

external line 1

vertex i δnLδϕn

∣∣∣ϕ=0

δ4 (p+ p′ + . . .)

which is now really very close to our presentation of the Feynman rules in theIntroductions/Conclusions22.

defined limit T → ∞. Of course, avoiding problems by omitting the trouble-making pieces isat least nasty, but what we are doing here is not that bad. Our sin is just a sloppiness. Weshould consider, from the very beginning, the limit T → ∞ for the probability and not forthe amplitude.

22In the Introductions/Conclusions the Feynman rules were used to calculate the scatteringamplitude Mfi rather than the S-matrix. These two quantities are, however, closely related:

Sfi = 1+ iMfi (2π)4 δ(4)

(

Pf − Pi

)

or sfi = 1+ imfi (2π)4 δ(4)

(

Pf − Pi

)

.


connected diagrams

IIThe Mfi, or rather mfi within our naive approach, is more appropriate forthe discussion of the cross-sections and decay rates, which is our next task.Atthis point, there are only three differences left:

• presence of dF (p) instead of the genuine Feynman propagator DF (p)

• no√Z factors corresponding to external legs

• presence of disconnected diagrams like in the perturbative expansions of the green function and the s-matrix23,while in the Introductions/Conclusions only connected Feynman diagramswere accounted for.

The differences are due to the fact that we are dealing with the s-matrixrather than the S-matrix. In the next section we will learn how the so-farmissed ingredients (replacement of dF by DF , appearance of

√Z and fadeaway

of disconnected diagrams) will enter the game in the standard approach.As to the comparison of the rules presented in the Introductions/Conclusions

to the ones derived here, let us remark that in the Introductions/Conclusions wedid not introduce the notion of the S-matrix explicitly. Neverthweless, it waspresent implicitly via the quantity Mfi, since S and M are very closely related

Sfi = 1+ iMfi (2π)4δ(4) (Pf − Pi)

The Mfi, or rather mfi within our naive approach, is more appropriate for thediscussion of the cross-sections and decay rates, which is our next task.

Remark: As we have seen, the green functions g and the s-matrix elements sfiare very closely related. The only differences are the external legs factors: dF (p)for the green function g and simply 1 (or something slightly more complicated incase of higher spins) for the s-matrix elements. This may be formulated in thefollowing way: sfi is obtained from the corresponding green function g by mul-tiplication of each external leg by the inverse propagator. Another, even morepopular, formulation: sfi is obtained from the corresponding g by amputationof the external legs.Actually, the relation between s and g is of virtually no interest whatsoever. Weare, after all, interested only in the s-matrix, so there is no reason to botherabout the green functions. Indeed, we could ignore the whole notion of the greenfunction and derive the rules directly for the s-matrix. Doing so, however, wewould miss the nice analogy between the naive and the standard approaches.The point is that similar relation holds also for the genuine Green functions Gand the S-matrix elements. Indeed, as we will see, Sfi is obtained from thecorresponding Green function G by amputation of the external leg and multi-plication by

√Z. And in the standard approach, unlike in the naive one, one

cannot easily avoid the Green functions when aiming at the S-matrix.

23Which term in the perturbative expansion of U(T,−T ) corresponds this diagram to?


remark on complex fields and arrows

The expansion of the complex scalar field in creation and annihilation operatorsreads

ϕI (~x, t) =

∫d3p

(2π)3

1√2ω~p

(a~p (t) e

i~p.~x + b+~p (t) e−i~p.~x)

ϕ∗I (~x, t) =

∫d3p

(2π)31√2ω~p

(a+~p (t) e−i~p.~x + b~p (t) e

i~p.~x)

with[a~p (t) , a

+~p′ (t)

]= (2π)

3δ (~p− ~p′)

[b~p (t) , b

+~p′ (t)

]= (2π)3 δ (~p− ~p′)

and all other commutators of creation and annihilation operators equal to zero.It is now straightforward to show that

T ϕI (x)ϕI (y) = N ϕI (x)ϕI (y)T ϕ∗

I (x)ϕ∗I (y) = N ϕ∗

I (x)ϕ∗I (y)

T ϕI (x)ϕ∗I (y) = N ϕI (x)ϕ

∗I (y)+ dF (x− y)

T ϕ∗I (x)ϕI (y) = N ϕ∗

I (x)ϕI (y)+ dF (x− y)

This means that the time ordered product of two ϕI -fields as well as of thetwo ϕ∗

I -fields is already in the normal form, i.e. the only contributions to theFeynman diagrams come from T ϕIϕ

∗I and T ϕ∗

IϕI.This result is typical for complex fields: they provide two types of propa-

gators, corresponding to products of the field and the conjugate field in twopossible orderings. In case of the complex scalar field the factors correspondingto the two different orderings are equal to each other, so there is no reasonto use two different graphical representations. This, however, is not a generalfeature. In other cases (e.g. in case of the electron-positron field) the differentorderings lead to different factors. It is therefore necessary to distinguish thesetwo possibilities also in their graphical representation, and this is usually doneby means of an arrow.


3.1.3 Cross-sections and decay rates

Because of the relativistic normalization of states 〈~p|~p′〉 = 2E~p (2π)3δ3 (~p− ~p′)

the S-matrix (s-matrix)24 elements do not give directly the probability ampli-tudes. To take care of this, one has to use the properly normalized vectors

(2π)−3/2

(2E)−1/2 |~p〉, which leads to the probability amplitude equal to Sfi

multiplied by∏n

j=1 (2π)−3/2 (2Ej)

−1/2. Taking the module squared one obtains

probability density = |Sfi|2n∏

j=1

1

(2π)3 2Ej

This expression presents an unexpected problem. The point is that Sfi turns

out to contain δ-functions, and so |Sfi|2 involves the ill-defined square of the δ-function. The δ-functions originate from the normalization of states and they arepotentially present in any calculation which involves states normalized to the δ-function. In many cases one is lucky enough not to encounter any such δ-functionin the result, but sometimes one is faced with the problem of dealing withδ-functions in probability amplitudes. The problem, when present, is usuallytreated either by switching to the finite volume normalization, or by exploitationof the specific set of basis vectors (”wave-packets” rather than ”plane waves”).

There are two typical δ-functions occurring in Sfi. The first one is just thenormalization δ-function, symbolically written as δ (f − i), which represents thecomplete result in the case of the free Hamiltonian (for initial and final statesbeing eigenstates of the free Hamiltonian). In the perturbative calculations,this δ (f − i) remains always there as the lowest order result. It is therefore acommon habit to split the S-matrix as (the factor i is purely formal)

Sfi = δ (f − i) + iTfi

and to treat the corresponding process in terms of Tfi, which contains the com-plete information on transition probability for any |f〉 6= |i〉. The probability for|f〉 = |i〉 can be obtained from the normalization condition (for the probability).Doing so, one effectively avoids the square of δ (f − i) in calculations.

But even the T -matrix is not ”δ-free”, it contains the momentum conser-vation δ-function. Indeed, both the (full) Hamiltonian and the 3-momentumoperator commute with the time-evolution operator 0 = [Pµ, U (T,−T )]. Whensandwiched between some 4-momentum eigenstates25 〈f | and |i〉, this implies(pµf −p

µi ) 〈f |U (T,−T ) |i〉 = 0, and consequently (pµf −p

µi )Sfi = 0. The same, of

course, must hold for Tfi. Now any (generalized) function of f and i, vanishingfor pµf 6= pµi , is either a non-singular function (finite value for pf = pi), or a

distribution proportional to δ4 (pf − pi), or even a more singular function (pro-portional to some derivative of δ4 (pf − pi)). We will assume proportionality tothe δ-function

Tfi = (2π)4δ4 (pf − pi)Mfi

where (2π)4 is a commonly used factor. Such an assumption turns out to lead tothe finite result. We will ignore the other two possibilities, since if the δ-functionprovides a final result, they would lead to either zero or infinite results.

24All definitions of this paragraph are formulated for the S-matrix, but they apply equallywell for the s-matrix, e.g. one have sfi = δ (f − i) + itfi and tfi = (2π)4 δ4

(

pf − pi)

mfi.25The notational mismatch, introduced at this very moment, is to be discussed in a while.


Now back to the notational mismatch. When discussing the first δ-functionδ (f − i), the states 〈f | and |i〉 were eigenstates of the free Hamiltonian H0. Onthe other hand, when discussing the second δ-function δ4 (pf − pi), the states〈f | and |i〉 were eigenstates of the full Hamiltonian H . We should, of course,make just one unambiguous choice of notation, and in this naive approach thechoice is: 〈f | and |i〉 are eigenstates of the free Hamiltonian H0. This choiceinvalidates a part of the above reasoning, namely the part leading to δ (Ef − Ei)

(the part leading to δ3 (~pf − ~pi) remains untouched, since H0 commutes with ~P

and so one can choose 〈f | and |i〉 to be eigenstates of both H0 and ~P ).Nevertheless, we are going to use δ4 (pf − pi) (rather then δ

3 (~pf − ~pi))in thedefinition of Mfi. The point is that δ (Ef − Ei) can be present in Tfi, even ifthe above justification fails. That this is indeed the case (to any order of theperturbation theory) can be understood directly from the Feynman rules. Recallthat in the p-representation every vertex contains the momentum δ-function.Every propagator, on the other hand, contains the momentum integration andafter all these integrations are performed, one is left with just one remainingδ-function, namely δ4 (pf − pi).

Proof: take a diagram, ignore everything except for momentum δ-functionsand integrations. Take any two vertices connected directly by an internal lineand perform the corresponding integration, using one of the vertices δ-functions.After integration the remaining δ-function contains momenta of all legs of theboth vertices. The result can be depicted as a diagram with two vertices shrunkinto a new one, and the new vertex contains the appropriate momentum δ-function. If the considered vertices were directly connected by more than oneinternal line, then the new diagram contains some internal lines going from thenew vertex back to itself. From the point of view of the present argument, such”daisy-buck loops” can be shrunk to the point, since they do not contain any δ-function. The procedure is then iterated until one obtains the simplest possiblediagram with one vertex with all external lines attached to it and with thecorresponding δ-function δ(Σ pext). The final point is to realize that in Feynmandiagrams momenta are oriented towards the vertices, while final state momentaare usually understood as flowing from the diagram, i.e. Σ pext = pf − pi.

After having identified the typical δ-function present in the T -matrix, weshould face the annoying issue of the undefined square of this δ-function in |Tfi|2.We will do this in a covardly manner, by an attempt to avoid the problem bythe standard trick with the universe treated as a finite cube (length L) with theperiodic boundary conditions. The allowed 3-momenta are ~p = 2π

L (nx, ny, nz)and the 3-momentum δ-function transfers to

δ3(~p− ~p′) =

∫d3x

(2π)3 e

i(~p−~p′).~x → V

(2π)3 δ~p~p′

def= δ3V (~p− ~p′)

while the same trick performed with the time variable gives the analogous result

δ(E − E′) → T2π δEE′

def= δT (E − E′). The calculation of |Tfi|2 in the finite 4-

dimensional cube presents no problem at all: |Tfi|2 = V 2T 2δ~pf ~piδEfEi |Mfi|2.

For reasons which become clear soon, we will write this as

|Tfi|2 = V T (2π)4 δ4V T (pf − pi) |Mfi|2

where δ4V T = δ3V δT .


The finite volume normalization affects also the normalization of states, since〈~p|~p′〉 = 2E~p (2π)

3δ3 (~p− ~p′) → 2E~pV δ~P ~P ′ . The relation from the beginning of

this paragraph between |Sfi|2 (or |Tfi|2 for f 6= i) and the corresponding proba-

bility therefore becomes: probabilityfor f 6=i = |Tfi|2∏n

j=11

2EjV. Note that using

the finite volume normalization, i.e. having discrete rather than continuous la-beling of states, we should speak about probabilities rather than probabilitydensities. Nevertheless, for the volume V big enough, this discrete distributionof states is very dense — one may call it quasi-continuous. In that case it istechnically convenient to work with the probability quasi-density, defined as theprobability of decay into any state |f〉 = |~p1, . . . , ~pm〉 within the phase-spaceelement d3p1 . . . d

3pm. This is, of course, nothing else but the sum of the prob-abilities over all states within this phase-space element. If all the probabilitieswithin the considered phase-space element were equal (the smaller the phase-space element is, the better is this assumption fulfilled) one could calculate thesaid sum simply by multiplying this probability by the number of states withinthe phase-space element. And this is exactly what is commonly used (the un-derlying reasoning is just a quasi-continuous version of the standard reasoningof integral calculus). And since the number of states within the interval d3p

is ∆nx∆ny∆nz = V d3p/ (2π)3one comes to the probability quasi-density (for

f 6= i) being equal to |Tfi|2∏n

j=11

2EjV

∏mk=1

V d3p(2π)3

. So for f 6= i one has

probability

quasidensity= V T (2π)

4δ4V T (pf − pi) |Mfi|2

m∏

j=1

d3p

(2π)32Ej

n∏

j=m+1

1

2EjV.

Comparing this to the expressions for dΓ and dσ given in the Introductions(see p.12) we realize that we are getting really close to the final result. We justhave to get rid of the awkward factors T and T/V in the probability quasi-densities for one and two initial particles respectively (and to find the relation

between 1/EAEB and [(pA.pB)2−m2

Am2B]

−1/2 in case of the cross section). Thisstep, however, is quite non-trivial. The point is that even if our result looks asif we are almost done, actually we are almost lost. Frankly speaking, the resultis absurd: for the time T being long enough, the probability exceeds 1.

At this point we should critically reexamine our procedure and understandthe source of the unexpected obscure factor T . Instead, we are going to followthe embarrassing tradition of QFT textbooks and use this evidently unreliableresult for further reasoning, even if the word reasoning used for what follows isa clear euphemism26. The reason for this is quite simple: the present author isboldly unable to give a satisfactory exposition of these issues27.

26A fair comment on rates and cross sections is to be found in the Weinberg’s book (p.134):The proper way to approach these problems is by studying the way that experiments areactually done, using wave packets to represent particles localized far from each other before acollision, and then following the time history of these superpositions of multiparticle states.In what follows we will instead give a quick and easy derivation of the main results, actuallymore a mnemonic than a derivation, with the excuse that (as far as I know) no interestingopen questions in physics hinge on getting the fine points right regarding these matters.

27This incapability seems to be shared by virtually all authors of QFT textbooks (whichperhaps brings some relief to any of them). There are many attempts, more or less relatedto each other, to introduce decay rates and cross sections. Some of them use finite volume,some use wave-packets (but do not closely follow the whole time evolution, as suggested byWeinberg’s quotation), some combine the two approaches. And one feature is common to allof them: they leave much to be desired.


After having warned the reader about the unsoundness of what follows, wecan proceed directly to the interpretation of the result for one-particle initialstate in terms of the decay rate: From the linear time dependence of the prob-ability density one can read out the probability density per unit time, this isequal to the time derivative of the probability density and for the exponentialdecay exp(−Γt) this derivative taken at t = 0 is nothing else but the decay rateΓ (or dΓ if we are interested only in decays with specific final states).

The previous statement is such a dense pack of lies that it would be hardlyoutmatched in an average election campaign. First of all, we did not get thelinear time dependence, since T is not the ”flowing time”, but rather a singlemoment. Second, even if we could treat T as a variable, it definitely appliesonly to large times and, as far as we can see now, has absolutely nothing to sayabout infinitesimal times in the vicinity of t = 0. Third, if we took the lineartime dependence seriously, then why to speak about exponential decay. Indeed,there is absolutely no indication of the exponential decay in our result.

Nevertheless, for the reasons unclear at this point (they are discussed in theappendix ??) the main conclusion of the above lamentable statement remainsvalid: decay rate is given by the probability quasi-density divided by T . Afterswitching back to the infinite volume, which boils down to δ4V T (pf − pi) →δ4(pf − pi), one obtains

dΓ = (2π)4δ4 (Pf − Pi)

1

2EA|Mfi|2

m∏

i=1

d3pi

(2π)3 2Ei

where EA is the energy of the decaying particle.

Remark: The exponential decay of unstable systems is a notoriously knownmatter, perhaps too familiar to realize how non-trivial issue it becomes in thequantum theory. Our primary understanding of the exponential decay is basedon the fact that exp(−Γt) is the solution of the simple differential equationdN/dt = −ΓN(t), describing a population of individuals diminishing indepen-dently of a) each other b) the previous history. In quantum mechanics, however,the exponential decay should be an outcome of the completely different time evo-lution, namely the one described by the Schrodinger equation.Is it possible to have the exponential decay in the quantum mechanics, i.e. canone get |〈ψ0 |ψ (t)〉|2 = e−Γt for |ψ (t)〉 being a solution of the Schrodinger equa-tion with the initial condition given by |ψ0〉? The answer is affirmative, e.g. onecan easily convince him/herself that for the initial state |ψ0〉 =

∑cλ |ϕλ〉, where

H |ϕλ〉 = Eλ |ϕλ〉, one obtains the exponential decay for∑ |cλ|2 δ (E − Eλ)

def=

p (E) = 12π

Γ(E−E0)

2+Γ2/4(the so-called Breit-Wigner distribution of energy in

the initial state).The Breit-Wigner distribution could nicely explain the exponential decay inquantum mechanics, were it not for the fact that this would require understand-ing of why this specific distribution is so typical for quantum systems. And thisis very far from being obvious. Needless to say, the answer cannot be that theBreit-Wigner is typical because the exponential decay is typical, since this wouldimmediately lead to a tautology. It would be nice to have a good understandingfor the exponential decay in the quantum mechanics, but (unfortunately) we arenot going to provide any.


For two particles in the initial state the reasoning is a bit more reasonable.The main trick is to use another set of the initial and final states, the one whichenables semiclassical viewpoint, which further allows to give some sense to thesuspicious factor T/V . We are speaking about well localized wave-packets withwell defined momentum (to the extend allowed by the uncertainty principle).Let us call one particle the target, while the second one the beam. The targetwill be localized in all three space dimensions, the beam is localized just in onedirection — the one of their relative momentum ~pA − ~pB. In the perpendic-ular directions the particle is completely unlocalized, the state corresponds tothe strictly zero transverse momentum, in this way the particle simulates thetransversely uniform beam. Let us note that if we want (as we do) to simulate abeam with a constant density independent of the extensiveness of the universebox, the beam particle state is to be normalized to L2 rather than to 1.

Now in the finite-box-universe with the periodic boundary conditions thebeam particle leaves the box from time to time, always simultaneously enteringon the other side of the world. During the time T the scattering can thereforetake place repeatedly. In the rest frame of the target particle the number ofscatterings is T

L/v , where v is the velocity of the beam particle in this frame. The

reasonable quantity (the cross-section28) in the target rest frame is therefore theprobability quasi-density as obtained above, divided by the ”repetition factor”vT/L and multiplied by the beam normalization factor L2

dσ = (2π)4δ4V T (pf − pi)

1

4EAEBv|Mfi|2

m∏

j=1

d3p

(2π)32Ej

Finally, one switches back to the infinite universe by δ4V T (pf −pi) → δ4(pf −pi).The energies EA and EB, as well as the beam particle velocity v, is under-

stood in the target rest frame. To have a formula applicable in any frame, oneshould preferably find a Lorentz scalar, which in the target rest frame becomesequal to EAEBv. Such a scalar is provided by [(pA.pB)

2 −m2Am

2B ]

1/2, since for

pA = (mA,~0) it equals to mA(E2B −m2

B)1/2 = mA |~pB| and v = |~pB| /EB, and

so we have come to the final result

dσ = (2π)4δ4 (Pf − Pi)

|Mfi|2

4√(pA.pB)

2 −m2Am

2B

n∏

i=1

d3pi

(2π)32Ei

28Reacall that the cross-section is a (semi)classical notion, defined as follows. For a uniformbeam of (semi)classical particles interacting with a uniformly distributed target particles, thenumber dn of beam particles scattered into an element of phase space dPf is proportional tothe flux j of the beam (density×velocity), the number N of the particles in target and dPf

itself: dn ∝ jNdPf . The coefficient of proportionality is the cross-section.


3.2 Standard approach

3.2.1 S-matrix and Green functions

Double role of the free fields

Let us consider a single-particle state in a theory of interacting quantum fields.How would this state evolve in time? The first obvious guess is that a singleparticle have nothing to interact with, so it would evolve as a free particle con-trolled by the hamiltonian with the interaction switched off. While the first partof this guess turns out to be correct, the second part does not. A single particlestate indeed evolves as a free particle state, but the hamitonian controlling thisevolution is not the full hamiltonian with the interaction switched off.

At the first sight, the last sentence looks almost self-contradictory. Theparticle described by the hamiltonian without interactions is free by definition,so how come? The point is that one can have free hamiltonians which are quitedifferent from each other. They can describe different particles with differentmasses, spins and other quantum numbers. Well, but can two such different freehamiltonians play any role in some realistic theory? Oh yes, they can. Take thequantum chromodynamics (QCD) as an example. The theory is formulated interms of quark and gluon fields with a specific interaction, with the hamiltonianH = H0 +Hint. With the interaction switched off, one is left with the theorywith free quarks and gluons. No such free particles were ever observed in nature.The single particles of QCD are completely different particles, called hadrons(pions, protons, ...) and the free hamitonian describing a single hadron is someeffective hamiltonian Heff

0 , completely different from H0. In another realistictheory, namely in the quantum electrodynamics (QED), the difference betweenH0 and Heff

0 is not that dramatic, but is nevertheless important.In the best of all worlds, the two free Hamiltonians might simply coincide.

Such a coincidence was assumed in the naive approach, and this very assumptionwas in the heart of that approach. Relaxation from this assumption is in theheart of the standard approach, which seems to be better suited to our world.

The free hamiltonian Heff0 is relevant not only for description of single-

particle states, it plays a very important role in the formulation of any quantumfield theory (QFT). We are speaking about a remarkable experimental fact that,in spite of presence of interactions in any realistic particle theory, there is a hugeamount of states in the Fock space (or even larger playground) behaving like ifthey were described by the free QFT. The said states, e.g. more or less localizedstates of particles far apart from each other, evolve according to the dynamicsof free particles, at least for some time. This time evolution is governed by theeffective free Hamiltonian Heff

0 . The relations of the two free Hamiltonians tothe full Hamiltonian H are

H = H0 +Hint∼= Heff

0

where the symbol ∼= stands for the effective equality (for a limited time) on asubset of the space of states of the complete theory.

Not only are these two free Hamiltonians as a rule different, they even enterthe game in a quite different way. The H0 is an almost inevitable ingredient ofthe formal development of any relativistic QFT. The Heff

0 , on the other hand,is an inevitable feature of any realistic QFT — it is dictated by what we see inNature.

3.2. STANDARD APPROACH 99

Since Heff0 is the Hamiltonian of noninteracting particles, one can naturally

define corresponding creation and annihilation operators a+eff~ps , aeff~ps , where the

discrete quantum number s distinguishes between different particle species (e.g.electron, proton, positronium), as well as between different spin projections etc.These operators create and annihilate the physical particles observed in the realworld and in terms of these operators one has

Heff0 =

∑

s

∫d3p

(2π)3E

eff~ps a

+eff~ps aeff~ps

where

Eeff~ps =

√m2

ph + ~p2

The parameter mph is the physical mass of the particles, i.e. the mass whichis measured in experiments. The value of this parameter can be (and usuallyindeed is) different from the mass parameter entering the free hamiltonian H0

(which is the parameter from the lagrangian of the theory). This raise aninteresting question about how is the value of the mass parameter from thelagrangian pinned down. We will come back to this interesting and importantissue later on, when discussing the renormalization of quantum field theories.

At this point it should be enough to state explicitly that once the quantumfield theory is defined by some lagrangian, everything is calculable (at least inprinciple) from this lagrangian. And so it happens that when one calculatesthe masses of free particles, it turns out that they are not equal to the valuesof mass parameters in the lagrangian (this applies to the masses in scalar fieldtheories, to the masses of charged particles in QED, etc.). In many theories itis just the calculation of masses where the naive approach breaks down. Thereason is, that this approach assumes implicitly that the mass parameter in thelagrangian is equal to the physical mass of particles and this assumption turnsout to be self-contradictory.

The difference between the two mass parameters is, however, not the onlymain difference between H0 and Heff

0 . Let us emphasize once more also thedifference between the operators a+~p and a~p and the operators a+eff

~ps and aeff~ps . In

the naive approach we have assumed implicitly a+eff~ps = a+~p and aeff~ps = a~p. In the

standard approach we do not know the exact relation between the two sets ofcreation and annihilation operators. This lack of knowledge is quite a seriousobstacle, which can be, fortunately, avoided by clever notions of S-matrix andGreen functions.

Remark: We have considered just the free effective hamiltonian, but the sameconsiderations apply to all 10 Poincare generators. For the states for which H0+Hint

∼= Heff0 , the analogous relations hold for all 10 generators. Each of these

effective generators have the form of the free field generators. If desired, onecan define effective fields as the appropriate linear combinations of the effectivecreation and annihilation operators

ϕeff (~x, t)ϕ (x) =

∫d3p

(2π)3

1√2ω~p

(aeff~p e−ipx + a+eff

~p eipx)

and write down the generators explicitly using the relations derived previouslyfor the free fields.


S-matrix

The S-matrix elements are defined in the following way: one takes a state ob-served in the real world by some inertial observer in the remote past as a stateof particles far apart from each other, evolve this state to the remote future andcalculate the scalar product with some other state observed in the real world inthe remote future as a state of particles far apart from each other. Individualparticles in the real world are described as superpositions of eigenstates of theeffective Hamiltonian Heff

0 , rather than H0. So the definition of the S-matrix(in the interaction picture) is

Sfi = 〈f |U (∞,−∞) |i〉

〈f | =∫d3p1 . . . d

3pm 〈Ω| f∗(~p1)aeff~p1. . . f∗(~pm)aeff~pm

|i〉 =∫d3pm+1 . . . d

3pn f(~pm+1)a+eff~pm+1

. . . f(~pn)a+eff~pn

|Ω〉

where |Ω〉 is the ground state of the effective Hamiltonian Heff0 , f(~p) are some

functions defining localized wave-packets, and US (t, t′) is the time-evolutionoperator in the Schrodinger picture29.

In practice, one usually does not work with wave-packets, but rather withproperly normalized plane-waves: f(~p) = (2Eeff

~p )1/2. This may come as a kindof surprise, since such states are not localized at all, still the matrix elements

Sfi = 〈f |U (∞,−∞) |i〉

〈f | = 〈Ω|√2Eeff

~p1aeff~p1

. . .√2Eeff

~pmaeff~pm

|i〉 =√2Eeff

~pm+1a+eff~pm+1

. . .√2Eeff

~pna+eff~pn

|Ω〉

are well defined, since they can be unambiguously reconstructed from sufficientlyrich set of matrix elements for localized states. Not that they are reconstructedin this way, the opposite is true: the main virtue of the plane-waves matrixelements is that they are calculable directly from the Green functions definedentirely in terms of a+and a.

The last sentence contains the core of the standard approach to the inter-acting quantum fields. The point is that the above definition odf the S-matrixcontains both a+eff

~ps , aeff~ps , and a+~p , a~p. One set of the creation and annihila-tion operators, namely the effective ones, is explicitly present, the second oneis hidden in the evolution operator US (∞,−∞), which is determined by thefull Hamiltonian defined in terms of a+ and a. Without the explicit knowledgeof the relations between the two sets of operators, we seem to be faced with aserious problem. The clever notion of the Green function enables one to avoidthis problem in a rather elegant way.

29The more common, although a bit tricky, definition within the Heisenberg picture readssimply Sfi = 〈fout|iin〉 (see page 78). Let us recall that the tricky part is labeling of thestates. One takes the states which look in the Schrodinger picture in the remote past likestates of particles far apart from each other, then switch to the Heisenberg picture and callthese states the in-states. The same procedure in the remote future leads to the definitionof the out-states. In spite of what their definitions and names may suggest, both in- andout-states stay put during the time evolution.

As to the interaction picture, we will try to avoid it for a while. The reason is that theinteraction picture is based on the splitting H0 +Hint, and simultaneous use of Heff

0 and H0

within one matrix element could be a bit confusing.


Green functions

Before presenting the definition of Green functions it seems quite reasonableto state explicitly, what they are not. The widespread use of the in- and out-states usually involves also the use of in- and out- creation and annihilationoperators, although there is no logical necessity for this. Operators do dependon time in the Heisenberg picture, but they can be unambiguously determinedby their appearance at any instant (which plays a role of initial conditionsfor the dynamical equations). The ones which coincide with our aeff~ps ,a

+eff~ps for

t → −∞ or t → ∞ are known as ain~ps, a+in~ps and aout~ps , a

+out~ps respectively. Their

appropriate superpositions30 are called the in- and out -fields, and they use toplay an important role in the discussion of renormalization in many textbooks31.

One might quess that the Green functions are obtained from the green func-tions by replacing the fields ϕ(x) by the fields ϕeff(x) (with a, a+ replaced byaeff , a+eff) and the vacuum state |0〉 (usually called the perturbative vacuum)by the vacuum state |Ω〉 (usually called the non-perturbative vacuum). Such anobject would be a reasonable quantity with a direct physical interpretationonthe interpretation of a propagator of the real particles. Such objects, however,are not the Green functions. The genuine Green functions are obtained fromthe green functions just by the replacement |0〉 → |Ω〉, with the fields ϕ(x)remaining untouched

G (x1, . . . , xn) = 〈Ω|out T ϕH(x1) . . . ϕH(xn) |Ω〉inwhere32 |Ω〉in = eiHti |Ω〉 and 〈Ω|out = 〈Ω| e−iHtf (compare with page 79).

Let us emphasize, that the Green functions do not have the direct physicalmeaning of propagators. Frankly speaking, they do not have any direct phys-ical meaning at all. They are rather some auxiliary quantities enabling trickycalculation of the plane-waves S-matrix elements33. This tricky caclulation isbased on two steps:

• the iε-trick, enabling the perturbative calculation of the Green functions

• the so-called LSZ reduction formula, relating the plane-wave S-matrix el-ements to the Green functions

We are now going to discuss in quite some detail these two famous tricks.

30ϕin,out (x) =∫ d3p

(2π)31

√

2Ein,out~p

(

ain,out~p

e−ipx + a+in,out~p

eipx)

, where p0 = Ein,out~p

.

31The popularity of the in- and out-fields is perhaps due to the fact, that they are relatedto the effective Hamiltonian, which is, paradoxically, somehow more fundamental then thefull one. The point is that about the effective Hamiltonian we know for sure — whateverthe correct theory is, the isolated particles are effectively described by the effective free fieldtheory. From this point of view, ϕin and ϕout are well rooted in what we observe in the nature,while the existence of ϕ relies on the specific assumptions about the corresponding theory.

32Note that since the state |Ω〉 is an eigenstate of the effective hamiltonian Heff0 as

well as of the full hamiltonian H (for the state with no particles these two Hamilto-nians are completely equivalent), one has |Ω〉in = expiHti |Ω〉 = expiEΩti |Ω〉 and〈Ω|out = 〈Ω| exp−iHtf = 〈Ω| exp−iEΩtf. The factors exp−iEΩti,f can be ig-nored as an overall phase, so one can write the Green function also in the following form:G (x1, . . . , xn) = 〈Ω|TϕH (x1) . . . ϕH (xn) |Ω〉

33The tricky techniques are of the utmost importance, since they relate something whichwe are able to calculate (the Green functions) to something which we are really interested in(the S-matrix elements).


the iε-trick

Due to our ignorance on how do the ϕ-fields (defined in terms of a+ and aoperators) act on the non-perturbative vacuum |Ω〉, a direct calculation of theGreen functions would be a very difficult enterprise. There is, however, a wayhow to express |Ω〉 via |0〉, and once this is achieved, the calculation of the Greenfunctions is even simpler than it was in case of the green functions.

The trick is to go back to the green functions

g (x1, . . . , xn) = limT→∞

〈0| e−iHTT ϕH(x1) . . . ϕH(xn) e−iHT |0〉

and to allow the time T to have a negative imaginary part. Decomposing the|0〉 state in eigenstates of the Hamiltonian one obtains34

e−iHT |0〉 =∑

n

e−iEnT cn |n〉

where H |n〉 = En |n〉 and |0〉 = ∑n cn |n〉. The formal change T → T (1 − iε)

leads to the exponential suppression or growth of each term in the sum. Theground state, i.e. the state of the lowest energy, is either suppressed most slowlyor grows most rapidly.

In the limit T → ∞ (1− iε) everything becomes negligible compared to theground state contribution (which by itself goes either to zero or to infinity).To get a finite limit, one multiplies both sides of the equation by an by anappropriate factor, namely by eiEΩT , where EΩ is the ground state energy.Doing so, one gets limT→∞(1−iε) e

iEΩT e−iHT |0〉 = cΩ |Ω〉 and therefore

|Ω〉 = limT→∞(1−iε)

e−iHT |0〉〈Ω| e−iHT |0〉

where we have used cΩe−iEΩT = 〈Ω|0〉 e−iEΩT = 〈Ω| e−iHT |0〉. Using the same

procedure for the 〈Ω| state, one can rewrite the Green function in the followingform

G (x1, . . . xn) = limT→∞(1−iε)

〈0| e−iHT T ϕH(x1) . . . ϕH(xn) e−iHT |0〉〈0| e−iHT |Ω〉〈Ω| e−iHT |0〉

The denominator can be written in an even more convenient form utilizing

limT→∞(1−iε)

〈0| e−iHT |Ω〉〈Ω| e−iHT |0〉 = limT→∞(1−iε)

∑

n

〈0| e−iHT |n〉〈n| e−iHT |0〉 = limT→∞(1−iε)

〈0| e−iHT e−iH

The Green function written in this form can be calculated using the tech-niques developed for the green functions. One use the interaction picture, whereone obtains

G (x1, . . . xn) = limT→∞(1−iε)

〈0|T U (T,−T )ϕI(x1) . . . ϕI(xn) |0〉〈0|U (T,−T ) |0〉

and now one uses the iterative solution for U (T,−T ) together with the Wicktheorem. The point is that the Wick theorem, which does not work for the

34This is the notorious relation describing the time evolution in the Schrodinger picture.Here it is used in the Heisenberg picture, where it holds as well, even if it does not have themeaning of time evolution of the state.


non-perturbative vacuum |Ω〉, works for the perturbative vacuum |0〉 (the non-perturbative vacuum expentation value of the normal product of the ϕ-fieldsdoes not vanish, while the perturbative vacuum expentation value does). Theonly new questions are how to take the limit T → ∞(1−iε) and how to calculatethe denominator.

Ako urobim limitu? Vvezmem t → t(1 − iε) vsade, nielen pre velke t, vkonecnej oblasti od tmin po tmax to neurobı nijaku zmenu, takze limita ε → 0da povodne vysledky.

dF (ξ) = ϑ(ξ0) ∫ d3p

(2π)3

1

2ω~pe−ipξ + ϑ

(−ξ0

) ∫ d3p

(2π)3

1

2ω~peipξ

DF (ξ) = ϑ(ξ0) ∫ d3p

(2π)3

1

2ω~pe−iω~pξ0(1−iε)ei~p

~ξ + ϑ(−ξ0

) ∫ d3p

(2π)3

1

2ω~peiω~pξ0(1−iε)e−i~p~ξ

= ϑ(ξ0) ∫ d3p

(2π)31

2ω~pe−ω~pξ0εe−ipξ + ϑ

(−ξ0

) ∫ d3p

(2π)31

2ω~peω~pξ0εeipξ

trik∫

d4p

(2π)4

i

p2 −m2 + iεe−ipξ =

∫d3p

(2π)3

1

2ω~p

ϑ(ξ0)e−ipξ−εξ0 + ϑ(−ξ0)eipξ+εξ0


3.2.2 the spectral representation of the propagator

We have just convinced ourselves that the relation between the Feynman rules inthe x- and p-representations is not the simple Fourier transformation, but rathera ”Fourier-like” transformation with the time variable possessing an infinitesimalimaginary part.35 This transformation relates the vacuum-bubble–free Feynmandiagrams calculated with the standard Feynman rules in the p-representation,and the (genuine) Green functions in the x-representation

G(x1, . . . , xn) = 〈Ω|T ϕ (x1) . . .

ϕ (xn) |Ω〉

As an intermediate step in the further analysis of this quantity, we shallinvestigate the special case n = 2. The reason for this is twofold. First of all,this is the proper definition of the 2-point Green function, i.e. of the dressedpropagator, which is an important quantity on its own merits. Second, it seemsfavorable to start with introducing all the technicalities in this simple case, andthen to generalize to an arbitrary n. The reader should be aware, however, thatthere is no logical necessity for a specific discussion of the case n = 2. If wewished, we could just skip this and go directly to a general n.

The main trick, which will do the remaining job in our search for the relationbetween the Feynman diagrams and the S-matrix elements, is a very simple one.It amounts just to an insertion of the unit operator, in a clever (and notoriouslyknown) form, at an appropriate place. The clever form is

1 = |Ω〉〈Ω|+∑

N

∫d3p

(2π)3

1

2EN~p|N, ~p〉〈N, ~p|

where (2EN~p)−1/2 |N, ~p〉 are the full Hamiltonian eigenstates (an orthonormal

basis). Recall that since the Hamiltonian commutes with 3-momentum, theseeigenstates can be labelled by the overall 3-momentum (plus additional quan-tum number N). Moreover, the eigenstates with different ~p are related by aLorentz boost and the eigenvalues satisfy EN~p =

√m2

N + ~p2, where mN is theso-called rest energy, i.e. the energy of the state with the zero 3-momentum.All this is just a consequence of the Poincare algebra (i.e. of the relativisticinvariance of the theory) and has nothing to do with presence or absence ofan interaction. Writing the ground state contribution |Ω〉〈Ω| outside the sumreflects the natural assumption that the physical vacuum is Lorentz invariant,i.e. the boosted vacuum is equal to the original one.

The appropriate place for the insertion of this unit operator in the dressedpropagator is (not surprisingly) between the two fields. The main trick whichmakes this insertion so useful, is to express the x-dependence of the fields viathe space-time translations generators Pµ

ϕ (x) = eiPx

ϕ (0) e−iPx

and then to utilize that both |Ω〉 and |N, ~p〉 are eigenstates of Pµ (this is theplace, where one needs |Ω〉 rather than |0〉): e−iPx |Ω〉 = |Ω〉 and e−iPx |N, ~p〉 =e−ipx |N, ~p〉, where p0 = EN~p (and we have set the ground state energy to zero).

35Let us call the reader’s attention to a potential misconception, which is relatively easy toadopt. Being faced with texts like our ”first look on propagators”, one may readily get animpression that the Feynman rules in the p-representation do give the Fourier transforms ofthe Green functions. This is simply not true, as should be clear from the above discussion.


Having prepared all ingredients, we can now proceed quite smoothly. If thetime ordering is ignored for a while, one gets immediately

〈Ω| ϕ (x)

ϕ (y) |Ω〉 =

∣∣∣〈Ω| ϕ (0) |Ω〉

∣∣∣2

+∑

N

∫d3p

(2π)3

e−ip(x−y)

2EN~p

∣∣∣〈Ω| ϕ (0) |N, ~p〉

∣∣∣2

Now one inserts another unit operator written as U−1~p U~p, where U~p is the repre-

sentation of the Lorentz boost transforming ~p to ~0. Making use of the Lorentzinvariance of both the nonperturbative vacuum 〈Ω|U−1

~p = 〈Ω| and the scalar

field U~pϕ (0)U−1

~p =ϕ (0), one can get rid of the explicit ~p-dependence in the

matrix element36

〈Ω| ϕ (0) |N, ~p〉 = 〈Ω|U−1

~p U~pϕ (0)U−1

~p U~p |N, ~p〉 = 〈Ω|ϕ (0) |N,~0〉

With the time ordering reinstalled, one can therefore write

〈Ω|T ϕ (x)

ϕ (y) |Ω〉 = v2ϕ +

∑

N

∫d4p

(2π)4

ie−ip(x−y)

p2 −m2N + iε

∣∣∣〈Ω|ϕ (0) |N,~0〉∣∣∣2

with vϕ = | 〈Ω| ϕ (0) |Ω〉 | (this constant usually vanishes because of symmetries

involved, but there is no harm in keeping it explicitly). So the dressed propa-gator can be written as a superposition of the bare propagators corresponding

to masses mN , with the weight given by∑

N |〈Ω|ϕ (0) |N,~0〉|2. This is usuallywritten in the form (the Kallen–Lehmann spectral representation)

G(x, y) = v2ϕ +

∫ ∞

0

dM2

2πG

0(x, y;M2)ρ(M2)

where the so-called spectral density function ρ(µ2)is defined as

ρ(M2)=∑

N

2πδ(M2 −m2

N

) ∣∣∣〈Ω|ϕ (0) |N,~0〉∣∣∣2

Now what was the purpose of all these formal manipulations? Is there any-thing useful one can learn from the spectral representation of the dressed prop-agator? Frankly, without further assumptions about the spectral function, notthat much. But once we adopt a set of plausible assumptions, the spectralrepresentation will tell us quite some deal about the structure of the dressedpropagator. It will shed some new light on the structure of G(x, y) anticipatedwithin the first look on the dressed propagator, which is not very important, butnevertheless pleasant. On top of that, it will enable us to make some nontrivialstatements about the analytic properties of the dressed propagator. This againis not that important, although interesting. The real point is that virtuallythe same reasoning will be used afterwards in the case of the n-point Greenfunction, and then all the work will really pay off.

36For higher spins one should take into account nontrivial transformation properties ofthe field components. But the main achievement, which is that one gets rid of the explicit3-momentum dependence, remains unchanged.


The usefulness of the spectral representation of the dressed propagator isbased on the following assumptions:1. the spectrum of the Hamiltonian at ~p = 0 contains discrete and continuousparts, corresponding to one-particle and multi-particle states respectively37

2. the lowest energy eigenstate (its energy will be denoted by m) with the

nonvanishing matrix element 〈Ω|ϕ (0) |N,~0〉 is a 1-particle one38

3. the minimum energy of the continuous spectrum is somewhere around 2m,

and if there are any other 1-particle states with nonvanishing 〈Ω|ϕ (0) |N,~0〉,their energy is also around or above this threshold39

Within these assumptions one gets

G(x, y) = v2ϕ + Z G0(x, y;m

2) +

∫ ∞

∼4m2

dM2

2πG

0(x, y;M2)ρ(M2)

where Z =∣∣∣〈Ω|ϕ (0) |N,~0〉

∣∣∣2

. In the p-representation this corresponds to

G(p) = v2ϕ (2π)4δ(p) +

iZ

p2 −m2 + iǫ+

∫ ∞

∼4m2

dM2

2π

iρ(M2)

p2 −M2 + iǫ

Apart from the first term (which is usually not present, because of the vanish-ing vϕ) this is just the form we have guessed in our ”first look” on propagators.But now we understand better its origin, we know from the very beginning thatthe parameter m is the physical mass (the rest energy) of a 1-particle state, wehave learned about how is the parameter Z related to the particular matrix ele-ment (which corresponds to the probability of the 1-particle state to be createdby acting of the field on the nonperturbative vacuum) and finally we have analmost explicit (not quite, because of an explicitly unknown spectral function)expression for what we have called ”analytic part” or ”irrelevant terms”.

As to the analycity, the third term is clearly analytic (one can differentiatewith respect to the complex p2) everywhere except of the straight line from∼ 4m2-iε to ∞-iε. On this line, the behavior of the integral depends on thestructure of ρ

(M2). If there are any additional 1-particle states with non-

vanishing 〈Ω|ϕ (0) |N,~0〉, their contribution to ρ(M2)is proportional to the

δ-function, leading to additional poles in G(p). Otherwise ρ(M2)is supposed

to be a decent function, in which case G(p) has a branch cut along the line40.

37A multiparticle state can be characterized by an overall 3-momentum (which is zero inour case), relative 3-momenta (which are the continuous parameters not present in single-particle states) and discrete quantum numbers like mass, spin, charges etc. It is natural toassume that energy of a multiparticle state is a continuous function of the relative 3-momenta.And since for one-particle states there is no known continuous characteristics, except of 3-momentum which is here fixed to be zero, there is no room for continuous change of energy inthis case. Note that adopting this philosophy, bound states of several particles are consideredone-particle states. This may seem as a contradiction, but is not, it is just a matter ofterminology.

38This assumption is not of the vital importance, and it can be relaxed easily. One mayview it as a typical case and stay open-minded for more exotic possibilities (e.g. no 1-particle

state with non-vanishing 〈Ω|ϕ (0) |N,~0〉, which may occur in theories with confinement).39This again is not very important and can be relaxed if needed. On the other hand, it

is quite natural assumption, reflecting the intuitive expectation of two-particle states havingmass approximately twice as big as one particle. While this is something to be expected ifthe interaction is small enough for perturbation theory to be useful, it cannot be taken forgranted in every circumstances. Also here one better stays open-minded.

40Proof: The imaginary part of the integral exhibits a discontinuity along this line, which


3.2.3 the LSZ reduction formula

We are now going to apply the technique from the previous paragraph to then-point Green function G(x1, . . . , xn). We will consider just a specific region inthe space of xi-variable, namely x0i corresponding to either the remote past orfuture, and ~xi being far apart from each other (in a particular reference frame,of course). For sake of notational convenience (and without a loss of generality)we will assume x01 ≤ x02 ≤ . . . ≤ x0n, with x0i → ±∞, where the upper (lower)sign holds for i ≤ j (i > j) respectively41. In this region the Hamiltonian canbe replaced by the effective one, which happens to be a free one. The basis ofthe free Hamiltonian eigenstates can be taken in the standard form42

|Ω〉 |~p〉 =√2E~p a

+eff~p |Ω〉 |~p, ~p′〉 =

√2E~p a

+eff~p |~p′〉 . . .

Note that we are implicitly assuming only one type of particles here, otherwisewe should make a difference between 1-particle states with the same momentum,but different masses (i.e. to write e.g. |m, ~p〉 and |m′, ~p〉). This corresponds, inthe language of the previous paragraph, to the assumption of a single-valueddiscrete part of the spectrum at ~p = 0, i.e. just one 1-particle state with the

vanishing 3-momentum and non-vanishing matrix element 〈Ω|ϕ (0) |N, ~p〉. It is,however, easy to reinstall other 1-particle states, if there are any.

Our starting point will be an insertion of the unit operator, now in the form43

1 = |Ω〉〈Ω|+∫

d3p1

(2π)3

1

2E~p1

|~p1〉〈~p1|+∫d3p1 d

3p2

(2π)6

1

2E~p12E~p2

|~p1, ~p2〉〈~p2, ~p1|+. . .

between the first two fields. Proceeding just like in the previous case, one gets

G(x1, . . . , xn) = 〈Ω|T ϕ (x1)

ϕ (x2) . . .

ϕ (xn) |Ω〉 =

=

∫d4p1

(2π)4ie−ip1x1

p21 −m2N + iε

〈Ω|ϕ (0) |~p1〉〈~p1|ϕ (x2) . . .ϕ (xn) |Ω〉+ . . .

where we have set vϕ = 0 to keep formulae shorter (it is easy to reinstall vϕ 6= 0,if needed) and the latter ellipses stand for multiple integrals

∫d3p1 . . . d

3pm.The explicitly shown term is the only one with projection on eigenstates witha discrete (in fact just single valued) spectrum of energies at a fixed value of3-momentum. The rest energy of this state is denoted by m and its matrix

element 〈Ω|ϕ (0) |~p〉 = 〈Ω|ϕ (0) |N,~0〉 is nothing but√Z, so one obtains for the

Fourier-like transform (in the first variable) of the Green function

G(p1, x2, . . . , xn) =i√Z

p21 −m2 + iε〈~p1|

ϕ (x2) . . .

ϕ (xn) |Ω〉+ . . .

where the latter ellipses stand for the sum of all the multi-particle continuousspectrum contributions, which does not exhibit poles in the p21 variable (reason-ing is analogous to the one used in the previous paragraph).

follows directly from the famous relation limε→0

∫

dxf(x)x±iε

= ∓iπf(0) + v.p.∫

dxf(x)x

41A note for a cautious reader: these infinite times are ”much smaller” than the infinite timeτ considered when discussing the relation between |Ω〉 and |0〉, nevertheless they are infinite.

42The normalization of the multiparticle states within this basis is different (and bettersuited for what follows) from what we have used for |N, ~p〉 in the previous paragraph.

43We should add a subscript in or out to every basis vector, depending on the place where(actually when) they are inserted (here it should be out). To make formulae more readable,we omit these and recover them only at the end of the day


Provided 3 ≤ j, the whole procedure is now repeated with the unit operator

inserted between the next two fields, leading to 〈~p1|ϕ (x2) . . .

ϕ (xn) |Ω〉 =

= . . .+

∫d3q d3p2

(2π)6

e−ip2x2

2E~q2E~p2

〈~p1|ϕ (0) |~q, ~p2〉〈~p2, ~q| ϕ (x3) . . .

ϕ (xn) |Ω〉+ . . .

with the ellipses now before as well as after the explicitly given term. The former

stand for terms proportional to 〈~p1|ϕ (x2) |Ω〉 and 〈~p1|ϕ (x2) |~q〉 which exhibitno singularities in p2 (because of no 1/2E~p2

present), the latter stand for terms

proportional to 〈~p1|ϕ (x2) |~q, ~p2, . . .〉, which again do not exhibit singularities (inaccord with the reasoning used in the previous paragraph). As to the remainingterm, one can write

〈~p1|ϕ (0) |~q, ~p2〉 =

√2E~q 〈~p1| a+eff

~q

ϕ (0) |~p2〉+

√2E~q 〈~p1|

[ϕ (0) , a+eff

~q

]|~p2〉

and then use 〈~p1| a+eff~q = 〈Ω|

√2E~q (2π)

3δ3 (~p1 − ~q) to get

〈~p1|ϕ (0) |~q, ~p2〉 = 2E~q (2π)

3δ3 (~p1 − ~q) 〈Ω|

ϕ (0) |~p2〉+ . . .

The alert reader have undoubtedly guessed the reason for writing ellipses insteadof the second term: the δ-function in the first term lead to an isolated singularity(namely the pole) in the p22 variable when plugged in the original expression,the absence of the δ-function in the second term makes it less singular (perhapscut), the reasoning is again the one from the previous paragraph — the oldsong, which should be already familiar by now. All by all we have

〈~p1|ϕ (x2) . . .ϕ (xn) |Ω〉 =

∫d3p2

(2π)3

e−ip2x2

2E~p2

√Z 〈~p2, ~p1|

ϕ (x3) . . .ϕ (xn) |Ω〉+ . . .

(recall 〈Ω| ϕ (0) |~p2〉 =

√Z).Writing the d3p integral in the standard d4p form

and performing the Fourier-like transformation in x2, one finally gets

G(p1, p2, x3, . . . , xn) =i√Z

p21 −m2 + iε

i√Z

p22 −m2 + iε〈~p2, ~p1|ϕ (x3) . . .

ϕ (xn) |Ω〉+. . .

Now we are almost finished. Repeating the same procedure over and overand reinstalling the subscripts in and out at proper places, one eventually comesto the master formula

G(p1, . . . , pn) =n∏

k=1

i√Z

p2k −m2 + iεout〈~pr, . . . , ~p1 |~pr+1, . . . , ~pn〉in + . . .

from where

out〈~pr, . . . , ~p1 |~pr+1, . . . , ~pn〉in = limp2k→m2

n∏

k=1

√Z

(iZ

p2k −m2 + iε

)−1

G(p1, . . . , pn)

where the on-shell limit p2k → m2 took care of all the ellipses. In words:

The S-matrix element out〈~pr, . . . , ~p1 |~pr+1, . . . , ~pn〉in is the on-shell limit

of G(p1, . . . , pn) (p-representation Feynman diagrams, no vacuum bubbles)multiplied by the inverse propagator for every external leg (amputation)and by

√Z for every external leg as well (wave-function renormalization)


Remark: If we wanted, we could now easily derive the basic assumption ofthe first look on propagators, namely the equivalence between the renormalizedand the effective Green functions. All we have to do is to take the masterformula for G (the next to the last equation) and Fourier transform it. TheLHS gives the Green function G(x1, x2, . . . , xn) (defined correctly with the non-perturbative vacuum, not the perturbative one as in the first look, which wasa bit sloppy in this respect). On the RHS one writes the S-matrix element

out〈~pr, . . . , ~p1 |~pr+1, . . . , ~pn〉in in terms of the effective creation and annihilationoperators, and then one rewrites the Fourier transform in terms of the effectivefields. Both these steps are straightforward and lead to the effective Green func-tion.All this, however, would be almost pointless. We do not need to repeat the rea-soning of the first look, since we have obtained all we needed within the closerlook. Once the closer look is understood, the first look can well be forgotten, orkept in mind as a kind of motivation.

Remark: The closer look reveals another important point, not visible in thefirst look, namely that one can calculate the S-matrix not only form the Greenfunctions of the fields, but also from the Green functions of any operators Oi (x)

〈Ω|TO1 (x1) . . .On (xn) |Ω〉

The only requirement is the nonvanishing matrix element between the nonper-turbative vacuum and a one-particle state < Ω|Oi (0) |N,~0 > 6= 0. This shouldbe clear from the above reasoning, as should be also the fact that the only changeis the replacement of the wave-function renormalization constant Z by the cor-responding operator renormalization constant

ZOi =∣∣∣〈Ω|Oi (0) |N,~0〉

∣∣∣2

in the master formula for G .

Remark: When formulating the basic assumptions of the closer look, we haveassumed a nonzero mass implicitly. For the zero mass our reasoning would notgo through, because there will be no energy gap between the one-particle andmultiparticle states. We do not have much to say about this unpleasant facthere. Anyway, one has better to be prepared for potential problems when dealingwith massless particles.


the LSZ reduction formula

The Lehman–Symanczik–Zimmerman reduction formula relates the S-matrixelements to the Green functions. It is derived by means of a notoriously knowntrick — inserting the unit operator in a proper form at a proper place.

Let us consider the n-point Green function G(x1, . . . , xn) with ~xi being farapart from each other. In this region the Hamiltonian can be replaced by theeffective one, i.e. by Heff

0 . The basis of the effective Hamiltonian eigenstatescan be taken in the standard form44

|Ω〉 |~p〉 =√2Eeff

~p a+eff~p |Ω〉 |~p, ~p′〉 =

√2Eeff

~p a+eff~p |~p′〉 . . .

Our starting point will be an insertion of the unit operator in the form

1 = |Ω〉〈Ω|+∫

d3p1

(2π)3

1

2Eeff~p1

|~p1〉〈~p1|+∫d3p1 d

3p2

(2π)6

1

2Eeff~p12Eeff

~p2

|~p1, ~p2〉〈~p2, ~p1|+. . .

between the first two fields. For sake of notational convenience (and without aloss of generality) we will assume x01 ≤ x02 ≤ . . . ≤ x0n and write

G(x1, . . . , xn) = 〈Ω|ϕH(x1) |Ω〉〈Ω|ϕH(x2) . . . ϕH(xn) |Ω〉

+

∫d3p1

(2π)31

2Eeff~p1

〈Ω|ϕH(x1) |~p1〉〈~p1|ϕH(x2) . . . ϕH(xn) |Ω〉+ . . .

The main trick which makes this insertion useful, is to express the x-dependenceof the fields via the space-time translations generators Pµ

ϕH (x) = eiPxϕH (0) e−iPx

and then to utilize that |Ω〉, |~p〉, |~p, ~p′〉, . . . are eigenstates of Pµ, i.e. thate−iPx |Ω〉 = |Ω〉 (where we have set EΩ = 0), e−iPx |~p〉 = e−ipx |~p〉 (where

p0 = Eeff~p ), e−iPx |~p, ~p′〉 = e−i(p+p′)x |~p, ~p′〉, etc. In this way one obtains

G(x1, . . . , xn) =

∫d3p1

(2π)3e−ip1x1

2Eeff~p1

〈Ω|ϕH(0) |~p1〉〈~p1|ϕH(x2) . . . ϕH(xn) |Ω〉+ . . .

where the ellipses contain the vacuum and the multiparticle-states contributions.At the first sight the matrix element 〈Ω|ϕH(0) |~p〉 depends on ~p, but in

fact it is a constatnt independent of ~p. To see this one just need to insertanother unit operator, now written as U−1

~p U~p, where U~p is the representation of

the Lorentz boost which transfors the 3-momentum ~p to ~0. Making use of theLorentz invariance of both the nonperturbative vacuum 〈Ω|U−1

~p = 〈Ω| and the

scalar field U~pϕH (0)U−1~p = ϕH (0), one can get rid of the explicit ~p-dependence

in the matrix element45

〈Ω|ϕH(0) |~p〉 = 〈Ω|U−1~p U~pϕH(0)U−1

~p U~p |~p〉 = 〈Ω|ϕH(0)|~0〉44Note that we are implicitly assuming only one type of particles here, otherwise we should

make a difference between 1-particle states with the same momentum, but different masses(i.e. to write e.g. |m, ~p〉 and |m′, ~p〉). It is, however, easy to reinstall other 1-particle states, ifthere are any.

45For higher spins one should take into account nontrivial transformation properties ofthe field components. But the main achievement, which is that one gets rid of the explicit3-momentum dependence, remains unchanged.


The constant 〈Ω|ϕH(0)|~0〉 or rather∣∣∣〈Ω|ϕH(0)|~0〉

∣∣∣2

plays a very important role

in QFT and as such it has its own name and a reserved symbol

Z =∣∣∣〈Ω|ϕH(0)|~0〉

∣∣∣2

and the constant Z is known as the field renormalization constant (for reasonswhich are not important at this moment). With the use of this constant we canrewrite our result as46

G(x1, . . . , xn) =

∫d3p1

(2π)3

√Ze−ip1x1

2Eeff~p1

〈~p1|ϕH(x2) . . . ϕH(xn) |Ω〉+ . . .

If one would, at the very beginning, insert the unit operator between the lasttwo rather than first two fields, the result would be almost the same, the onlydifference would be the change in the sign in the exponent.

Now one proceeds with the insertion of the unit operator over and over again,utill one obtains

G(x1, . . . , xn) =

∫d3p1

(2π)3

√Ze−ip1x1

2Eeff~p1

. . .d3pn

(2π)3

√Zeipnxn

2Eeff~pn

〈~p1 . . . ~pm| ~pm+1 . . . ~pn〉+. . .

Proceeding just like in the previous case, one getsG(x1, . . . , xn) = 〈Ω|T ϕ (x1)

ϕ (x2) . . .

ϕ (xn) |Ω〉 =

=

∫d4p1

(2π)4

ie−ip1x1

p21 −m2N + iε

〈Ω|ϕ (0) |~p1〉〈~p1|ϕ (x2) . . .ϕ (xn) |Ω〉+ . . .

where we have set vϕ = 0 to keep formulae shorter (it is easy to reinstall vϕ 6= 0,if needed) and the latter ellipses stand for multiple integrals

∫d3p1 . . . d

3pm.The explicitly shown term is the only one with projection on eigenstates witha discrete (in fact just single valued) spectrum of energies at a fixed value of3-momentum. The rest energy of this state is denoted by m and its matrix

element 〈Ω|ϕ (0) |~p〉 = 〈Ω|ϕ (0) |N,~0〉 is nothing but√Z, so one obtains for the

Fourier-like transform (in the first variable) of the Green function

G(p1, x2, . . . , xn) =i√Z

p21 −m2 + iε〈~p1|ϕ (x2) . . .

ϕ (xn) |Ω〉+ . . .

where the latter ellipses stand for the sum of all the multi-particle continuousspectrum contributions, which does not exhibit poles in the p21 variable (reason-ing is analogous to the one used in the previous paragraph).

Provided 3 ≤ j, the whole procedure is now repeated with the unit operator

inserted between the next two fields, leading to 〈~p1|ϕ (x2) . . .

ϕ (xn) |Ω〉 =

= . . .+

∫d3q d3p2

(2π)6

e−ip2x2

2E~q2E~p2

〈~p1|ϕ (0) |~q, ~p2〉〈~p2, ~q|

ϕ (x3) . . .

ϕ (xn) |Ω〉+ . . .

46Note that Z is by definition a real number, while√Z is a complex number with some

phase. It is a common habit, however, to treat√Z as a real number too, which usually means

nothing but neglecting an irrelevant overall phase of the matrix element.


with the ellipses now before as well as after the explicitly given term. The former

stand for terms proportional to 〈~p1|ϕ (x2) |Ω〉 and 〈~p1|

ϕ (x2) |~q〉 which exhibit

no singularities in p2 (because of no 1/2E~p2present), the latter stand for terms

proportional to 〈~p1|ϕ (x2) |~q, ~p2, . . .〉, which again do not exhibit singularities (in

accord with the reasoning used in the previous paragraph). As to the remainingterm, one can write

〈~p1| ϕ (0) |~q, ~p2〉 =

√2E~q 〈~p1| a+eff

~q

ϕ (0) |~p2〉+

√2E~q 〈~p1|

[ϕ (0) , a+eff

~q

]|~p2〉

and then use 〈~p1| a+eff~q = 〈Ω|

√2E~q (2π)

3δ3 (~p1 − ~q) to get

〈~p1| ϕ (0) |~q, ~p2〉 = 2E~q (2π)

3 δ3 (~p1 − ~q) 〈Ω| ϕ (0) |~p2〉+ . . .

The alert reader have undoubtedly guessed the reason for writing ellipses insteadof the second term: the δ-function in the first term lead to an isolated singularity(namely the pole) in the p22 variable when plugged in the original expression,the absence of the δ-function in the second term makes it less singular (perhapscut), the reasoning is again the one from the previous paragraph — the oldsong, which should be already familiar by now. All by all we have

〈~p1|ϕ (x2) . . .ϕ (xn) |Ω〉 =

∫d3p2

(2π)3e−ip2x2

2E~p2

√Z 〈~p2, ~p1|

ϕ (x3) . . .ϕ (xn) |Ω〉+ . . .

(recall 〈Ω| ϕ (0) |~p2〉 =

√Z).Writing the d3p integral in the standard d4p form

and performing the Fourier-like transformation in x2, one finally gets

G(p1, p2, x3, . . . , xn) =i√Z

p21 −m2 + iε

i√Z

p22 −m2 + iε〈~p2, ~p1|ϕ (x3) . . .

ϕ (xn) |Ω〉+. . .

Now we are almost finished. Repeating the same procedure over and overand reinstalling the subscripts in and out at proper places, one eventually comesto the master formula

G(p1, . . . , pn) =n∏

k=1

i√Z

p2k −m2 + iεout〈~pr, . . . , ~p1 |~pr+1, . . . , ~pn〉in + . . .

from where

out〈~pr, . . . , ~p1 |~pr+1, . . . , ~pn〉in = limp2k→m2

n∏

k=1

√Z

(iZ

p2k −m2 + iε

)−1

G(p1, . . . , pn)

where the on-shell limit p2k → m2 took care of all the ellipses. In words:

The S-matrix element out〈~pr, . . . , ~p1 |~pr+1, . . . , ~pn〉in is the on-shell limit

of G(p1, . . . , pn) (p-representation Feynman diagrams, no vacuum bubbles)multiplied by the inverse propagator for every external leg (amputation)and by

√Z for every external leg as well (wave-function renormalization)


Remark: If we wanted, we could now easily derive the basic assumption ofthe first look on propagators, namely the equivalence between the renormalizedand the effective Green functions. All we have to do is to take the masterformula for G (the next to the last equation) and Fourier transform it. TheLHS gives the Green function G(x1, x2, . . . , xn) (defined correctly with the non-perturbative vacuum, not the perturbative one as in the first look, which wasa bit sloppy in this respect). On the RHS one writes the S-matrix element

out〈~pr, . . . , ~p1 |~pr+1, . . . , ~pn〉in in terms of the effective creation and annihilationoperators, and then one rewrites the Fourier transform in terms of the effectivefields. Both these steps are straightforward and lead to the effective Green func-tion.All this, however, would be almost pointless. We do not need to repeat the rea-soning of the first look, since we have obtained all we needed within the closerlook. Once the closer look is understood, the first look can well be forgotten, orkept in mind as a kind of motivation.

Remark: The closer look reveals another important point, not visible in thefirst look, namely that one can calculate the S-matrix not only form the Greenfunctions of the fields, but also from the Green functions of any operators Oi (x)

〈Ω|TO1 (x1) . . .On (xn) |Ω〉

The only requirement is the nonvanishing matrix element between the nonper-turbative vacuum and a one-particle state < Ω|Oi (0) |N,~0 > 6= 0. This shouldbe clear from the above reasoning, as should be also the fact that the only changeis the replacement of the wave-function renormalization constant Z by the cor-responding operator renormalization constant

ZOi =∣∣∣〈Ω|Oi (0) |N,~0〉

∣∣∣2

in the master formula for G .

Remark: When formulating the basic assumptions of the closer look, we haveassumed a nonzero mass implicitly. For the zero mass our reasoning would notgo through, because there will be no energy gap between the one-particle andmultiparticle states. We do not have much to say about this unpleasant facthere. Anyway, one has better to be prepared for potential problems when dealingwith massless particles.

Documents

QuantumFieldTheoryI - uniba.sksophia.dtp.fmph.uniba.sk/~mojzis/qft.pdf · QuantumFieldTheoryI Martin Mojˇziˇs. ... We chose An Introduction to Quantum Field Theory by Peskin and