FieldFormulationofMany-Body QuantumPhysicsusers.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-2...nuclei with an even atomic number. Examples for fermions are electrons, neutrinos,

A common mistake that people make

when trying to design something completely foolproof

is to underestimate the ingenuity of complete fools.

Douglas Adams (1952-2001)

2Field Formulation of Many-Body

Quantum Physics

A piece of matter composed of a large number of microscopic particles is called amany-body system. The microscopic particles may either be all identical or of dif-ferent species. Examples are crystal lattices, liquids, and gases, all of these beingaggregates of molecules and atoms. Molecules are composed of atoms which, in turn,consist of an atomic nucleus and electrons, held together by electromagnetic forces,or more precisely their quanta, the photons. The mass of an atom is mostly due tothe nucleus, only a small fraction being due to the electrons and an even smallerfraction due to the electromagnetic binding energy. Atomic nuclei are themselvesbound states of nucleons, held together by mesonic forces, or more precisely theirquanta, the mesons. The nucleons and mesons, finally, consist of the presently mostfundamental objects of nuclear material, called quarks, held together by gluonicforces. Quarks and gluons are apparently as fundamental as electrons and photons.It is a wonderful miracle of nature that this deep hierarchy of increasingly funda-mental particles allows a common description with the help of a single theoreticalstructure called quantum field theory.

As a first step towards developing this powerful theory we shall start from thewell-founded Schrödinger theory of nonrelativistic spinless particles. We show thatthere exists a completely equivalent formulation of this theory in terms of quantumfields. This formulation will serve as a basis for setting up various quantum fieldtheoretical models which can eventually explain the physics of the entire particlehierarchy described above.

2.1 Mechanics and Quantum Mechanicsfor n Distinguishable Nonrelativistic Particles

For a many-body system with only one type of nonrelativistic spinless particles ofmass M , which may be spherical atoms or molecules, the classical Lagrangian hasthe form

L(xν , ẋν ; t) =n∑

ν=1

M

2ẋ2ν − V (x1, . . . ,xn; t), (2.1)

82

2.1 Mechanics and Quantum Mechanics for n Nonrelativistic Particles 83

where the arguments xν , ẋν in L(xν , ẋν ; t) stand, pars pro toto, for all positions xνand velocities ẋν , ν = 1, . . . , n. The general n-body potential V (x1, . . . ,xn; t) canusually be assumed to consist of a sum of an external potential V1(xν ; t) and a pairpotential V2(xν − xµ; t), also called one- and two-body-potentials , respectively:

V (x1, . . . ,xn; t) =∑

ν

V1(xν ; t) +1

2

∑

ν,µ

V2(xν − xµ; t). (2.2)

The second sum is symmetric in µ and ν, so that V2(xν − xµ; t) may be taken asa symmetric function of the two spatial arguments — any asymmetric part wouldnot contribute. The symmetry ensures the validity of Newton’s third law “actio estreactio”. The two-body potential is initially defined only for µ 6= ν, and the sum isrestricted accordingly, but for the development to come it will be useful to includealso the µ = ν -terms into the second sum (2.2), and compensate this by an appro-priate modification of the one-body potential V1(xν ; t), so that the total potential re-mains the same. Such a rearrangement excludes pair potentials V2(xν −xµ; t) whichare singular at the origin, such as the Coulomb potential between point chargesV2(xν − xµ; t) = e2/4π|xν − xµ|. Physically, this is not a serious obstacle sinceall charges in nature really have a finite charge radius. Even the light fundamen-tal particles electrons and muons possess a finite charge radius, as will be seen inChapter 12.

At first we shall consider all particles to be distinguishable. This assumptionis often unphysical and will be removed later. For the particle at xν , the Euler-Lagrange equation of motion that extremizes the above Lagrangian (2.1) reads [recallEq. (1.8)]

M ẍν = −∂V1(xν ; t)

∂xν−∑

µ

∂

∂xνV2(xν − xµ; t). (2.3)

The transition to the Hamiltonian formalism proceeds by introducing the canonicalmomenta [see (1.10)]

pν =∂L

∂ẋν=M ẋν , (2.4)

and forming the Legendre transform [see (1.9)]

H(pν ,xν ; t) =

[

∑

ν

pνẋν − L(xν , ẋν ; t)]

ẋν=pν/M

=∑

ν

p2ν2M

+∑

ν

V1(xν ; t) +1

2

∑

ν,µ

V2(xν − xµ; t). (2.5)

From this, the Hamilton equations of motions are derived as [see (1.17)]

ṗν = {H,pν} = −∂V1(xν ; t)

∂xν−∑

µ

∂

∂xνV2(xν − xµ; t), (2.6)

ẋν = {H,xν} =∂H

∂pν=

pν

M, (2.7)

84 2 Field Formulation of Many-Body Quantum Physics

with {A,B} denoting the Poisson brackets defined in Eq. (1.20):

{A,B} =n∑

ν=1

(

∂A

∂pν

∂B

∂xν− ∂B∂pν

∂A

∂xν

)

. (2.8)

An arbitrary observable F (pν ,xν ; t) changes as a function of time according to theequation of motion (1.19):

dF

dt= {H,F}+ ∂F

∂t. (2.9)

It is now straightforward to write down the laws of quantum mechanics for thesystem. We follow the rules in Eqs. (1.236)–(1.238), and take the local basis

|x1, . . . ,xn〉 (2.10)

as eigenstates of the position operators x̂ν :

x̂ν |x1, . . . ,xn〉 = xν |x1, . . . ,xn〉, ν = 1, . . . , n. (2.11)

They are orthonormal to each other:

〈x1, . . . ,xn|x′1, . . . ,x′n〉 = δ(3)(x1 − x′1) · · · δ(3)(xn − x′n), (2.12)

and form a complete basis in the space of localized n-particle states:

∫

d3x1 · · · d3xn |x1, . . . ,xn〉〈x1, . . . ,xn| = 1. (2.13)

An arbitrary state is denoted by a ket vector and can be expanded in this basis bymultiplying the state formally with the unit operator (2.13), yielding the expansion

|ψ(t)〉 ≡ 1× |ψ(t)〉 =∫

d3x1 · · · d3xn |x1, . . . ,xn〉〈x1, . . . ,xn|ψ(t)〉. (2.14)

The scalar products

〈x1, . . . ,xn|ψ(t)〉 ≡ ψ(x1, . . . ,xn; t) (2.15)

are the wave functions of the n-body system. They are the probability amplitudesfor the particle 1 to be found at x1, particle 2 at x2, etc. .

The Schrödinger equation reads, in operator form,

Ĥ|ψ(t)〉 = H(p̂ν , x̂ν ; t)|ψ(t)〉 = ih̄∂t|ψ(t)〉, (2.16)

where p̂ν are Schrödinger’s momentum operators, whose action upon the wave func-tions is specified by the rule

〈x1, . . . ,xn|p̂ν = −ih̄∂xν 〈x1, . . . ,xn|. (2.17)

2.2 Identical Particles: Bosons and Fermions 85

Multiplication of (2.16) from the left with the bra vectors 〈x1, . . . ,xn| yields via(2.17) the Schrödinger differential equation for the wave functions:

H(−ih̄∂xν ,xν ; t)ψ(x1, . . . ,xn; t)

=

[

−∑

ν

h̄2

2M∂2xν +

∑

ν

V1(xν ; t) +1

2

∑

ν,µ

V2(xν − xµ; t)]

ψ(x1, . . . ,xn; t)

= ih̄∂tψ(x1, . . . ,xn; t). (2.18)

In many physical systems, the potentials V1(xν ; t) and V2(xν−xµ; t) are independentof time. Then we can factor out the time dependence of the wave functions withfixed energy as

ψ(x1, . . . ,xn; t) = e−iEt/h̄ψE(x1, . . . ,xn), (2.19)

and find for ψE(x1, . . . ,xn) the time-independent Schrödinger equation

H(−ih̄∂xν ,xν ; t)ψE(x1, . . . ,xn) = EψE(x1, . . . ,xn). (2.20)

2.2 Identical Particles: Bosons and Fermions

The quantum mechanical rules in the last section were written down in the previ-ous section under the assumption that all particles are distinguishable. For mostrealistic n-body systems, however, this is an unphysical assumption. For example,there is no way of distinguishing the electrons in an atom. Thus not all of thesolutions ψ(x1, . . . ,xn; t) of the Schrödinger equation (2.18) can be physically per-missible. Consider the case where the system contains only one species of identicalparticles, for example electrons. The Hamilton operator in Eq. (2.18) is invariantunder all permutations of the particle labels ν. In addition, all probability ampli-tudes 〈ψ1|ψ2〉 must reflect this invariance. They must form a representation spaceof all permutations. Let Tij be an operator which interchanges the variables xi andxj.

Tijψ(x1, . . . ,xi, . . . ,xj, . . . ,xn; t) ≡ ψ(x1, . . . ,xj, . . . ,xi, . . . ,xn; t). (2.21)

It is called a transposition. The invariance may then be expressed as 〈Tijψ1|Tijψ2〉 =〈ψ1|ψ2〉, implying that the wave functions |ψ1〉 and |ψ2〉 can change at most by aphase:

Tijψ(x1, . . . ,xn; t) = eiαijψ(x1, . . . ,xn; t). (2.22)

But from the definition (2.21) we see that Tij satisfies

T 2ij = 1. (2.23)

Thus eiαij can only have the values +1 or −1. Moreover, due to the identity of allparticles, the sign must be the same for any pair ij.

The set of all multiparticle wave functions can be decomposed into wave functionstransforming in specific ways under arbitrary permutations P of the coordinates.


It will be shown in Appendix 2A that each permutation P can be decomposedinto products of transpositions. Of special importance are wave functions whichare completely symmetric, i.e., which are obtained by applying, to an arbitraryn-particle wave function, the operation

S =1

n!

∑

P

P. (2.24)

The sum runs over all n! permutations of the particle indices. Another importanttype of wave function is obtained by applying the antisymmetrizing operator

A =1

n!

∑

P

ǫPP (2.25)

to the arbitrary n-particle wave function. The symbol ǫP is unity for even permu-tations and −1 for odd permutations. It is called the parity of the permutation.

By applying S or A to an arbitrary n-particle wave function one obtains com-pletely symmetric or completely antisymmetric n-particle wave functions. In nature,both signs can occur. Particles with symmetric wave functions are called bosons,the others fermions. Examples for bosons are photons, mesons, α-particles, or anynuclei with an even atomic number. Examples for fermions are electrons, neutrinos,muons, protons, neutrons, or any nuclei with an odd atomic number.

In two-dimensional multi-electron systems in very strong magnetic fields, aninteresting new situation has been discovered. These systems have wave functionswhich look like those of free quasiparticles on which transpositions Tij yield a phasefactor eiαij which is not equal to ±1: These quasiparticles are neither bosons norfermions. They have therefore been called anyons . Their existence in two dimensionsis made possible by imagining each particle to introduce a singularity in the plane,which makes the plane multisheeted. A second particle moving by 3600 around thissingularity does not arrive at the initial point but at a point lying in a second sheetbelow the initial point. For this reason, the equation (2.23) needs no longer to befulfilled.

Distinguishing the symmetry properties of the wave functions provides us withan important tool to classify the various solutions of the Schrödinger equation (2.18).Let us illustrate this by looking at the simplest nontrivial case in which the particleshave only a common time-independent external potential V1(xν) but are otherwisenoninteracting, i.e., their time-independent Schrödinger equation (2.20) reads

∑

ν

[

− h̄2

2M∂2xν + V1(xν)

]

ψE(x1, . . . ,xn) = EψE(x1, . . . ,xn). (2.26)

It can be solved by the factorizing ansatz in terms of single-particle states ψEαν ofenergy Eαν ,

ψE(x1, . . . ,xn) =n∏

ν=1

ψEαν (xν), (2.27)


with the total energy being the sum of the individual energies:

E =n∑

ν=1

Eαν . (2.28)

The single-particle states ψEαν (xν) are the solutions of the one-particle Schrödingerequation

[

− h̄2

2M∂2x + V1(x)

]

ψEα(x) = EαψEα(x). (2.29)

If the wave functions ψEα(x) form a complete set of states, they satisfy the one-particle completeness relation

∑

α

ψEα(x)ψ∗Eα(x

′) = δ(3)(x− x′). (2.30)

The sum over the labels α may, of course, involve an integral over a continuous partof the spectrum. It is trivial to verify that the set of all products (2.27) is completein the space of n-particle wave functions:

∑

α1,...,αn

ψEα1 (x1) · · ·ψEαn (xn)ψ∗Eαn

(x′n) · · ·ψ∗Eα1 (x′1)

= δ(3)(x1 − x′1) · · · δ(3)(xn − x′n). (2.31)

For identical particles, this Hilbert space is greatly reduced. In the case of bosons,only the fully symmetrized products correspond to physical energy eigenstates. Weapply the symmetrizing operation S of Eq. (2.24) to the product of single-particlewave functions

∏nν=1 ψEαp(ν) (xν) and normalize the result to find

ψS{Eα}(x1, . . . ,xn) = N S{Eα}Sn∏

ν=1

ψEαp(ν) (xν) = NS{Eα}

1

n!

∑

P

n∏

ν=1

ψEαp(ν) (xν). (2.32)

The sum runs over n! permutations of the indices ν = 1, . . . , n, the permuted indicesbeing denoted by p(ν).

Note that ψS{Eα}(x1, . . . ,xn) no longer depends on the order of labels of the en-ergies Eα1 , . . . , Eαn . This is a manifestation of the indistinguishability of the parti-cles in the corresponding one-particle states indicated by the curly-bracket notationψS{Eα}. Also the normalization factor is independent of the order and has been

denoted by N S{Eα}.For fermions, the wave functions are

ψA{Eα}(x1, . . . ,xn) = NA{Eα}An∏

ν=1

ψEαp(ν) (xν) = NA{Eα}

1

n!

∑

P

ǫPn∏

ν=1

ψEαp(ν) (xν), (2.33)

where ǫP has the values ǫP = ±1 for even or odd permutations, respectively.


Instead of the indices on the labels αν of the energies, we may just as wellsymmetrize or antisymmetrize the labels of the position variables:

ψS,A{Eα}(x1, . . . ,xn) = NS,A{Eα}

1

n!

∑

P

{

1ǫP

}

n∏

ν=1

ψEαν (xp(ν)). (2.34)

See Appendix 2A for more details.The completely antisymmetrized products (2.33) can also be written in the form

of a determinant first introduced by Slater:

ψA{Eα}(x1, . . . ,xn) = NA{Eα}1

n!

∣

∣

∣

∣

∣

∣

∣

∣

ψEα1 (x1) ψEα1 (x2) . . . ψEα1 (xn)...

......

ψEαn (x1) ψEαn (x2) . . . ψEαn (xn)

∣

∣

∣

∣

∣

∣

∣

∣

. (2.35)

To determine the normalization factors in (2.32)–(2.35), we calculate the integral∫

d3x1 · · · d3xn ψS ∗{Eα}(x1, . . . ,xn)ψS{Eα}(x1, . . . ,xn)

= N S 2{Eα}1

n!2∑

P,Q

n∏

ν=1

∫

d3xνψ∗Eαp(ν)

(xν)ψEαq(ν) (xν). (2.36)

Due to the group property of permutations, the double-sum contains n! identicalterms with P = Q and can be rewritten as

∫

d3x1 · · ·d3xn ψS ∗{Eα}(x1, . . . ,xn)ψS{Eα}(x1, . . . ,xn)

= N S 2{Eα}1

n!

∑

P

n∏

ν=1

∫

d3xνψ∗Eαν

(xν)ψEαp(ν) (xν). (2.37)

If all single-particle states ψEαν (xν) are different from each other, then only theidentity permutation P = 1 with p(ν) = ν survives, and the normalization integral(2.37) fixes

N S 2{Eα} = n!. (2.38)Suppose now that two of the single-particle wave functions ψEαν , say ψEα1 and ψEα2 ,coincide, while all others are different from these and each other. Then only twopermutations survive in (2.37): those with p(ν) = ν, and those in which P is atransposition Tij of two elements (see Appendix 2A). The right-hand side of (2.38)is then reduced by a factor of 2:

N S 2{Eα} =n!

2. (2.39)

Extending this consideration to n1 identical states ψEα1 , . . . , ψEαn1 , all n1! permuta-

tions among these give equal contributions to the normalization integral (2.37) andlead to

N S 2{Eα} =n!

n1!= 1. (2.40)


Finally it is easy to see that, if there are groups of n1, n2, . . . , nk identical states, thenormalization factor is

N S 2{Eα} =n!

n1!n2! · · ·nk!. (2.41)

For the antisymmetric states of fermion systems, the situation is much simpler.Here, no two states can be identical as is obvious from the Slater determinant (2.35).Similar considerations as in (2.36), (2.37) for the wave functions (2.33) lead to

NA 2{Eα} = n!. (2.42)

The projection into the symmetric and antisymmetric subspaces has the followingeffect upon the completeness relation (2.31): Multiplying it by the symmetrizationor antisymmetrization operators

P̂ S,A =1

n!

∑

P

{

1ǫP

}

, (2.43)

which may be applied upon the particle positions x1, . . . ,xn as in (2.34), we calculate

∑

α1,...,αn

[

N S,A{Eα}]2ψS,A{Eα}(x1, . . . ,xn)ψ

S,A ∗

{Eα}(x′1, . . . ,x

′n) = δ

(3)S,A(x1, . . . ,xn;x′1, . . . ,x

′n),

(2.44)

where the symmetrized or antisymmetrized δ-function is defined by

δ(3)S,A(x1, . . . ,xn;x′1, . . . ,x

′n) ≡

1

n!

∑

P

{

1ǫP

}

δ(3)(x1 − x′p(1)) · · · δ(3)(xn − x′p(n)).

(2.45)

Since the left-hand side of (2.44) is independent of the order of Eα1 , . . . , Eαn , wemay sum only over a certain order among the labels

∑

α1,...,αn

−−−→ n!∑

α1>...>αn

. (2.46)

If there are degeneracy labels in addition to the energy, these have to be ordered inthe same way.

For antisymmetric states, the labels α1, . . . , αn are necessarily different from eachother. Inserting (2.42) and (2.46) into (2.45), this gives directly

∑

α1>...>αn

ψA{Eα}(x1, . . . ,xn)ψA ∗

{Eα}(x′1, . . . ,x

′n) = δ

(3)A(x1, . . . ,xn;x′1, . . . ,x

′n). (2.47)

For the symmetric case we can order the different groups of identical states and de-note their common labels by αn1, αn2 , . . . , αnk , with the numbers nν indicating howoften the corresponding label α is present. Obviously, there are n!/n1!n2! · · ·nk!permutations in the completeness sum (2.44) for each set of labels αn1 > αn2 >


. . . > αnk , the divisions by nν ! coming from the permutations within each groupof n1, . . . , nk identical states. This combinatorial factor cancels precisely the nor-malization factor (2.41), so that the completeness relation for symmetric n-particlestates reads

∑

n1,...,nk

∑

αn1>...>αnk

ψS{Eα}(x1, . . . ,xn)ψS ∗

{Eα}(x′1, . . . ,x

′n)

= δ(3)S(x1, . . . ,xn;x′1, . . . ,x

′n). (2.48)

The first sum runs over the different breakups of the total number n of states intoidentical groups so that n = n1 + . . .+ nk.

It is useful to describe the symmetrization or antisymmetrization procedure di-rectly in terms of Dirac’s bra and ket formalism. The n-particle states are directproducts of single-particle states multiplied by the operator P̂ S,A, and can be writtenas

|ψS,A〉 = P̂ S,A|Eα1〉 · · · |Eαn〉 = N S,A1

n!

∑

P

{

1ǫP

}

|Eαp(1)〉 · · · |Eαp(n)〉. (2.49)

The wave functions (2.32) and (2.33) consist of scalar products of these states withthe localized boson states |x1, . . . ,xn〉, which may be written as direct products

|x1,x2, . . . ,xn〉 = |x1〉 ×⊙ |x2〉 ×⊙ . . . ×⊙ |xn〉. (2.50)

In this state, the particle with number 1 sits at x1, the particle with number 2 at x2,. . . , etc. The symmetrization process wipes out the distinction between the particles1, 2, . . . , n.

Let us adapt the symbolic completeness relation to the symmetry of the wavefunctions. The general relation

∫

d3x1 · · · d3xn |x1〉〈x1| ×⊙ . . . ×⊙ |xn〉〈xn| = 1 (2.51)

covers all square integrable wave functions in the product space. As far as the phys-ical Hilbert space is concerned, it can be restricted as follows:

∫

d3x1 · · · d3xn |x1, . . . ,xn〉S,A S,A〈x1, . . . ,xn| = P̂ S,A, (2.52)

where

|x1, . . . ,xn〉S,A =1

n!

∑

P

{

1ǫP

}

|xp(1), . . . ,xp(n)〉. (2.53)

The states are orthonormal in the sense

S,A〈x1, . . . ,xn|x′1, . . . ,x′n〉S,A = δ(3)S,A(x1, . . . ,xn;x′1, . . . ,x′n). (2.54)

This basis will play an essential role for the introduction of quantum fields.

2.3 Creation and Annihilation Operators for Bosons 91

While the formalism presented so far is applicable to any number of particles,practical calculations usually present a tremendous task. The number of particles isoften so large, of the order 1023, that no existing computer could even list the wavefunctions. On the other hand, macroscopic many-body systems containing such alarge number of microscopic particles make up our normal environment, and ourexperience teaches us that many global phenomena can be predicted quite reliably.They should therefore also be calculable in simple terms. For example, for mostpurposes a crystal follows the laws of a rigid body, and nothing in theses laws recordsthe immense number of degrees of freedom inherent in a microscopic description.If the solid is excited, there are sound waves in which all the many atoms in thelattice vibrate around their equilibrium positions. Their description requires only afew bulk parameters such as elastic constants and mass density. Phenomena of thistype are called collective phenomena.

To describe such phenomena, an economic way had to be found which doesnot require the solution of the Schrödinger differential equation with 3n ∼ 1023coordinates. We shall later see that field theory provides us with an elegant andefficient access to such phenomena. After a suitable choice of field variables, simplemean-field approximations will often give a rough explanation of many collectivephenomena. In the subsequent sections we shall demonstrate how the Schrödingertheory of any number of particles can be transformed into the quantum field theoryof a single field.

There is further motivation at a more fundamental level for introducing fields.They offer a natural way of accounting for the symmetry properties of the wavefunctions, as we shall now see.

2.3 Creation and Annihilation Operators for Bosons

When dealing with n-particle Schrödinger equations, the imposition of symmetryupon the Schrödinger wave functions ψ(x1, . . . ,xn; t) seems to be a rather artificialprocedure. There exists an alternative formulation of the quantum mechanics ofn particles in which the Hilbert space automatically carries the correct symmetry.This formulation may therefore be viewed as a more “natural” description of suchquantum systems. The basic mathematical structure which will serve this purposewas first encountered in a particular quantum mechanical description of harmonicoscillators which we now recall. It is well-known that the Hamilton operator of anoscillator of unit mass

Ĥ =1

2p̂2 +

ω2

2q̂2 (2.55)

can be rewritten in the form

Ĥ = h̄ω(

â†â+1

2

)

, (2.56)

where

â† =

√ωq̂ − ip̂/√ω√

2h̄, â =

√ωq̂ + ip̂/

√ω√

2h̄(2.57)


are the so-called raising and lowering operators. The canonical quantization rules

[p̂, p̂] = [x̂, x̂] = 0,

[p̂, x̂] = −ih̄ (2.58)

imply that â, â† satisfy

[â, â] = [â†, â†] = 0,

[â, â†] = 1. (2.59)

The energy spectrum of the oscillator follows directly from these commutation rules.We introduce the number operator

N̂ = â†â, (2.60)

which satisfies the equations[N̂, â†] = â†, (2.61)

[N̂ , â] = −â. (2.62)These imply that â† and â raise and lower the eigenvalues of the number operatorN̂ by one unit, respectively. Indeed, if |ν〉 is an eigenstate with eigenvalue ν,

N̂ |ν〉 = ν|ν〉, (2.63)

we see that

N̂ â†|ν〉 = (â†N̂ + â†)|ν〉 = (ν + 1)â†|ν〉, (2.64)N̂ â|ν〉 = (âN̂ − â)|ν〉 = (ν − 1)â|ν〉. (2.65)

Moreover, the eigenvalues ν must all be integer numbers n which are larger or equalto zero. To see this, we observe that â†â is a positive operator. It satisfies for everystate |ψ〉 in Hilbert space the inequality

〈ψ|â†â|ψ〉 = ||â†|ψ〉||2 ≥ 0. (2.66)

Hence there exists a state, usually denoted by |0〉, whose energy cannot be loweredby one more application of â. This state will satisfy

â|0〉 = 0. (2.67)

As a consequence, the operator N̂ applied to |0〉 must be zero. Applying the raisingoperator â† any number of times, the eigenvalues ν of N̂ will cover all integer numbersν = n with n = 0, 1, 2, 3, . . . . The corresponding states are denoted by |n〉:

N̂ |n〉 = n|n〉, n = 0, 1, 2, 3, . . . . (2.68)

Explicitly, these states are given by

|n〉 = Nn(â†)n|0〉, (2.69)

2.3 Creation and Annihilation Operators for Bosons 93

with some normalization factor Nn, that can be calculated using the commutationrules (2.59) to be

Nn =1√n!. (2.70)

By considering the commutation rules (2.59) between different states |n〉 and insert-ing intermediate states, we derive the matrix elements of the operators â, â†:

〈n′|â|n〉 = √n δn′,n−1, (2.71)〈n′|â†|n〉 =

√n + 1 δn′,n+1. (2.72)

In this way, all properties of the harmonic oscillator are recovered by purely algebraicmanipulations, using (2.59) with the condition (2.67) to define the ground state.

This mathematical structure can be used to describe the complete set of sym-metric localized states (2.32). All we need to do is reinterpret the eigenvalue n ofthe operator N̂ . In the case of the oscillator, n is the principal quantum numberof the single-particle state that counts the number of zeros in the Schrödinger wavefunction. In quantum field theory, the operator changes its role and its eigenvaluesn count the number of particles contained in the many-body wave function. Theoperators â† and â which raise and lower n are renamed creation and annihilationoperators, which add or take away a single particle in the state |n〉. The groundstate |0〉 contains no particle. It constitutes the vacuum state of the n-body system.In the states (2.32), there are n particles at places x1, . . . ,xn. We therefore intro-duce the spatial degree of freedom by giving â†, â a spatial label and defining theoperators

â†x, âx,

which permit the creation and annihilation of a particle localized at the position x.1

The operators at different locations are taken to be independent, i.e., they com-mute as

[âx, âx′] = [â†x, â

†x′] = 0, (2.73)

[âx, â†x′ ] = 0, x 6= x′. (2.74)

The commutation rule between â†x and âx′ for coinciding space variables x and x′ is

specified with the help of a Dirac δ-function as follows:

[âx, â†x′] = δ

(3)(x− x′). (2.75)We shall refer to these x-dependent commutation rules as the local oscillator algebra.

The δ-function singularity in (2.75) is dictated by the fact that we want topreserve the raising and lowering commutation rules (2.61) and (2.62) for the particlenumber at each point x, i.e., we want that

[N̂, â†x] = â†x, (2.76)

1The label x in configuration space of the particles bears no relation to the operator q̂ in theHamiltonian (2.55), which here denotes an operator in field space, as we shall better understandin Section 2.8.


[N̂, âx] = −âx. (2.77)The total particle number operator is then given by the integral

N̂ =∫

d3x â†xâx. (2.78)

Due to (2.74), all parts in the integral (2.78) with x′ different from the x in (2.76)and (2.77) do not contribute. If the integral is supposed to give the right-handsides in (2.76) and (2.77), the commutator between âx and â

†x has to be equal to a

δ-function.The use of the δ-function is of course completely analogous to that in Subsec-

tion 1.4. [recall the limiting process in Eq. (1.160)]. In fact, we could have introducedlocal creation and annihilation operators with ordinary unit commutation rules ateach point by discretizing the space into a fine-grained point-lattice of a tiny latticespacing ǫ, with discrete lattice points at

xn = (n1, n2, n3)ǫ, nν = 0,±1,±2, . . . . (2.79)

And for the creation or annihilation of a particle in the small cubic box around xn wecould have introduced the operators â†n or ân, which satisfy the discrete commutationrules

[ân, ân′] = [â†n, â

†n′] = 0, (2.80)

[ân, â†n′ ] = δ

(3)nn′ . (2.81)

For these the total particle number operator is

N̂ =∑

n

â†nân. (2.82)

This would amount to identifying ân with a discrete subset of the continuous set ofoperators âx as follows:

ân =√ǫ3 âx

∣

∣

∣

x≡xn. (2.83)

Then the discrete and continuous formulations of the particle number operator wouldbe related by

N̂ =∑

n

â†nân ≡ ǫ3∑

xn

â†xn âxn −−−→ǫ→0∫

d3x â†xâx. (2.84)

In the same limit, the commutator

1

ǫ3[ân, â

†n′] ≡ [âxn , â†xn’ ] =

1

ǫ3δ(3)nn′ (2.85)

would tend to δ(3)(x− x′), which can be seen in the same way as in Eq. (1.160).We are now ready to define the vacuum state of the many-particle system. It is

given by the unique state |0〉 of the local oscillator algebra (2.73) and (2.74), whichcontains no particle at all places x, thus satisfying

âx|0〉 ≡ 0, 〈0|â†x ≡ 0. (2.86)

2.4 Schrödinger Equation for Noninteracting Bosons in Terms of Field Operators 95

It will always be normalized to unity:

〈0|0〉 = 1. (2.87)

We can now convince ourselves that the fully symmetrized Hilbert space of all lo-calized states of n particles may be identified with the states created by repeatedapplication of the local creation operators â†x:

|x1, . . . ,xn〉S ≡1√n!â†x1 · · · â†xn |0〉. (2.88)

The such-generated Hilbert space will be referred to as the second-quantized Hilbertspace for reasons to be seen below. It decomposes into a direct sum of n-particlesectors. The symmetry of these states in the position variables is obvious, due tothe commutativity (2.73) of all â†xν , â

†xµ

among each other.Let us verify that the generalized orthonormality relation is indeed fulfilled by

the single-particle states. Using the local commutation rules (2.73), (2.74), and thedefinition of the vacuum state (2.86), we calculate for a single particle

S〈x|x′〉S = 〈0|âxâ†x′ |0〉= 〈0|δ(3)(x− x′) + â†x′ âx|0〉= δ(3)(x− x′). (2.89)

For two particles we find

S〈x1,x2|x′1,x′2〉S =1

2!〈0|âx2âx1 â†x′1 â

†x′2|0〉

=1

2!

[

δ(3)(x1 − x′1)〈0|âx2 â†x′2 |0〉+ 〈0|âx2â†x′1âx1â

†x′2|0〉]

=1

2!

[

δ(3)(x1 − x′1)δ(3)(x2 − x′2) + δ(3)(x2 − x′1)δ(3)(x1 − x′2)]

= δ(3)S(x1,x2;x′1,x

′2). (2.90)

The generalization to n particles is straightforward, although somewhat tedious. Itis left to the reader as an exercise. Later in Section 7.17.1, rules will be derived in adifferent context by a procedure due to Wick, which greatly simplifies calculationsof this type.

2.4 Schrödinger Equation for Noninteracting Bosonsin Terms of Creation and Annihilation Operators

Expressing the localized states in terms of the local creation and annihilation oper-ators â†x, âx does not only lead to an automatic symmetrization of the states. It alsobrings about an extremely simple unified form of the Schrödinger equation, whichdoes not require the initial specification of the particle number n, as in Eq. (2.18).


This will now be shown for the case of identical particles with no two-body interac-tions V2(xν − xµ; t).

In order to exhibit the unified Schrödinger equation for any number of particles,let us first neglect interactions and consider only the motion of the particles in anexternal potential with the Schrödinger equation:

{

∑

ν

[

− h̄2

2M∂2xν + V1(xν ; t)

]}

ψ(x1, . . . ,xn; t) = ih̄∂tψ(x1, . . . ,xn; t). (2.91)

We shall now demonstrate that the â†, â -form of this equation, which is valid forany particle number n, reads

Ĥ(t)|ψ(t)〉 = ih̄∂t|ψ(t)〉, (2.92)

where Ĥ(t) is simply the one-particle Hamiltonian sandwiched between creation andannihilation operators â†x and âx and integrated over x, i.e.,

Ĥ(t) =∫

d3xâ†x

[

− h̄2

2M∂x

2 + V1(x; t)

]

âx. (2.93)

The operator (2.93) is called the second-quantized Hamiltonian, equation (2.92) thesecond-quantized Schrödinger equation, and the state |ψ(t)〉 is an arbitrary n-particlestate in the second-quantized Hilbert space, generated by multiple application of â†xupon the vacuum state |0〉, as described in the last section. The operator nature ofâx, â

†x accounts automatically for the many-body content of Eq. (2.92).This statement is proved by multiplying Eq. (2.92) from the left with

S〈x1, . . . ,xn| =1√n!〈0|âxn · · · âx1 ,

which leads to

1√n!〈0|âxn · · · âx1Ĥ(t)|ψ(t)〉 = ih̄∂t

1√n!〈0|âxn · · · âx1 |ψ(t)〉. (2.94)

Here we make use of the property (2.86) of the vacuum state to satisfy 〈0|â†x = 0.As a consequence, we may rewrite the left-hand side of (2.94) with the help of acommutator as

1√n!〈0|[âxn · · · âx1 , Ĥ]|ψ(t)〉.

This commutator is easily calculated using the operator chain rules

[Â, B̂Ĉ] = B̂[Â, Ĉ] + [Â, B̂]Ĉ, [ÂB̂, Ĉ] = Â[B̂, Ĉ] + [Â, Ĉ]B̂. (2.95)

These rules can easily be memorized, noting that their structure is exactly thesame as in the Leibnitz rule for derivatives. In the first rule we may imagine A tobe a differential operator applied to the product BC, which is evaluated by first

2.5 Second Quantization and Symmetrized Product Representation 97

differentiating B, leaving C untouched, and then C, leaving B untouched. In thesecond rule we imagine C to be a differential operator acting similarly to the leftupon the product AB. Generalizing this rule to products of more than two operatorswe derive

[âxn · · · âx1 , â†yâz] = [âxn · · · âx1 , â†y]âz + â†y[âxn · · · âx1 , âz]= âxn · · · âx3 âx2 [âx1 , â†y]âz + [âxn · · · âx3 âx2 , â†y]âx1 âz + . . .=

∑

ν

δ(3)(xν − y)âxn · · · âxν+1âzâxν−1 · · · âx1 .

Multiplying both sides by

δ(3)(y − z)[

−h̄2∂z2/2M + V1(z; t)]

,

and integrating over d3y d3z using (2.93) and (2.75) we find

[âxn · · · âx1, Ĥ(t)] =∑

ν

[

− h̄2

2M∂2xν + V1(xν ; t)

]

âxn · · · âxν+1 âxν âxν−1 · · · âx1 , (2.96)

so that (2.94) becomes

∑

ν

[

− h̄2

2M∂2xν + V1(xν ; t)

]

1√n!〈0|âxn · · · âx1|ψ(t)〉=ih̄∂t

1√n!〈0|âxn · · · âx1 |ψ(t)〉,

(2.97)which is precisely the n-body Schrödinger equation (2.91) for the wave function

ψ(x1, . . . ,xn; t) ≡1√n!〈0|âxn · · · âx1 |ψ (t)〉. (2.98)

2.5 Second Quantization and Symmetrized ProductRepresentation

It is worth pointing out that the mathematical structure exploited in the process ofsecond quantization is of a very general nature.

Consider a set of matrices Mi with indices α′, α

(Mi)α′α

which satisfy some matrix commutation rules, say

[Mi,Mj] = ifijkMk. (2.99)

Let us sandwich these matrices between creation and annihilation operators whichsatisfy

[âα, âα′ ] = [â†α, â

†α′ ] = 0,

[âα, â†α′ ] = δαα′ , (2.100)


and form the analogs of “second-quantized operators” (2.93) by defining

M̂i ≡ â†α′(Mi)α′αâα = â†Miâ. (2.101)

In expressions of this type, repeated indices α, α′ imply a summation over all α,α′. This is commonly referred to as Einstein’s summation convention. On the right-hand side of (2.101) we have suppressed the indices α, α′ completely, for brevity. Itis now easy to verify, using the operator chain rules (2.95), that the operators M̂isatisfy the same commutation rules as the matrices Mi:

[

M̂i, M̂j]

=[

â†Miâ, â†Mj â

]

=[

â†Miâ, â†]

Mj â+ â†Mj

[

â†Miâ, â]

= â†MiMj â− â†MjMiâ= â† [Mi,Mj ] â = ifijkâ

†Mkâ = ifijkM̂k. (2.102)

Thus the “second-quantized” operators M̂i generate an operator representation ofthe matrices Mi. They can be sandwiched between states in the “second-quantized”Hilbert space generated by applying products of creation operators â†n upon thevacuum state |0〉. Thereby they are mapped into an infinite-dimensional matrixrepresentation. On each subspace spanned by the products of a fixed number ofcreation operators, they generate the symmetrized part of the direct product repre-sentation.

The action of the “second-quantized” operators M̂i upon the large Hilbert spaceis very simple to calculate. The only commutation rules required are

[

M̂i, â†α

]

= â†α′(Mi)α′α,[

âα′ , M̂i]

= (Mi)α′αâα. (2.103)

From this property we calculate directly the action upon “single-particle states”:

M̂iâ†α|0〉 =

[

M̂i, â†α

]

|0〉 = â†α′ |0〉(Mi)α′α,〈0|âα′M̂i = 〈0|

[

âα′ , M̂i]

= (Mi)α′α〈0|âα. (2.104)

Thus the states â†α|0〉 span an invariant subspace and are transformed into eachother via the matrix (Mi)α′α.

Consider now a state with two particles:

â†α1 â†α2|0〉. (2.105)

Applying M̂i to this state yields[

M̂i, â†α1â†α2

]

=[

M̂i, â†α1

]

â†α2 + â†α1

[

M̂i, â†α2

]

= â†α′1â†α′2

[

(Mi)α′1α1δα′2α2 + δα′1α1(Mi)α′2α2]

. (2.106)

2.5 Second Quantization and Symmetrized Product Representation 99

Multiplying Eq. (2.106) by |0〉 from the right, we find the transformation law for thetwo-particle states â†α1 â

†α2|0〉. They are transformed via the representation matrices

(Mi)α′1α′2,α1α2 = (Mi)α′1α1δα′2α2 + δα′1α1(Mi)α′2α2 . (2.107)

Omitting the indices, we may also write the matrices as

M(2)i =Mi × 1 + 1×Mi, (2.108)

which is the well-known way of forming representations of a matrix algebra in adirect product space.

Since a†α1 and a†α2

commute with each other, the invariant space constructed inthis way contains only symmetric tensors, and only the symmetrized part of thematrices (Mi)α′1α′2,α1α2 contribute.

The alert reader will have realized that the same operator structure can be ob-tained for the antisymmetrized parts of the matrices (Mi)α′1α′2,α1α2 by using creation

and annihilation operators a†α1 and aα2 which satisfy the fermionic version of thealgebraic rules (2.100):

{âα, âα′} = {â†α, â†α′} = 0,{âα, â†α′} = δαα′ . (2.109)

In these, curly brackets are used to abbreviate anticommutators

{A,B} ≡ AB +BA. (2.110)The first two lines in (2.102) are unchanged since the operator chain rules (2.95)hold for both Bose and Fermi operators. To derive the third line we must use theadditional rules

[Â, B̂Ĉ] = {Â, B̂}Ĉ − B̂{Â, Ĉ},[ÂB̂, Ĉ] = Â{B̂, Ĉ} − {Â, Ĉ}B̂. (2.111)

This fermionic version of the commutation relation (2.102) will form, in Section 2.10,the basis for constructing a second-quantized representation for the n-particle wavefunctions and Schrödinger operators of fermions.

If Mi are chosen to be representation matrices Li of the generators of the rota-tion group (to be discussed in detail in Section 4.1), the law (2.108) represents thequantum mechanical law of addition of two angular momenta:

L(2)i = Li × 1 + 1× Li.

The generalization to any number of angular momenta is obvious.Incidentally, any operator which satisfies the same commutation rules with M̂i,

as âα in (2.103), i.e., satisfies the commutation rules[

M̂i, Ô†α

]

= Ô†α′(Mi)α′α,[

Ôα′ , M̂i]

= (Mi)α′αÔα, (2.112)


will be referred to as a spinor operator. Generalizing this definition, an opera-tor Ôα1α2...αn which commutes with M̂i like a product Ôα1Ôα2 · · · Ôαn , generalizingEq. (2.106), is called a multispinor operator of rank n.

Another type of operators which frequently occurs in quantum mechanics andquantum field theory is a vector operator. This is any operator Ôj which commutes

with M̂i in the same way as M̂j does in (2.102), i.e.,[

M̂i, Ôj]

= ifijkÔk. (2.113)

Its generalization Ôj1j2...jn , which commutes with M̂i as the product of operators

M̂j1M̂j2 . . . M̂jn , is called a tensor operator of rank n.The many-particle version of the Schrödinger theory is obtained if we view the

one-particle Schrödinger equation

H(−ih̄∂x,x)ψ (x, t) = ih̄∂tψ (x, t) (2.114)as a matrix equation in the discretized x-space with the lattice positions x =(n1, n2, n3)ǫ, so that the wave functions ψ (x, t) correspond to vectors ψn(t). Thenthe differential operator ∂iψ(x) becomes simply ∇iψ(x) = [ψ (x + iǫ)− ψ(x)] /ǫ =[ψn+i − ψn] /ǫ where i is the unit vector in the ith direction, and ǫ the lattice spacing.The Laplacian may be viewed as the continuum limit of the matrix

∇̄i∇iψ(x) =1

ǫ2

3∑

i=1

[ψ (x+ iǫ)− 2ψ(x) + ψ (x− iǫ)]

=1

ǫ2

3∑

i=1

[ψn+i − 2ψn + ψn−i] . (2.115)

The Schrödinger equation (2.114) is then the ǫ→ 0 -limit of the matrix equationHnn′ψn′(t) = ih̄∂tψn(t). (2.116)

The many-particle Schrödinger equation in second quantization form reads

Ĥ|ψ(t)〉 = ih̄∂t|ψ(t)〉, (2.117)with the Hamiltonian operator

Ĥ = â†n′Hn′nân. (2.118)

It can be used to find the eigenstates in the symmetrized multispinor representationspace spanned by

â†n1 . . . â†nN

|0〉. (2.119)

Applying Ĥ to it we see that this state is multiplied from the right by a direct-product matrix

H × 1× . . .× 1 + 1×H × . . .× 1 + . . . 1× 1× . . .×H. (2.120)Due to this very general relation, the Schrödinger energy of a many-body systemwithout two- or higher-body interactions is the sum of the one-particle energies.

2.6 Bosons with Two-Body Interactions 101

2.6 Bosons with Two-Body Interactions

We now include two-body interactions. For simplicity, we neglect the one-bodypotential V1(x; t) which can be added at the end and search for the second-quantizedform of the Schrödinger equation[

−∑

ν

h̄2

2M∂2xν +

1

2

∑

µ,ν

V2(xν − xµ; t)]

ψ(x1, . . . ,xn; t) = ih̄∂tψ(x1, . . . ,xn; t). (2.121)

It is easy to see that such a two-body potential can be introduced into the second-quantized Schrödinger equation (2.92) by adding to the Hamilton operator in (2.93)the interaction term

Ĥint(t) =1

2

∫

d3xd3x′â†xâ†x′V2(x− x′; t)âx′ âx. (2.122)

To prove this we work out the expectation value

1√n!〈0|âxn . . . âx1Ĥint(t)|ψ(t)〉 =

1√n!〈0|[âxn . . . âx1, Ĥint(t)]|ψ(t)〉. (2.123)

We do this by using the local commutation rules (2.73), (2.75), and the vacuumproperty (2.86). First we generalize Eq. (2.96) to

[âxn · · · âx1 , â†y2â†y1âz1âz2]

= [âxn · · · âx1 , â†y2 â†y1]âz1âz2 + â

†y2â†y1[âxn · · · âx1 , âz1âz2]

= [âxn · · · âx1 , â†y2 ]â†y1âz1âz2 + â

†y2[âxn · · · âx1 , â†y1 ]âz1âz2

=∑

ν

δ(3)(xν − y2)âxn · · · âxν+1 âxν−1 · · · âx1â†y1 âz1âz2

+ â†y2

∑

ν

δ(3)(xν − y1)âxn · · · âxν+1 âxν−1 · · · âx1 âz1 âz2. (2.124)

The second piece does not contribute to Eq. (2.123) since â†y2 annihilates the vacuumon the left. For the same reason, the first piece can be written as

∑

ν

δ(3)(xν − y2)[âxn · · · âxν+1 âxν−1 · · · âx1, â†y1 ]âz1âz2, (2.125)

as long as it stands to the right of the vacuum. Using the commutation rule (2.96),this leads to

〈0|âxn · · · âx1 â†y2 â†y1âz1 âz2 |ψ(t)〉 =

∑

µ,ν

δ(3)(xµ − y1)δ(3)(xν − y2)

×〈0|âxn · · · âxν+1 âxν−1 · · · âxµ+1âxµ−1 . . . âx1 âz1 âz2|ψ(t)〉. (2.126)

After multiplying this relation by V2(y2 − y1; t)δ(3)(y1 − z1)δ(3)(y2 − z2)/2, andintegrating over d3y1 d

3y2 d3z1 d

3z2, we find

〈0|âxn · · · âx1Ĥint|ψ(t)〉 =1

2

∑

µ,ν

V2(xν − xµ; t)〈0|âxn · · · âx1 |ψ (t) 〉, (2.127)


which is precisely the two-body interaction in the Schrödinger equation (2.121).Adding now the one-body interactions, we see that an n-body Schrödinger equa-

tion with arbitrary one- and two-body potentials can be written in the form of asingle operator Schrödinger equation

Ĥ(t)|ψ(t)〉 = ih̄∂t|ψ(t)〉, (2.128)

with the second-quantized Hamilton operator

Ĥ(t) =∫

d3xâ†x

[

− h̄2

2M∂x

2 + V1(x; t)

]

âx +1

2

∫

d3x d3x′ â†xâ†x′V2(x− x′; t)âx′ âx.

(2.129)

The second-quantized Hilbert space of the states |ψ(t)〉 is constructed by repeatedmultiplication of the vacuum vector |0〉 with particle creation operators â†x. Theorder of the creation and annihilation operators in this Hamiltonian is such that thevacuum, as a zero-particle state, has zero energy:

Ĥ(t)|0〉 = 0, 〈0|Ĥ(t) = 0, (2.130)

as in the original Schrödinger equation.A Hamiltonian which is a spatial integral over a Hamiltonian density H(x) as

H =∫

d3xH(x), (2.131)

is called a local Hamiltonian. In (2.129), the free part is local, but the interactingpart is not. It consists of an integral over two spatial variables, thus forming a bilocaloperator.

2.7 Quantum Field Formulation of Many-BodySchrödinger Equations for Bosons

The annihilation operator âx can now be used to define a time-dependent quantumfield ψ̂(x, t) as being the Heisenberg picture of the operator âx (which itself is alsoreferred to as the Schrödinger picture of the annihilation operator). According toEq. (1.285), the Heisenberg operator associated with âx is

axH(t) ≡ [Û(t, ta)]−1âxÛ(t, ta). (2.132)

Thus we defineψ̂(x, t) ≡ axH(t). (2.133)

Choosing the time variable ta = 0, the quantum field ψ̂(x, t) coincides with âx att = 0:

ψ̂(x, 0) ≡ âx. (2.134)

2.7 Quantum Field Formulation of Many-Body Schrödinger Equations for Bosons 103

The time dependence of ψ̂(x, t) is ruled by Heisenberg’s equation of motion (1.280):

∂tψ̂(x, t) =i

h̄[ĤH(t), ψ̂(x, t)]. (2.135)

For simplicity, we shall at first assume the potentials to have no explicit time de-pendence, an assumption to be removed later. Then Eq. (2.135) is solved by

ψ̂(x, t) = eiĤt/h̄ψ̂(x, 0)e−iĤt/h̄ = eiĤt/h̄âxe−iĤt/h̄. (2.136)

The Hermitian conjugate of this determines the time dependence of the Heisenbergpicture of the creation operator:

ψ̂†(x, t) = eiĤt/h̄ψ̂†(x, 0)e−iĤt/h̄ = eiĤt/h̄â†xe−iĤt/h̄. (2.137)

At each given time t, the field ψ̂(x, t) fulfills the same commutation rules (2.73) and(2.74) as âx:

[ψ̂(x, t), ψ̂(x′, t)] = 0,

[ψ̂†(x, t), ψ̂†(x′, t)] = 0, (2.138)

[ψ̂(x, t), ψ̂†(x′, t)] = δ(3)(x− x′).

Consider now the Hamiltonian operator (2.129) in the Heisenberg representation.Under the assumption of no explicit time dependence in the potentials we may simplymultiply it by eiĤt/h̄ and e−iĤt/h̄ from the left and right, respectively, and see that

ĤH(t) =∫

d3x ψ̂†(x, t)

[

− h̄2

2M∂x

2 + V1(x)

]

ψ̂(x, t)

+1

2

∫

d3xd3x′ ψ̂†(x, t)ψ̂†(x′, t)V2(x− x′)ψ̂(x′, t)ψ̂(x, t). (2.139)

Since Ĥ commutes with itself, the operator ĤH(t) is time independent, so that

ĤH(t) ≡ Ĥ. (2.140)

The important point about the expression (2.139) for Ĥ is now that by contain-ing the time-dependent fields ψ̂(x, t), it can be viewed as the Hamilton operator of acanonically quantized Heisenberg field. This is completely analogous to the Hamil-tonian operator Ĥ ≡ H(p̂H(t), x̂H(t), t) in (1.278). Instead of pH(t) and qH(t), weare dealing here with generalized coordinates and their canonically conjugate mo-menta of the field system. They consist of the Hermitian and anti-Hermitian partsof the field, ψ̂R(x, t) and ψ̂I(x, t), defined by

ψ̂R ≡(ψ̂ + ψ̂†)√

2, ψ̂I ≡

(ψ̂ − ψ̂†)√2i

. (2.141)


They commute like

[ψ̂I(x, t), ψ̂R(x′, t)] = −iδ(3)(x− x′),

[ψ̂I(x, t), ψ̂I(x′, t)] = 0, (2.142)

[ψ̂R(x, t), ψ̂R(x′, t)] = 0.

These commutation rules are structurally identical to those between the quasi-Cartesian generalized canonical coordinates q̂iH(t) and p̂iH(t) in Eq. (1.97).

In fact, the formalism developed there can be generalized to an infinite set ofcanonical variables labeled by the space points x rather than i, i.e., to canonicalvariables px(t) and qx(t). Then the quantization rules (1.97) take the form

[p̂x(t), q̂x′(t)] = −ih̄δ(3)(x− x′),[p̂x(t), p̂x′(t)] = 0, (2.143)

[q̂x(t), q̂x′(t)] = 0,

which is a local version of the algebra (2.58). The replacement i→ x can of coursebe done on a lattice with a subsequent continuum limit as in Eqs. (2.79)–(2.85).When going from the index i to the continuous spatial variable x, the Kronecker δijturns into Dirac’s δ(3)(x− x′), and sums become integrals.

By identifying

p̂x(t) ≡ h̄ψ̂I(x, t), q̂x(t) ≡ ψ̂R(x, t), (2.144)we now obtain the commutation relations (2.142). In quantum field theory it iscustomary to denote the canonical momentum variable px(t) by the symbol πx(t),and write

p̂x(t) = h̄ψ̂I(x, t) ≡ π̂(x, t). (2.145)

Thus the many-body nature of the system may be considered as a consequenceof quantizing the fields qx(t) = ψ̂R(x, t) and p̂x(t) = h̄ψ̂I(x, t) canonically viaEq. (2.143).

2.8 Canonical Formalism in Quantum Field Theory

So far, the commutation rules have been imposed upon the fields ψ̂(x, t) and ψ̂†(x′, t)by the particle nature of the n-body Schrödinger theory. It is, however, possible toderive these rules by applying the standard canonical formalism to the fields ψR(x, t)and ψI(x, t), treating them as generalized Lagrange coordinates. To see this, let usrecall once more the general procedure for finding the quantization rules and theSchrödinger equation for a general Lagrangian system with an action

A =∫

dt L(q(t), q̇(t)), (2.146)

2.8 Canonical Formalism in Quantum Field Theory 105

where the Lagrangian L is some function of the independent variables q(t) =(q1(t), . . . , qN(t)) and their velocities q̇(t) = (q̇1(t), . . . , q̇N(t)). The conjugate mo-menta are defined, as usual, by the derivatives

pi(t) =∂L

∂q̇i(t). (2.147)

The Hamiltonian is given by the Legendre transformation

H(p(t), q(t)) =∑

i

pi(t)qi(t)− L(q(t), q̇(t)). (2.148)

If q(t) are Cartesian or quasi-Cartesian coordinates, quantum physics is imposed inthe Heisenberg picture by letting pi(t), qi(t) become operators p̂iH(t), q̂iH(t) whichsatisfy the canonical equal time commutation rules

[p̂iH(t), q̂jH(t)] = −iHh̄δij ,[p̂iH(t), p̂jH(t)] = [q̂iH(t), q̂jH(t)] = 0, (2.149)

and postulating the Heisenberg equation of motion

d

dtÔH =

i

h̄[ĤH , ÔH] +

∂

∂tÔH (2.150)

for any observableÔH(t) ≡ O(p̂H(t), q̂H(t), t). (2.151)

This formalism holds for any number of Cartesian or quasi-Cartesian variables.It can therefore be generalized to functions of space variables xn lying on a latticewith a tiny width ǫ [see (2.79)]. Suppressing the subscripts of xn, the canonicalmomenta (2.147) read

px(t) =∂L

∂q̇x(t), (2.152)

and the Hamiltonian becomes

H =∑

x

px(t)q̇x(t)− L(qx, q̇x). (2.153)

The canonical commutation rules (2.149) become the commutation rules (2.143) ofsecond quantization.

In quantum field theory, the formalism must be generalized to continuous spacevariables x. For a Hamiltonian (2.153), the action (2.146) is

A =∫

dt L(t) =∫

dt∫

d3xψI(x, t)h̄∂tψR(x, t)−∫

dtH [ψI , ψR], (2.154)

where H [ψI , ψR] denotes the classical Hamiltonian associated with the operatorHH(t) in Eq. (2.139). The derivative term can be written as an integral over akinetic Lagrangian Lkin(t) as

Akin=∫

dtLkin(t)=∫

dt∫

d3xLkin(x, t)≡∫

dt∫

d3xψI(x, t)h̄∂tψR(x, t). (2.155)


Then the lattice rule (2.152) for finding the canonical momentum has the followingfunctional generalization to find the canonical field momentum:

px(t) =∂L

∂q̇x(t)→ π(x, t) ≡ ∂L

kin

∂∂tψR(x, t)= h̄ψI(x, t), (2.156)

in agreement with the identification (2.152) and the action (2.155). The canonicalquantization rules

[π(x, t), ψ̂R(x′, t)] = −iδ(3)(x− x′),

[π(x, t), π(x′, t)] = 0, (2.157)

[ψ̂R(x, t), ψ̂R(x′, t)] = 0

coincide with the commutation rules (2.142) of second quantization. Obviously, theLegendre transformation (2.153) turns L into the correct Hamiltonian H .

More conveniently, one expresses the classical action in terms of complex fields

A =∫

dt L(t) =∫

dt∫

d3xψ∗(x, t) ih̄∂tψ(x, t)−∫

dtH [ψ, ψ∗], (2.158)

and defines the canonical field momentum as

π(x, t) ≡ ∂Lkin

∂∂tψ(x, t)= h̄ψ∗(x, t). (2.159)

Then the canonical quantization rules become

[ψ(x, t), ψ†(x′, t)] =−iδ(3)(x− x′),[ψ(x, t), ψ(x′, t)] = 0, (2.160)

[ψ†(x, t), ψ†(x′, t)] = 0.

We have emphasized before that the canonical quantization rules are applicableonly if the field space is quasi-Cartesian (see the remark on page 15). For this, thedynamical metric (1.94) has to be q-independent. This condition is violated by theinteraction in the Hamiltonian (2.139). There are ambiguities in ordering the fieldoperators in this interaction. These are, however, removed by the requirement that,after quantizing the field system, one wants to reproduce the n-body Schrödingerequation, which requires that the zero-body state has zero energy and thus satisfiesEq. (2.130).

The equivalence of the n-body Schrödinger theory with the above-derived canoni-cally quantized field theory requires specification of the ordering of the field operatorsafter having imposed the canonical commutation rules upon the fields.

By analogy with the definition of a local Hamiltonian we call an action A localif it can be written as a spacetime integral over a Lagrangian density L(x, t):

A =∫

dt∫

d3xL(x, t), (2.161)

2.8 Canonical Formalism in Quantum Field Theory 107

where L(x, t) depends only on the fields ψ(x, t) and their first derivatives. Thekinetic part in (2.158) is obviously local, the interacting part is bilocal [recall (2.139)].

For a local theory, the canonical field momentum (2.162) becomes

π(x, t) ≡ ∂L∂∂tψ(x, t)

= h̄ψ∗(x, t). (2.162)

The formal application of the rules (2.143) leads again directly to the commutationrules (2.138) without prior splitting into kinetic part and remainder.

In the complex-field formulation, only ψ(x, t) has a canonical momentum, notψ∗(x, t). This, however, is an artifact of the use of complex field variables. Later,in Section 7.5.1 we shall encounter a more severe problem, where the canonicalmomentum of a component of the real electromagnetic vector field vanishes as aconsequence of gauge invariance, requiring an essential modification of the quanti-zation procedure.

Let us calculate the classical equations of motion for the continuous field theory.They are obtained by extremizing the action with respect to ψ(x) and ψ∗(x). Todo this we need the rules of functional differentiation. These rules are derivedas follows: we take the obvious differentiation rules stating the independence ofgeneralized Lagrange variables qi(t), which read

∂qi(t)

∂qj(t)= δij , (2.163)

and generalize them to lattice variables

∂qx(t)

∂qx′(t)= δxx′ . (2.164)

For continuous field variables, these become

∂ψ(x, t)

∂ψ(x′, t)= δ(x− x′). (2.165)

The entire formalism can be generalized, thus considering the action as a lo-cal functional of fields living in continuous four-dimensional spacetime. Then thederivative rules must be generalized further to functional derivatives whose varia-tions satisfy the basic rules

δψ(x, t)

δψ(x′, t′)= δ(3)(x− x′)δ(t− t′) = δ(4)(x− x′). (2.166)

The functional derivatives of actions which depend on spacetime-dependent fieldsψ(x, t) are obtained by using the chain rule of differentiation together with (2.166).The formalism of functional differentiation and integration will be treated in detailin Chapter 14.


For a local theory, where the action has the form (2.161), and the fields andtheir canonical momenta (2.162) are time-dependent Lagrange coordinates with dif-ferentiation rules (2.165), the extremality conditions lead to the Euler-Lagrangeequations

∂A∂ψ(x, t)

=∂L

∂ψ(x, t)− ∂t

δL∂ ∂tψ(x, t)

= 0, (2.167)

∂A∂ψ∗(x, t)

=∂L

∂ψ∗(x, t)− ∂t

δL∂ ∂∗t ψ(x, t)

= 0. (2.168)

The second equation is simply the complex-conjugate of the first.Note that these equations are insensitive to surface terms. This is why, in spite

of the asymmetric appearance of ψ and ψ∗ in the action (2.158), the two equations(2.167) and (2.168) are complex conjugate to each other. Indeed, the latter readsexplicitly

[

ih̄∂t +h̄2

2M∂x

2 − V1(x)]

ψ(x, t) =∫

dx′ ψ∗(x′, t)V2(x− x′; t)ψ(x′, t)ψ(x, t), (2.169)

and it is easy to verify that (2.167) produces the complex conjugate of this.After field quantization, the above Euler-Lagrange equation becomes an equation

for the field operator ψ(x′, t) and its conjugate ψ∗(x′, t) must be replaced by theHermitian conjugate field operator ψ†∗(x′, t).

Let us also remark that the equation of motion (2.168) can be used directly toderive the n-body Schrödinger equation (2.91) once more in another way, by workingwith time-dependent field operators. As a function of time, an arbitrary state vectorevolves as follows:

|ψ(t)〉 = e−iĤt/h̄|ψ(0)〉. (2.170)Multiplying this by the basis bra-vectors

1√n!〈0|âxn · · · âx1, (2.171)

we obtain the time-dependent Schrödinger wave function

ψ(x1, . . . ,xn; t). (2.172)

Inserting between each pair of âxν -operators in (2.171) the trivial unit factors 1 =

e−iĤt/h̄eiĤt/h̄, each of these operators is transformed into the time-dependent fieldoperators ψ̂(xν , t), and one has

ψ(x1, . . . ,xn; t) =1√n!〈0|e−iĤt/h̄ψ̂(xn, t) · · · ψ̂(x1, t)|ψ(0)〉. (2.173)

Using the zero-energy property (2.130) of the vacuum state, this becomes

ψ(x1, . . . ,xn; t) = 〈x1, . . . ,xn; t|ψ(0)〉. (2.174)

2.9 More General Creation and Annihilation Operators 109

The bra-states arising from the application of the time-dependent field operatorsψ̂(xi, t) to the vacuum state on the left

1√n!〈0|ψ̂(xn, t) · · · ψ̂(x1, t), (2.175)

define a new time-dependent basis

〈x1, . . . ,xn; t|, (2.176)

with the property

〈x1, . . . ,xn; t|ψ(0)〉 ≡ 〈x1, . . . ,xn|ψ(t)〉. (2.177)

If we apply to the states (2.175) the differential operator (2.169) and use thecanonical equal-time commutation rules (2.138), we may derive once more thatψ(x1, . . . ,xn; t) obeys the Schrödinger equation (2.18).

The difference between the earlier way (2.98) of defining the wave function andthe formula (2.174) is, of course, the second-quantized version of the differencebetween the Schrödinger and the Heisenberg picture for the ordinary quantum me-chanical wave functions. In Eq. (2.98), the states |ψ(t)〉 are time-dependent butthe basis ket vectors 〈x1, ..,xn| are not, and with them also the field operatorsψ̂(x, 0) = âx generating them. In Eq. (2.174), on the contrary, the states 〈ψ(0)|are time-independent (and may be called Heisenberg states), but the local basis brastates 〈x1, . . . ,xn; t| are not, and with them the field operators ψ̂(x, t) generatingthem. Whatever representation we use, the n-body wave function ψ(x1, . . . ,xn, t)remains the same and obeys the Schrödinger equation (2.18). The change of pictureis relevant only for the operator properties of the many-particle description.

Certainly, there is also the possibility of changing the picture in the Schrödingerwave function ψ(x1, . . . ,xn; t). But the associated unitary transformation wouldtake place in another Hilbert space, namely in the space of square integrable func-tions of n arguments, where p̂ and x̂ are the differential operators −ih̄∂x and x.

When going through the proof that (2.174) satisfies the n-body Schrödingerequation (2.18), we realize that at no place do we need the assumption of time-independent potentials. Thus we can conclude that the canonical quantizationscheme for the action (2.158) is valid for an arbitrary explicit time dependenceof the potentials in the Hamiltonian operator Ĥ [see (2.5)]. It is always equivalentto the Schrödinger description for an arbitrary number of particles.

2.9 More General Creation and Annihilation Operators

In many applications it is possible to solve exactly the Schrödinger equation withonly the one-body potential V1(x; t). In these cases it is useful to employ, insteadof the creation and annihilation operators of particles at a point, another equivalentset of such operators which refers, right away, to the corresponding eigenstates. We


do this by expanding the field operator into the complete set of solutions of theone-particle Schrödinger equation

ψ̂(x, t) =∑

α

ψEα(x, t)âα. (2.178)

If the one-body potential is time-independent and there is no two-body potential,the states have the time dependence

ψEα(x, t) = ψEα(x) e−iEαt/h̄. (2.179)

The expansion (2.178) is inverted to give

âα =∫

d3xψ∗Eα(x, t)ψ̂(x, t), (2.180)

which we shall write shorter in a scalar-product notation as

âα = (ψEα(t), ψ̂(t)). (2.181)

As opposed to the Dirac bracket notation to denote basis-independent scalar prod-ucts, the parentheses indicate more specifically a scalar product between spatialwave functions.

From the commutation rules (2.138) we find that the new operators âα, â†α satisfy

the commutation rules

[âα, âα′ ] = [â†α, â

†α′] = 0,

[âα, â†α′ ] = δα,α′ . (2.182)

Inserting (2.178) into (2.139) with V2 = 0, we may use the orthonormality relationamong the one-particle states ψEα(x) to find the field operator representation forthe Hamilton operator

Ĥ =∑

α

Eαâ†αâα. (2.183)

The eigenstates of the time-independent Schrödinger equation

Ĥ|ψ(t)〉 = E|ψ(t)〉 (2.184)

are now

|n1, . . . , nn〉 =1√

n1! · · ·nn!(â†α1)

n1 · · · (â†αk)nk |0〉, (2.185)

where the prefactor ensures the proper normalization. The energy is

E =k∑

i=1

Eαini. (2.186)

Finally, by forming the scalar products

1√n!〈0|âxn · · · âx1 (â†α1)n1 · · · (â†αk)

nk |0〉 1√n1! · · ·nk!

, (2.187)

2.10 Quantum Field Formulation of Many-Fermion Schrödinger Equations 111

we recover precisely the symmetrized wave functions (2.32) with the normalizationfactors (2.41).

Similar considerations are, of course, possible in the Heisenberg picture of theoperators â†α, âα which can be obtained from

âα(t) =∫

d3xψ∗Eα(x)ψ̂(x, t). (2.188)

In the field operator description of many-body systems, the Schrödinger wavefunction ψ(x, t) has become a canonically quantized field object. Observe that thefield ψ(x, t) by itself contains all relevant quantum mechanical information of thesystem via the derivative terms of the action (2.158),

ψ̂†(x, t)ih̄∂tψ̂(x, t) +h̄2

2Mψ̂†(x, t)∂2xψ̂(x, t). (2.189)

This fixes the relation between wavelength and momentum, and between frequencyand energy. The field quantization which introduces the additional processes ofparticle creation and annihilation is distinguished from this and often referred to assecond quantization.

It should be kept in mind that, for a given n-body system, second quantizationis completely equivalent and does not go beyond the usual n-body Schrödingertheory. It merely introduces the technical advantage of collecting the wave equationsfor any particle number n in a single operator representation. This advantage is,nevertheless, of great use in treating systems with many identical particles. Inthe limit of large particle densities, it gives rise to approximations which would bevery difficult to formulate in the Schrödinger formulation. In particular, collectiveexcitations of many-particle systems find their easiest explanation in terms of aquantum field formulation.

The full power of quantum fields, however, will unfold itself when trying toexplain the physics of relativistic particles, where the number of particles is nolonger conserved. Since the second-quantized Hilbert space contains any number ofparticles, the second-quantized formulation allows naturally for the description of theemission and absorption of fundamental particles, processes which the Schrödingerequation is unable to deal with.

2.10 Quantum Field Formulation of Many-FermionSchrödinger Equations

The question arises whether an equally simple formalism can be found which auto-matically leads to the correct antisymmetric many-particle states

|x1, . . . ,xn〉A =1√n!

∑

P

ǫP |xp(1)〉 ×⊙ . . . ×⊙ |xp(n)〉. (2.190)

This is indeed possible. Let us remember that the symmetry of the wave functionswas a consequence of the commutativity of the operators ψ̂(x, t) for different position


values x. Obviously, we can achieve an antisymmetry in the coordinates by formingproduct states

|x1, . . . ,xn〉A =1√n!â†x1 · · · â†xn |0〉 (2.191)

and requiring anticommutativity of the particle creation and annihilation operators:

{â†x, â†x′} = 0, {âx, âx′} = 0. (2.192)

The curly brackets denote the anticommutator defined in Eq. (2.110). To define aclosed algebra, we require in addition, by analogy with the third commutation rule(2.75) for bosons, the anticommutation rule

{âx, â†x′} = δ(3)(x− x′). (2.193)

As in the bosonic case we introduce a vacuum state |0〉 which is normalized as in(2.87) and contains no particle [cf. (2.86)]:

âx|0〉 = 0 , 〈0|â†x = 0. (2.194)

The anticommutation rules (2.192) have the consequence that each point can atmost be occupied by a single particle. Indeed, applying the creation operator twiceto the vacuum state yields zero:

â†xâ†x|0〉 = {â†x, â†x}|0〉 − â†xâ†x|0〉 = 0. (2.195)

This guarantees the validity of the Pauli exclusion principle.The properties (2.192), (2.193), and (2.194) are sufficient to derive the many-

body Schrödinger equations with two-body interactions for an arbitrary number offermionic particles. It is easy to verify that the second-quantized Hamiltonian hasthe same form as in Eq. (2.129). The proof proceeds along the same line as inthe symmetric case, Eqs. (2.94)–(2.129). A crucial tool is the operator chain rule(2.111) derived for anticommutators. The minus sign, by which anticommutatorsdiffer from commutators, cancels out in all relevant equations.

As for bosons we define a time-dependent quantum field for fermions in theHeisenberg picture as

ψ̂(x, t) = eiĤt/h̄ψ̂(x, 0)e−iĤt/h̄

= eiĤt/h̄âxe−iĤt/h̄, (2.196)

and find equal-time anticommutation rules of the same type as the commutationrelations (2.138):

{ψ̂(x, t), ψ̂(x′, t)} = 0,{ψ̂†(x, t), ψ̂†(x′, t)} = 0, (2.197){ψ̂(x, t), ψ̂†(x′, t)} = δ(3)(x− x′).

2.11 Free Nonrelativistic Particles and Fields 113

The Hamiltonian has again the form Eq. (2.139).There is only one place where the fermionic case is not completely analogous

to the bosonic one: The second-quantized formulation cannot be derived from astandard canonical formalism of an infinite number of generalized coordinates. Thestandard formalism of quantum mechanics applies only to true physical canonicalcoordinates p(t) and q(t), and these can never account for anticommuting propertiesof field variables.2 Thus an identification analogous to (2.144),

p̂x(t) ≡ ψ̂I(x, t), q̂x(t) ≡ ψ̂R(x, t), (2.198)

is at first impossible.The canonical formalism may nevertheless be generalized appropriately. We may

start out with exactly the same classical Lagrangian as in the boson case, Eq. (2.158),but treat the fields formally as anticommuting objects, i.e.,

ψ(x, t)ψ(x′, t′) = −ψ(x′, t′)ψ(x, t). (2.199)

In mathematics, such objects are called Grassmann variables. Using these, we defineagain classical canonical momenta

px(t) ≡δL

δψ̇(x, t)= −ih̄ψ†(x, t) ≡ π(x, t). (2.200)

Together with the field variable qx(t) = ψ†(x, t), this is postulated to satisfy the

canonical anticommutation rule

{px(t), qx′(t)} = −ih̄δ(3)(x− x′). (2.201)

2.11 Free Nonrelativistic Particles and Fields

An important way to approach interacting theories is based on perturbative meth-ods. Usually, these begin with the free theory and prescribe how to calculate suc-cessive corrections due to the interaction energies. A detailed discussion of howand when this works will be given later. It seems intuitively obvious, however, thatat least for weak interactions, the free theory may be a good starting point for anapproximation scheme. It is therefore worthwhile to study a few properties of thefree theory in detail.

The free-field action is, according to Eqs. (2.158) and (2.139) for V1(x) = 0 andV2(x− x′) = 0:

A =∫

dtd3xψ∗(x, t)

[

ih̄∂t +h̄2

2M∂x

2

]

ψ(x, t). (2.202)

2For a detailed discussion of classical mechanics with supersymmetric Lagrange coordinates seeA. Kapka, Supersymmetrie, Teubner, 1997.


The quantum field ψ̂(x, t) satisfies the field operator equation

(

ih̄∂t +h̄2

2M∂x

2)

ψ̂(x, t) = 0, (2.203)

with the conjugate field satisfying

ψ̂†(x, t)(

− ih̄ ←∂ t +h̄2

2M

←

∂x2)

= 0. (2.204)

The equal-time commutation rules for bosons and fermions are

[ψ̂(x, t), ψ̂(x′, t)]∓ = 0,

[ψ̂†(x, t), ψ̂†(x′, t)]∓ = 0, (2.205)

[ψ̂(x, t), ψ̂†(x′, t)]∓ = δ(3)(x− x′),

where we have denoted commutator and anticommutator collectively by [ . . . , . . . ]∓,respectively.

In a finite volume V , the solutions of the free one-particle Schrödinger equationare given by the time-dependent version of the plane wave functions (1.185) [compare(2.179)]:

ψpm(x, t) = 〈x, t|pm〉 = 〈x, t|â†pm〉 =1√V

exp

{

i

h̄

(

pmx− pm2

2Mt

)}

. (2.206)

These are orthonormal in the sense∫

d3xψ∗pm(x, t)ψpm′ (x, t) = δpm,pm′ , (2.207)

and complete, implying that∑

pm

ψpm(x, t)ψ∗pm(x

′, t) = δ(3)(x− x′). (2.208)

As in Eq. (2.178), we now expand the field operator in terms of these solutions as

ψ̂(x, t) =∑

pm

ψpm(x, t)âpm. (2.209)

This expansion is inverted with the help of the scalar product (2.181) as

âpm = (ψpm(t), ψ̂(t)) =∫

d3xψ∗pm(x, t)ψ̂(x, t). (2.210)

In the sequel we shall usually omit the superscript of the momenta pm if theirdiscrete nature is evident from the context. The operators âp and â

†p, obey the

canonical commutation rules corresponding to Eq. (2.182):

[âp, âp′]∓ = 0,

[â†p, â†p′]∓ = 0,

[âp, â†p′]∓ = δpp′, (2.211)

2.11 Free Nonrelativistic Particles and Fields 115

where we have used the modified δ-functions introduced in Eq. (1.196).In an infinite volume, we use the time-dependent version of the continuous wave

functions (1.195)

ψp(x, t) = 〈x, t|p〉 = 〈x, t|â†p〉 =1√V

exp

{

i

h̄

(

pmx− pm2

2Mt

)}

, (2.212)

which are orthonormal in the sense∫

d3xψ∗p(x, t)ψp′(x, t) = (2πh̄)3δ(3)(p− p′)

≡ -δ (3)(p− p′), (2.213)

and complete, as expressed by

∫

d3p

(2πh̄)3ψp(x, t)ψ

∗p(x

′, t) ≡∫

-d3p ψp(x, t)ψ∗p(x

′, t) = δ(3)(x− x′). (2.214)

In terms of these continuum wave functions, we expand the field operator as

ψ̂(x, t) =∫

-d3p ψp(x, t)â(p), (2.215)

and have the inverse

â(p) = (ψp(t), ψ̂(t)) =∫

d3xψ∗p(x, t)ψ̂(x, t). (2.216)

The discrete-momentum operators âp, and â†p and the continuous ones â(p) and

â†(p), are related by [recall Eq. (1.190)]

â(p) =√V âp, â

†(p) =√V â†p. (2.217)

For the continuous-momentum operators â(p) and â(p)†, the canonical commutationrules in Eq. (2.182) take the form

[â(p), â(p′)]∓ = [â†(p), â†(p′)]∓ = 0,

[â(p), â†(p′)]∓ =-δ(3)(p− p′). (2.218)

The time-independent many-particle states are obtained, as in (2.185), by re-peatedly applying any number of creation operators â†p [or â

†(p)] to the vacuumstate |0〉, thus creating states

|np1 , np2 , . . . , npk〉 = N S,A(â†p1)np1 · · · (â†pk)

npk |0〉, (2.219)

where the normalization factor is determined as in Eq. (2.185). For bosons with np1identical states of momentum p1, with np2 identical states of momentum p2, etc.,the normalization factor is

N S = 1√np1! · · ·npk!

. (2.220)


The same formula can be used for fermions, only that then the values of npi arerestricted to 0 or 1, and the normalization constant NA is equal to 1. The time-independent wave functions are obtained as

〈x1, . . . ,xn|np1, np2 , . . . , npk〉S,A =N S,A√n!

(2.221)

× 〈0|ψ̂(xn, 0) · · · ψ̂(x1, 0)(â†p1)np1 · · · (â†pk)

npk |0〉,

and the time-dependent ones as

〈x1, . . . ,xn|np1 , np2, . . . , npk ; t〉S,A = 〈x1, . . . ,xn|np1, np2 , . . . , npk〉S,Ae−iEt/h̄

=N S,A√n!

〈0|ψ̂(xn, 0) · · · ψ̂(x1, 0)e−iĤt/h̄(â†p1)np1 . . .(â†pk)

npk|0〉

=N S,A√n!

〈0|ψ̂(xn, 0) · · · ψ̂(x1, 0)[â†p1(t)]np1 · · · [â†pk(t)]

npk |0〉,

with the time-dependent creation operators being defined by

â†p(t) ≡ eiĤt/h̄â†pe−iĤt/h̄. (2.222)

The energy of these states is

E =k∑

i=1

niεpi , (2.223)

where εp ≡ p2/2M are the energies of the single-particle wave functions (2.212). Themany-body states (2.219) form the so-called occupation number basis of the Hilbertspace. For fermions, ni can only be 0 or 1, due to the anticommutativity of theoperators âp and â

†p among themselves. The basis states are properly normalized:

〈np1np2np3 . . . npk |n′p1n′p2n′p3 . . . n

′pk〉 = δnp1n′p1δnp2n′p2δnp3n′p3 . . . δnpkn′pk . (2.224)

They satisfy the completeness relation:

∑

p1p2p3...

∑

np1np2np3 ...npk

|np1np2np3 . . . npk〉〈np1np2np3 . . . npk | = 1S,A, (2.225)

where the unit operator on the right-hand side covers only the physical Hilbert spaceof symmetric or antisymmetric n-body wave functions.

2.12 Second-Quantized Current Conservation Law

In Subsection 1.3.4 of Chapter 1 we have observed an essential property for theprobability interpretation of the Schrödinger wave functions: The probability cur-rent density (1.107) and the probability density (1.108) are related by the localconservation law (1.109):

∂tρ(x, t) = −∇ · j(x, t). (2.226)

2.13 Free-Particle Propagator 117

This followed directly from the Schrödinger equation (2.169). Since the same equa-tion holds for the field operators, i.e., with ψ∗(x′, t) replaced by ψ̂†(x′, t), the fieldoperators of charge and current density,

ρ̂(x, t) = ψ̂†(x, t)ψ̂(x, t),

ĵ(x, t) = −i h̄2M

ψ̂†(x, t)↔

∇ ψ̂(x, t) (2.227)

satisfy the same relation. When integrating (2.226) over all x, and using Green’stheorem as done in Eq. (1.110), we obtain a global conservation law that ensures thetime independence of the particle number operator

N̂ =∫

d3x ρ̂(x, t) =∫

d3x ψ̂†(x, t)ψ̂(x, t). (2.228)

Since this is time-independent, we can use (2.134) to rewrite

N̂ =∫

d3x ρ̂(x, 0) =∫

d3x ψ̂†(x, 0)ψ̂(x, 0) =∫

d3x â†xâx, (2.229)

so that N̂ coincides with the particle number operator (2.78). The original form(2.228) is the Heisenberg picture of the particle number operator, which coincideswith (2.229), since the particle number is conserved.

2.13 Free-Particle Propagator

The perturbation theory of interacting fields to be developed later in Chapter 10requires knowledge of an important free-field quantity called the free-particle prop-agator. It is the vacuum expectation of the time-ordered product of two free fieldoperators. As we shall see, the calculation of any observable quantities can be re-duced to the calculation of some linear combination of products of free propagators[see Section 7.17.1]. Let us first extend the definition (1.249) of the time-orderedproduct of n time-dependent operators to allow for fermion field operators. Supposethat the times in an operator product Ân(tn) · · · Â1(t1) have an order

tin > tin−1 > . . . > ti1 . (2.230)

Then the time-ordered product of the operators is defined by

T̂ Ân(tn) · · · Â1(t1) ≡ ǫP Âin(tin) · · · Âi1(ti1). (2.231)

With respect to the definition (1.250), the right hand side carries a sign factorǫP = ±1 depending on whether an even or an odd permutation P of the fermionfield operators is necessary to reach the time-ordered form. For bosons, εP ≡ 1.

The definition of the time-ordered products can be given more concisely usingthe Heaviside function Θ(t) of Eq. (1.313).


For two operators, we have

T̂ Â(t1)B̂(t2) = Θ(t1 − t2)Â(t1)B̂(t2)±Θ(t2 − t1)B̂(t2)Â(t1), (2.232)

with the upper and lower sign applying to bosons and fermions, respectively. Thefree-particle propagator can now be constructed from the field operators as thevacuum expectation value

G(x, t;x′, t′) = 〈0|T̂ ψ̂(x, t)ψ̂†(x′, t′)|0〉. (2.233)

Applying the free-field operator equations (2.203) and (2.204), we notice a remark-able property: The free-particle propagator G(x, t;x′, t′) coincides with the Greenfunction of the Schrödinger differential operator. Recall that a Green function of ahomogeneous differential equation is defined by being the solution of the inhomoge-neous equation with a δ-function source (see Section 1.6). This property may easilybe verified for the free-particle propagator, which satisfies the differential equations

(

ih̄∂t +h̄2

2M∂x

2)

G(x, t;x′, t′) = ih̄δ(t− t′)δ(3)(x− x′), (2.234)

G(x, t;x′, t′)(

− ih̄←

∂t′ +h̄2

2M

←

∂x2)

= ih̄δ(t− t′)δ(3)(x− x′), (2.235)

thus being a Green function of the free-particle Schrödinger equation: The right-hand side follows directly from the fact that the field ψ̂(x, t) satisfies the Schrödingerequation and the obvious formula

∂tΘ(t− t′) = δ(t− t′). (2.236)

With the help of the chain rule of differentiation and Eq. (2.232), we see that

(

ih̄∂t +h̄2

2M∂x

2)

〈0|T̂ ψ̂(x, t)ψ̂†(x′, t′)|0〉

= ih̄[

∂tΘ(t− t′)〈0|ψ̂(x, t)ψ̂†(x′, t′)|0〉 ± ∂tΘ(t′ − t)〈0|ψ̂†(x′, t′)ψ̂(x, t)|0〉]

= ih̄δ(t− t′)〈0|[ψ̂(x, t), ψ̂†(x′, t)]∓|0〉 = ih̄δ(t− t′)δ(3)(x− x′), (2.237)

where the commutation and anticommutation rules (2.138) and (2.197) have beenused, together with the unit normalization (2.87) of the vacuum state.

In the theory of differential equations, Green functions are introduced to findsolutions for arbitrary inhomogeneous terms. These solutions may be derived fromsuperpositions of δ-function sources. In quantum field theory, the same Green func-tions serve as propagators to solve inhomogeneous differential equations that involvefield operators.

Explicitly, the free field propagator is calculated as follows: Since ψ̂(x, t) anni-hilates the vacuum, only the first term in the defining Eq. (2.232) contributes, sothat we can write

G(x, t;x′, t′) = Θ(t− t′)〈0|ψ̂(x, t)ψ̂†(x′, t′)|0〉. (2.238)

2.13 Free-Particle Propagator 119

Inserting the expansion Eq. (2.215) with the wave functions (2.212), and using(2.218), the right-hand side becomes

Θ(t− t′)∫

-d3p -d3p′ ei[(px−p′x′)−(p2t/2M−p′2t′/2M)]/h̄〈0|â(p)â†(p′)|0〉

= Θ(t− t′)∫

-d3p ei[p(x−x′)−p2(t−t′)/2M ]/h̄. (2.239)

By completing the square and using the Gaussian integral

∫

-d3p e−ap2/2h̄ =

1√2πh̄a

3 , (2.240)

we find

G(x, t;x′, t′) = Θ(t− t′) 1√

2πih̄(t− t′)/M3 e

iM(x−x′)2/2h̄(t−t′)

= G(x− x′, t− t′). (2.241)

The right-hand side is recognized as the usual quantum-mechanical Green functionof the free-particle Schrödinger equation of Eq. (1.350). Indeed, the factor afterΘ(t− t′) is simply the one-particle matrix element of the time evolution operator

〈0|ψ̂(x, t)ψ̂†(x′, t′)|0〉 = 〈0|ψ̂(x)e−iĤ(t−t′)/h̄ψ̂†(x′)|0〉= 〈x|Û(t, t′)|x′〉. (2.242)

This is precisely the expression discussed in Eqs. (1.310)–(1.312). It describes theprobability amplitude that a single free particle has propagated from x to x′ in thetime t− t′ > 0. For t− t′ < 0, G vanishes.

There exists a more useful way of writing the Fourier representation of the prop-agator than that in Eq. (2.239). It is based on the integral representation (1.319)of the Heaviside function:

Θ(t− t′) =∫ ∞

−∞

-dE e−iE(t−t′)/h̄ ih̄

E + iη. (2.243)

As discussed in general in Eqs. (1.317)–(1.319), the iη in the denominator ensuresthe causality. For t > t′, the contour of integration can be closed by an infinitesemicircle below the energy axis, thereby picking up the pole at E = −iη, so thatwe obtain by the residue theorem

Θ(t− t′) = 1, t > t′. (2.244)

For t < t′, on the other hand, the contour may be closed above the energy axis and,since there is no pole in the upper half-plane, we have

Θ(t− t′) = 0, t < t′. (2.245)


Relation (2.243) can be generalized to

Θ(t− t′)e−iE0(t−t′)/h̄ =∫ ∞

−∞

-dE e−iE(t−t′)/h̄ ih̄

E − E0 + iη. (2.246)

Using this with E0 = p2/2M we find from (2.239) the integral representation

G(x− x′, t− t′) =∫

-d3p∫ ∞

−∞

-dE eip(x−x′)/h̄−iE(t−t′)/h̄ ih̄

E − p2/2M + iη . (2.247)

In this form we can trivially verify the equations of motion (2.234) and (2.235). Thisexpression agrees, of course, with the quantum mechanical time evolution amplitude(1.344).

The Fourier-transformed propagator

G(p, E) =∫

d3x∫ ∞

−∞dt e−i(px−Et)/h̄G(x, t)

=ih̄

E − p2/2M + iη

Documents

FieldFormulationofMany-Body QuantumPhysicsusers.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-2...nuclei with an even atomic number. Examples for fermions are electrons, neutrinos,