Luis Alvarez- GauméOCR Output

CERN

Quantum Field Theory

A Small Course in

topics with extensive references to the literature. OCR Output

ment of the subjects covered in this minicourse as well as many other

In these references the interested reader can find an authoritative treat

Field Theory. Addison Wesley.

• M.E. Peskin and D.V. Schroeder: An Introduction to Quantum

U. Press.

• S. Weinberg: The Quantum Theory of Fields Vol. LII; Cambridge

further reading I would recommend looking at the two recent books:

to keep the material as close as possible to the original lectures. For

a few other topics than those covered in these notes, but I preferredand to give a detailed presentation. The original plan was to includeare accustomed to but that we often have difficulties in understandingsonable option for such a course is to select a number of topics that wegood books with a modern presentation of the material. The only reasubject as vast as Quantum Field Theory. Fortunately there are now1996. In five hours it is difficult to give a reasonable introduction to aacademic training programme during the week of the 7th of november

These notes are a text transcription of a set of lectures given in CERN’s

OCR OutputINTRODUCTION

E’"zv 2 26?j;§2 jj Qj)4a3 - vJ. m)*IJ : 0 OCR Output

where T2. = c = 1):Using the correspondence principle (we use throughout natural units

E2 :52+7% E: i(/sum?.

particle the relation between energy and momentum is,

tum behaviour of particles in the relativistic domain. For a free scalar

the notion that relativistic wave equations should described the quan

below). Let us briefly see what happens if we insist on maintainingparticles (this can be seen in terms of the old Klein paradox describedsolution, and this always leads to problems with fixing the number of

function. The relativistic dispersion relation allows for negative energy

structures that cannot be simply described in terms of a one—body wave

look at energies E > 100 MeV, we also see the pion cloud and other

ourselves to Hu ® Hu ® Hd. The proton is much more than that. If we

state of three quarks und, but this does not mean that we can restrict

non-relativistic approximation we say that the proton p is a bound

to describe the system. We say H is a bound state of e“, p. In theleads to an electron and a proton: e` + p. Hence it suflices HE ® Hppossibly few well-deiined ‘bodies’. The hydrogen atom H if ionizedIn QM we are used to the notion that the Hilbert space contains one,

plest systems. There are several ways to argue this basic point.quantum fields and to a many (oo) body description of even the sim(QM) and Special Relativity we are led inevitably to the theory oftal laws of nature. When we try to put together Quantum MechanicsQFT is the most basic language known so far to express the fundamen

The need for Quantum Fields

LECTURE 1

paradox briefly explained below. OCR OutputA way of saying the same thing in equations is the so-called Klein

and we can no longer say that there is a single particle inside the box.

wavelength AE ~ m, particle pair production begins to be possible,

AE << m. As L is decreased, we eventually reach the Comptonslowly decrease L, Az ~ L,Ap ~ 1/L, AE ~ 1/L2. For L >> 1/mwe can say with confidence that there is a particle on the box. If we

For a single particle of mass m in a box of size L3 with L > 1/m,The spectrum is schematically drawn in the iigure.ones. Note that the scalar product is negative on the negative solutions.The fp are the positive energy solutions and the f.p the negative energy

<fplfpr> = 6(P - p') <f-plf-p/> = —6(2¤— P') <fp|f.;¢> = 0

(¢1li!*2} = ifd3$(*/G801/*2 ·· $230*/Jl)

Z f—p(¤¤) 1 ipx

, .».. ·_ fp($) — 1 —i x P "“‘*·‘. i. `->*’»==— N ) gg<;;»¤iv%%€§2é5é»;;. » .wigs -·-r~ · ~··=~t?Z3¤<’T”*’l’€<"°`7”;"""F>’§""'fY"°’§$`€?“""*“°i""7‘%" sw; ;:;v:;.s“'$ i+}$;=<»: ;"£=‘ g is=¥ ?’rér=5`§ \‘f$·_

3;,% = O jo not positive definite

it = $i(¢"@,n/¤ - 1/JQMV)

positive dennite and it cannot be used to deiine a probability density.construct a conserved current. However the time-like component is notsolutions to the wave equation. For complex solutions we can alwaysNote that both signs of cu are necessary to describe a complete set of

(x t) : 6-—ikua:p : €—i(wt—k·5:`) W2 : k2 + m2

whose plane-wave solutions arc:

x = 6 — V0 OCR Output(:1:(2m + a:)) 1/2pz = Mm + {6 — Vb))2 - mzl= (2m(¤— Wa) + (6 - Vb)2

l/2 1/2

reflected and transmitted waves.

E = m + e fixed. lf 6 > VO, the standard picture applies. We have

1/2 pz = (<E — VUV — mg)

p1:(E2 _ m2)1/2

I R = T I — R = T - I- 2 + P1( ) P2 R

‘I’1(0) = ‘I’H(0) \I*Q(0> = ‘I'i1(0) } T : piésl-lpz Mcoefficients.

Matching at the barrier, we compute the reflection and transmission

E= (pi+m2>1/2 E=(2¤3+m2)1/2+%

QH = TCZPQJC

*11; = e’p*“ + Re"""’

The energy of the wave is taken to be E.gion II where we have the transmitted wave. v

the incoming and reflected wave, and in re

Gordon equation we consider the wave function in region I containing

Consider a potential with the profile in the figure. To solve the Klein

Klein paradox

H = x/—V2 + m2 OCR Output

<2¤)3 -» -· iz?-z-¢»:\/'JEQF w(=v,0) = 6 (<¤) MM) =/-16

dgk

ISI

time with the relativistic Hamiltonian H = x/—V2 + m2. The resultfunction concentrated near if = 6, 1p(:E', O) ~ 63(E), and we evolve it inconhict with causality. Imagine we consider at time t : O a wave

If we insist still on the one-particle interpretation, we find a direct

infinite number of degrees of freedom.a: = (ty?) is a label and not an operator as in QM. Thus we have an

A(:c)

sentation will be of the form

the basic operators on the Heisenberg repre

sition. Since we can localize measurements,

Observables must know about space-time po

t (R1- 1-22)** < 0 \ $ R2[ARN Ami = U

ments taken at space-like separated regions, they must commute:in special relativity. If ARL, are two operators associated to measureity. This is a restriction imposed by the causal structure of space-time

surements can be localized, but we have to respect microscopic causal

In QM any Hermitian operator is an observable. Not so in QF T. Mea

a pair. We cannot avoid a many body interpretation.

mitted wave. This is interpreted by saying that the barrier has created

If 33 < O, but 2m + :2: < O, (Vb > 2m), pg is real, and we have a transIf :2: < 0, but 2m + :1: > O, we have total reilexion.

imate momentum states we want to construct a multiparticle Hilbert OCR Output

By analogy, since we observe weakly interacting particles on approx

3·*·133U·1(R)5U(12) Z / dk|121E>1§(Rk| Z /dk|12>R1§(k| Z R1

P Z fd3k[l;)l;(l;

1 Z J dgkllg) (E

Some basic relations are:

v<R>w?> Z use ; P1/€> Z ¤€v€>

The rotation group acts unitarity:

’r?5> . <¤€n%> Z MF — in

with normalization:

free moving particle of given momentum k is described by a ket lk),by wave packets made of free particles. In the non-relativistic case a

ing process we assume that the initial and final states are well describedphysical system we do it through a scattering experiment. In a scatter

way or another when we obtain information about a

Let us take a different point of view. After all in one

Fock Space

—lfT\interpretation.

the figure. This is again bad new for the one-body

seen by evaluating the integral using the contour on

light-cone, i.e. ¢(v~,t) 96 O for r > t > O, as can beFor any t > U the wave function spreads outside the

_, _it(k2+ 2)1/2 1b(:c,t) - —§;l$7j5£ dk coskr e m]. 8 00

2<¤p6(px — PQ) OCR Output

Consider motion along the :0-direction:

ance it suffices to prove invariance under boosts along the :1:-direction.

rotations. To show invariance under boosts, using rotational invari2wp63(;$’ -— gi") is a relativistic invariant. It is clearly invariant under

<z>Iz>’> = (2¤>32wp63(2?— zi")

Pl = (2¢r>3/2(2<¤p)‘/QI?}

Starting with the non-relativistic states we construct:

+m212>>{ 1f_°lz>> = x/F P|;¤> = Flr)We have states lp) which are eigenstates of fourqnornenturn

-- = we + 2 2wk(2*rr)3 wk mdi/~c 4% —A

The measure is:

• An invariant normalization

count states.

• An invariant measure with respect to Lorentz transformations to

group we need.

tion group above, to construct a unitary representation of the Lorentz

to be the basic object to describe physical processes. As with the rota·space with a unitary representation 0f the Poincaré group (ISO (3,1))

[<><(k)»<>¢“`(k’)l = (2r)2wk6(k — k') OCR Output33

o4(k) E (2¢r)3/2(Qwk)1/2a(l;) ¤+<k) = (QW)3/2<2<¤k)*/2¤*<i?>

Now introduce:

k> = ¤“'<k)I0><0|0> = 1

¤»<i€>, ¤<¤€>+ im, ¤<¤€*>+1 = 6<3><% — liv

erators a(k), a(k)+ satisfying:In the non—relativistic case we introduce creation and annihilation op

using Fock operators.

Multiparticle states follow in the same way, but there is a simple way

U"(A)P"U(A) = MP"

Unitarity: (p”|U+(A)U(A))lp’) = (Ap|Ap’) = (p|p’) (inv. measure)

f ia"; P "d P" =On one particle states

U (A)|z>> = lAp>

P”|1>> = ;¤"l;v>The action of the Lorentz group is:

The Lorentz boost acts by cb —>¢ + const, and invariance follows.

= 2¤¤pg;rf@6(¢p ·¢pr) = 26(¢p —<f>p/)} px = msh <Z>pwp Z mch Qbp 2wp6(pI —— pg) = 2wp6(sh<;§p — shqbp)

Since

ii) l¢(¤¤)»¢>(y)l = 0 (rv — y)2 < 0OCR Output

i) ¢+(¢) = Mw)

ables of the form We make some assumptions.

Since measurements can be localized, we can try to construct observ

Simple observable: Scalar Field

U(A)o4+(k)U+(A) = o4+(Ak)

U(A)oe(k)U+(A) = o4(Ak:)

U(A)<>¤““(k)|0> = U<¤)¤+(k)U“1(A)U(A)|0> = IM} = ¤+(Ak)l0>

¤+(k)l0> = Ik}

vacuum:

linear combinations of polynomials in the creation operators on the

This way we obtain the Fock space generated by acting with arbitrary

i + + f) — /Hi%%;f(/<i,···,/~¤n)¤ (/~¤r)···c¤ (MIO).dgk

for multiparticle states:

()kI __ =•< I <f|f>— /~———27T32UJf (klf (kl ,d3k

]+ fl — /6T·§‘5(j;O¢ (k)|0>f(k)

wk

General 1—particle state:

locality, and relativistic invariance. OCR Output

far we can go if we insist on the existence of free asymptotic states,

ical quantization of classical fields. It is nevertheless useful to see howThe same results can be obtained more straightforwardly using canon

· iA(:1: — = ——e”'kIekD6k2 — m2y) ()(f <2¤>3 ’d4k

Mw — yl = A+(=¤ - yl — A+(y —¤=l A+(<v — yl /¥——e(2¢r)32wkZj€‘”°("‘”’

l¢<¤¤l»<z>(yll = iA(¤¤— yl

bl l¢>(<i3¢l, <b(y3¢ll = ¢63(5¤’· yl

al (¤ + m2l¢(¤¢l = 0

This field satisfies:

k.k A = 1 :2: k zk-x + k ¢<¤¤) [ (2W)32w(6 el l + 6 Oz ( >)d3k

then:

U+(A)q5(O)U(}\) = $(0) :> f(k,U), g(k,O) independent of lc

k= -—;— kl k k + k ¢(=¤) /(2W)32w(f( »¤¤)<>¤( l + 9( ,¢v)¢¤ ( ))d3l~c

in terms of them. These constraints lead to a unique answerform a complete set of operators, any observable can be expressedAs a simplifying assumption take <Z> linear on a,<x+. Since a, a+

U +<Al¢(¤¤lU (Al = ¢<A`1=v)

Lorentz invariance:iv)

an assumption it means that PU, P generate t,:1c translations.iii) Translational invariance eiP‘“¢(:c)e_”iP‘“ = qz5(:c — a), is not really

_1 WM 1 2 2{(¤+m’)4>=0 OCR OutputThe simplest example is a free massive scalar fields:

of any operator

Time evolution = [A, H] = [A,H]

Hamiltonian H = Zjpiql —— L H = fd35:°(l'Ia(a?)¢“(aE`) — LZ)

relation

commutation [qmpj] = MU [¢“(:E’, t), m,(Q`, t)] = i6§6(:E — Q')Canonical

Momenta

Canonical H = a(:E', t) =

Motion

FJ W· QQ Z GL M : M? EC]U.3,JE1OI1S of

Action S = fdtL(q, cj) S = fd4x£(<;5, Glyqb)

<1i(#) —·>¢“(<?'¢) ii-> (il C1)

Lagrangian L(qi, qi) L(q>“(:E, t), 8,,¢>“ (if, t))

becomes a Lagrangian density

context. We start with a Lagrangian L(qi,cji) which in field theoryquantization in particle mechanics and its extension to the field theoryIn the following table we summarize the standard rules of canonical

LECTURE 2

10

tematic approach to set up these relationships. OCR Output

renormalization and the renormalization group provides a more sys

observables is not necessarily so straightforward, and the machinery ofthis identification between parameters in the Lagrangian and physicalphysical mass of the associated particles. For interacting Lagrangiansin the Lagrangian for the free massless scalar can be identified with theAs a final remark we should point out that the parameter m appearingSimilar arguments can be carried out for more complicated fields.

V w.Z g}; l»

which in finite volume V is

6 (0) / d kw.,3 3_°1

(iniinite) zero point energythe right and the annihilation operators to the left. This removes theNormal ordering is the instruction to place all creation operators toIn the last line we have defined the notion of a normal ordered operator.

H — (O|H|O) = /d*‘k wka+(k)a(k)

4% wk a+(k)a(k) + gwk63(0)

d3k% (a,+(k)a(k) —|— a(k)a+(k))

·x k—i -2: + i <U](k)€ k +G (k)€disk

j dag (w + my + mwg)gg

[¢>(¤?3 1), H(27»¢)] = i63(<F — 17)

11

• SO(3,1) D(S+~S·)(A) with dimension (25+ —l— 1)(2S- —|— 1) OCR Output

• SO(3) D(S)(R) with dimension 25+1

conjugation J (J') <—> J l"),Under parity J —> J M —>—M J> Jwhile under complexH) —<‘>have dimension (25+ —l— 1)(2S- -l- 1).the Lorentz group are labelled by two spins Si., S- = 0,1/2,1,- ·· anddimension 2S —l— 1, the finite dimensional irreducible representations ofof this group are labelled in terms of a spin S = O, 1/ 2, 1, - - ·; withcopies of the angular momentum algebra. Since the representations

new generators Ju). In terms of the latter the algebra becomes twowhile on the right hand side we have the commutators in terms of the

On the left hand side we have the commutators in terms of J ,M ,

[Mo Mil '-T -7; Gijk J]; [J:-, JJ?] = OUs Mil [4]}, 1

: iw Jkl-In Jil J“<*> = (fi iii)/2

representation:

pute the commutation relations of J and M by looking at a particularVVe also have translations for the inhornogeneous group. We can com

(rotations) (boosts)» .R(€¢) = €`“°€"’ B(€cb) = ¢`“*€‘M

(resp. rapidity) gb can be written as:finite rotations (resp. boosts) with axis the unit vector é', and angleby J (resp. M) the generators of infinitesimal rotations (resp. boosts);represented as a composition of a rotation and a boost. If we denote

tion theory of the Lorentz group. A Lorentz transformation can bein the theory of relativistic fields it is useful to review the representa

To understand better the type of fields and free Wave equations involved

Lorentz Group Representations

12

ufagu- 0’j = (1,-5) , OCR Outputuf0_’fu+ af; = (1,6) ,

We can construct four—vectors:

ill') = O0, flrl = ge? (0,1/2) ji: ga M = —l-éid

f<+> = %6’ j`) = O (1/2,0) Je: éc? M = -55

Respectively

in particular the spinors. We can consider either (1 / 2,0) or (0,1/2).To illustrate the previous classification we study some important cases;

Spinors

with negative polarization.

S+ = O, S- = 1 (O, 1) anti-dual antisymmetry tensor E— , ideml

cularly polarized electromagnetic wave with positive polarization.

S- = 1,S- = O (1, O) se1f—dua1 antisymmetry tensor E—I— , cir,

S- = S- = 1/2 (1/2, 1/2) ordinary vector.

S+ = O, S- = 1/2 (O, 1/2) left-handed spinor.

S+ = 1/2,S- = O (1/2, 0) right·handed spinor.

S+ = S- = O scalar field.

f: fir) + ft), M = ;(Ji+) — fl-)

Some examples:properties follow from the C1ebsch—Gordan decomposition of S+ ® S-.

f+ are the rotation group generators. Hence the rotationsw j`)

13

integer spin should be quantized according to Fermi-Dirac statistics OCR Output

to Bose—Einstein statistics (i.e. with commutators) while Helds of halffield theory that fields of integer spin should be quantized accordinging objects: c—numbers. It is a fundamental consequence of quantumtheory you would be tempted to believe that u+A,uQLB are commuteABu+AuQrB(e12 = —c12 = 1) is certainly an invariant. In classical Heldspinors. ui, has two components u.,,A, A = 1, 2; and the combination

can form a Lorentz scalar by an antisymmetric combination of two u+

spin (1/2, O) representation, recalling that 1/2 ® 1/2 = 1 ® Q, weone possible extra term in the quadratic Lagrangian. Since re., is a

Notice that the requirement of symmetry under u+ —> emur excludes

field.

For u- we have negative helicity —1 /2. (O, 1/2) represents the neutrinois the projection of angular momentum along the direction of motion.Hence 5 · = +1, and we have positive helicity +1/2. Helicity

(ku — k - 6‘)u+(k) = O

u+(¤¤) = u+(k)¢` "°‘””

Hence the plane wave solutions are massless kx = O. Further,

(80+6'·‘)u+=0=>¤u+=0@(3O—6’·V)u-=O=>\Ilu-=O

With equations of motion:

L= :l;tuf(8O - is · v)u_<·>

Li"`) = (:l;)iuj(30 + 5 · V)u+

number of derivatives

quadratic in ui, symmetric under ut —> ewui and the lowest possibleand the free Lagrangian follows if we require, that the Lagrangian be

14

i0Si,Gj,u+ = mu- Ip _ (ure) i0’i3,ju- == mur. _ u_ OCR Output

of motion

clude a pair of spinors, (u+, u-). This is a Dirac spinor, with equationsIf we want a parity invariant theory the simplest thing to do is to in

LECTURE 3

where M is a complex N >< N symmetric matrix.

MijuQAu{LBe—I— c.c.AB

u§rAi = 1, · - - ,N is a l\/lajorana mass:most general mass term we can write down for a collection of spinorsIf we do not worry about symmetries of the form ui, —> e‘°‘u+, the-1- and — spinors in terms of a theory with only -4- or only — spinors.formations). In other words, we can write a theory containing both(ex. check that this spinor transforms like ui, under Lorentz transThis means that instead of working with u-, we can work with iagufchanges J (J') and J(‘), it also exchanges left and right-handed fields.a Majorana mass. Finally, notice that since complex conjugation ex

limit for fermionic fields. The mass term we have just written is calledA of a given state cannot exceed one, there is no well—defined classical

ory. In fact since in the Fermi-Dirac statistics the occupation numberm is the mass parameter. This term would vanish in a classical the

“gTrL(€U+AU+B ‘l' C.C.) ,1 AB

single Weyl spinor:

orem. This also opens now the possibility of writing a mass term for a

(i.e. with anti-commutators); this is the celebrated Spin-Statistics The

15

(K — m)u = O —%- m)v = 0 with properties OCR Outputs = ;i;1/2u(/c, s)e""” 11(lc, s)"°`”

states:

The positive and negative energy solutions of — m)zp = O give 4

iw, ww, www, zigwwiw, wap

Basic biiinears are :

1/ L/ ' {#2 v }= 2n" v5 = —wOv1v2v3 Pr = ————,, 151 Zi:

And the Dirac matrices satisfy:

w — mw v" 5O ai USL, U U ai cr'}, OL = iw @0 - WMM =O 1 ai O 1 O O 05

The Lagrangian becomes

w E wvv“ E U E_ ““°0 1 1 O

Defining the Dirac conjugate:

L : up ip - mz/1+ 0 1 1 0JZOH O O o’f6’,,and Lagrangian:

16

H Z /dm;(EZ B), OCR Output1 .. .. 32 2

The Hamiltonian is:

,

where Eand H are the standard electric and magnetic fields.

Fuy T- GMA, · 6,,.A,u F0; 2 80Ai — 8iAO = Ei

c Z Zp FW Z Zi@B) 4 *‘" 2$2 '?

Lagrangian describing free photons is:

its properties are described by the electromagnetic theory. The basic

The best known example of a spin one particle is the photon, and

Spin 1

with the electric charge current in Quantum Electrodynamics.zpq/’*1/1 is a conserved current i.e. BMW = O. This current is associatedFurthermore, a consequence of the equations of motion is that j"

• (n + 1/2)·spin quantized with anticommutators

• Integer spin quantized with commutators

reminds us again of the spin statistic theorem:

Only if we use ACCR the Hamiltonian is bounded from below. This

1p(:z:) = (2¢r)32wk S"°`”(u(k s)b(k s)eik“”” +v(k s)d+(k s)e°’’’

In quantizing we have

{jsut]: .§é+m ZSv6=§é—m

UU = -2771 6*%**0 = 2k"

’l]')’/LU = 2.%**

17

L : -§Fu,,F’”’ -l— 1/1(v11D—— m)1lz OCR Output

Lagrangian is:

namics (QED) is obtained by coupling photons to Dirac ferrnions. Theof freedom while the vector potential has four. Quantum Electrody—

the description of a physical system. A physical photon has two degreesIt should be noticed that a gauge symmetry is more a redundancy in

Z E-i¢Qx»(/)

29 -> 9% ; 9 = €`”"(’”")

1

Ap —> AH + BMX

transformations

function (or the fields) of particles of charge Q also change under gaugeon whether we use one vector potential or the other. In QM the waveand AH + GMX. This is gauge invariance. Physics should not dependnotice that the same gauge field strength Fw, can obtained with Auvided by the Aharonov—Bohm effect. As soon as we work with Au wenecessary. A dramatic conhrmation of the need of using AM is proever, in order to write down the Schrodinger equation A is absolutely

equation of motion involve only E and B. In the quantum theory howIn classical physics we do not need to use the vector potential Ap. TheEIA;. == O showing the masslessness of photons.In the Landau gauge GMA" = O, the photon’s equation of motion is

AM : 2 <€LA)(k)G,(A)<k>€-ikx _‘_ EELA)(k)*a(A)+(k)€ikz> (2W) 2IkI

and the oscillator quantization of AH is given by

18

is: OCR Output

The general Lagrangian coupling gauge fields to scalars and fermions

Fw, E Fj,,T“———> UF,,,,U`

/,, ,,,F3*= 8/ia — 3,,AZ — gf°bCAZAf

The generalization of the gauge field strength is:

T“><“A, -> U(A,, + ga,,)U·1; U Z €*¢>—> U1/J

Gauge covariance is achieved by the following transformations:

EA,,

D,,<I> = 6,,<I>—l— ig AfT“<I>

for each generator of G and define:ant derivative of the field <I>, we introduce a photon like vector field Aj1,-- ·,dimG are the generators of the group G, to obtain a covari

transforming as some irreducible representations of G. If T“ ct =

ory we begin with a continuous group G, and choose a set of fieldsinteractions is indeed based on these theories. To set up a gauge thetheories. The standard model of strong, weak and electromagnetic

,__ An important generalization of QED is provided by non-abelian gaugecontract the equation with 8,, and use the antisymmetry of FW).which is consistent only if the electromagnetic current is conserved (just

@,iF"” = ¢Qv>v"¢

The equations of motion for the photon field are:

D;. ·y’*D,, , D,, = 3,, + ieA,,Q

19

EOMmI 6L 6L L; 6L : Kp ar: Eauaa + wap aaqsap =>p Qxm) OCR Output

then there is a conserved current:

6£=8NK"=>6S=O,

transformation:

grangian density changes by a total divergence under an infinitesimalIf without using the equations of motion one can show that the La

The statement of N oether’s theorem is:

:c’ = xv — cz

¢’<¤¤') = ¢(¤¤) 4>’(m) —¢<<¤> = —¤»“6,.¢

Consider the active action of symmetries, for instance for translations:

conservation law and a conserved current (Noether’s Theorem).tinuous symmetries. To each continuous symmetry we can associate a

In field theory there is a general consequence of the existence of con

Symmetries. Noether Theorem

LECTURE 4

M1, M2 at most linear in the scalar fields.

For a renormalizable Lagrangian V should be at most quartic and

¢(M1(¢) + We-M2(¢))i/J

L = —iFj,,F“"" +i1/>D¢ + D[$D”¢>— I/(gb)

20

j" = ¢@"1/J OCR Output

JW — mw 1/) —> eww

• Dirac fermion

Two important currents associated to phase rotations are:

symmetry acting on the fields.

• After canonical quantization i[Q, gb] = —6q5, hence Q generates the

• Q = f dgas j°(i', t) is conserved = O, and it is a Lorentz scalar.Q

If 3,,j** = O, there are some important consequences

T 00 = energy-density | P” == f d2x T 0**

newTALL]! Z GU G _ [LVL ‘l’ "

3L`

W_ _ GL (SL — QM

GMLZ a2" —> ;r*‘ + a" 6q5 = —a”3

current is the energy-momentum tensor.

One important example is translational invariance. The associated

6,,.]/* = O

6L` u = y .. u

21

+` '"ec·—> ;f;.r OCR Output

notion of running coupling constants comes about consider the process

including some comments. To understand how renormalization and the

for lack of time; however, the subject is important enough to justify

This section was not covered in the material presented in the lectures

Renormalization, Renormalization Group

fact that the centre of mass of the system moves with constant velocity.

Lorentz invariance implies conservation of angular momentum and the

etc.

jg : wTa'Y;i'¢E = — Gil} = i€aT“1b

Flavour symmetries

b+(/c) creates a state of charge -1

a*(k) creates a state of charge +1

(awk) (kl ?>*(k)b(/rl)/ (27T)32wkd3k

. . k —·zk:z b+ k zk·:c /(2T{)32wk<¤< le + < le ldgk

Q zi / d`$¤$(¢*80¢°· @0¢*¢)

2L = new — m¢*¢ cw = —m¢

• Complex scalar

22

gator. In the gauge or = 1 OCR Output

We look at the last diagram, and analyze the modified photon propa

O (94)(3 mOl'6)

Tree level O (ez)

order oz2 we have the following graphs.If we compute the scattering amplitude in perturbation theory up to

23

('qaqg + 77cx5q)H(q) OCR Output22

(ze + tara + qv + taTrlév (lf + vim 11 : - 2 (142 2 4 ° QM) 6 / ( ")

infrared divergence to worry about.

keep q2 large we can work in the limit me = O, as we will not have anq2 > mg, (the photon is time-like (off-shell) in our process) hence if weWe have arbitrary e+e" intermediate states with any energy. Take

<X|¢>~ 2 ——————-——— " (Ex _ En)(En " Ev)<x|Hmt|¤><¤|Hmt|¢><2>

the propagator in perturbation theory. Schematically:

This last diagram can be thought of as the second order correction to

H°"’(q)

vertices

factor two looppropagator | fermion

<2r>’* lkz — mi + iélllk + <1)2 — mi + ie]

iw QB agar —<··——+ = P · P ’q (6 + e+)G+ze < I q2+zek+q

24

at that scale. This coupling constant is called the running coupling OCR Output

at some scale, and everything can be expressed with respect to e2(,u)Since we do not know in principle what A is, we need to measure e

e(;.¢) = e(A) l + ln* 222 %E 311A

of Coulomb’s law at some scale pz

measure it. We can define the renormalized coupling to be the strengthwas a parameter in the Lagrangian but now we have to know how to

stand what is going on we must ask what is the coupling constant e. e

We have correction to Coulomb’s law (1/q2 term), however, to under

(1 —I—& ln jgtree + loop

= (vv”u)(uw»)·

we are computing is:

We seem to be in trouble. Do we have to abandon? The full process

4..3vr AH 2 l - , <q > H 2 G2 i2

physics could appear) and do the integral to obtainintegral (for instance with a cut·off A which could mark where newpoint—like interaction is too idealized. What to do'? Regularize thecoupling will have the same form all the way to infinite energies. Our

However, this is unreasonable. We do not really know if the *ye+e`fact that we are including intermediate states of arbitrarily high energy.

H(q2) is divergent logarithmically. The divergence is induced by the

25

(A.F.), while the right hand side corresponds to infrared freedom. OCR Outputfigures below. The left hand side corresponds to asymptotic freedom

At weak coupling the possible qualitative behaviour of B is shown in the

E — @(00dor

or using ln it = tz

/.4;- = B(o4) (6 — function)d

The logarithmic change of cr with the scale it is knows as the B—function

~ 1/128 it grows!

¤¢(Mz) = Mme) 1 + ZQ?1¤2 E M($

all charged particles:

if we want to know the value at the Z -mass, include contributions fromthe electron mass, (Thompson scattering), o4(m€) ~ 1/ 137. However,Normally we define o4(me), the fine structure constant at the scale of

6 (M) 3?uVQ/Jil _ 204 ,1; - 1 —I—ln;

Z Uf /10 (H) 1 — l2¤¢l11¤l73¢r)1¤(u/Mo)

32 2 M§¤»<~> = 2;-Q

constant. To see how it runs:

26

systems where limit cycles or strange attractors are also possible. In OCR Output

tions that appear are fixed points in contrast to the case of dynamicalIn QFT it seems that the only singularities of the ,6-function equa

contrast the infrared may be very complicated).trary (large) energy, although AF theories may have this property (inIn general it is too naive to imagine that a theory can exist to arbi

(if they exist).

High energy physics is dominated by ultraviolet fixed points.

Low energy physics is dominated by infrared fixed points.

izing the coefficients of leading operators (also anomalous dimensions).In principle low energy states influence low-energy physics by renormal

(IF).If H > 0 at ,@(040) = O => as E —> O the coupling becomes Weak

(AF).If ,O" < 0 at U(040) = O => as E —> oo the coupling becomes weak

tune 04 = 040, then cz(;r) = 040 for all ,u.The origin in the two pictures corresponds to a fixed point; i.e. if we

A.F. LE

27

All coupling can be given dimensions.

gauge fields [Au] : 1fermions [rp] = 3 / 2scalars [Q5] == 1

For:

l¢l = I

PX OCR Output3u¢8#¢ [gu] = 1[L] = 4 (energy units)

Given some Lagrangian we can count dimensions.

iii) Marginal.

ii) Relevant.

i) Irrelevant.

respect to a given fixed point into:

In modern RG parlance we can divide operators (interactions) with

is to a large extent independent of the specific microscopic interactions.

systems. There is an universality in thermodynamic behaviour which

Think of thermodynamics and the microscopic description of different

compared to their intrinsic scale.

All QF T look renormalizable at energy scales smallImportant principle (Wilson

proof (which may not exist).the only singularities. In four dimensions we are still lacking a similarpositivity of the energy A. Zamolodchikov showed that fixed points aretwo-dimensions, using locality, unitarity and Lorentz invariance and

28

tive. Although the machinery of the RG is complicated, the physical OCR Output

tions at low E are really independent of high energies, or very insensiirrelevant (non-renorrnalizable) couplings are indeed irrelevant. Predicthe appropriate running. Below the characteristic scale E < M theThe corrections can be systematically taken into account by including

XV;~ < V Getc.

coupling constants

we make all coefficients on the Lagrangian at low energies be runningthe coefficients of irrelevant operators, but they are normally small ifHigher order corrections by light fields also introduce corrections inmg/MgutrGUT theories p —+ 7i`U€+, O ~ uud e`, it is suppressed by Mg§T FFor instance in Fermi theory e` —l— ue —> e` + uc 0 ~ G§¤E2. In

45 = mw/¤)=> G ~ W1 - E2 2

Irrelevant operators are weak at low energies

called relevant operators.

of dimension 4 are called marginal, and operators of dimension < 4 are

from our point of view they are called irrelevant operators. OperatorsOperators of dimension > 4 are supposed to be non-renormalizable and

[g]=—2 em9¢1¢2¢z¢4 [@04] = 6 =>]= -2 g:§/M2Q6 _ [@56] = 6 =>

[>~]=0)~<Z>4 [Q54 = 4 =.>milnb [IW] = 3 => [m]=1

29

Consider a scalar theory with Lagrangian:

global chiral rotations of fermions.

variance. The other example we will deal with later has to do with

depend on the scale. Thus the quantum theory violates scale in·

our brief analysis of renormalization that the coupling constants

quantum theory generically this is not the case. We have seen in

scale invariant, physics will look the same at every scale. In theeters or dimension-full coupling constants, the classical theory issymmetries. If a classical theory does not contain mass param

fluctuations. Two classic examples are scale invariance and axial

classical theory respects a symmetry that is violated by quantum

These are the anomalous symmetries. This is the case when the

There is a third "away” of understanding symmetries in QFT.

generator there is a massless scalar particle associated.

symmetry is not manifest on the spectrum, and for every broken2) Nambu—Goldstone: [Q, H] = O, but QIO ># 0. In this case the

explicit multiplets of the symmetry group.

1) Wigner-Weyl: [Q,H] = (),Q|() >= 0 and the spectrum falls in

There are two modes of realizing the symmetry:

the Hamiltonian, [Q,H] = O, we are not guaranteed that Q|O >= 0.that the charge Q we obtain from Noether’s Theorem commutes with

A OCR OutputThere are two ways of realizing symmetric on a QFT. Even if we have

Goldstone’s Theorem: General remarks

ory.

just described. This ends our brief interlude into renormalization the

ing with effective Lagrangian if you understand the physical pictureideas are simple and intuitive. You should no longer be afraid of work

30

Take one more derivative with respect to q§“ and then set qz5’ = (q5’) OCR Output

6V = ·5(T°)¢y¢j I OGV

invariance of V(<j>) under the symmetry group means:

V ¢ = = Mz fl) 1*

massless particle in the spectrum. The mass matrix is given by:

vacuum invariant K A(<;’>} # O. We now show that for each K A there isas the unbroken subgroup. The broken generators K A do not leave theH = U, the H ’s generate a subgroup of the symmetry group knownsubsets {H i, K A}. The H i’s correspond to unbroken symmetries; i.e.Goldstone to 6 # O. We can divide the generators {T°} into twoWigner—Weyl mode corresponds to the case 6(gb’} = O. The NambuV(¢) == 0. Call these minima vacuum expectation values (VEV’s). TheThe minima are given by those configurations such that Vqff = O

V(¢) = d rv (Vd>) + V(<f>)fi§2. 1 " 1 2

Hence the potential energy is:

H : 01-Y+ (@gb)+ vw)2 li2/% (L2 2

this classical system are obtained as follows. The Hamiltonian for LZ is:

Invariance means that 6l/(gb) = O. The ground state configurations for

@$1 Z €“<T“>rj¢j

invariant under some symmetry group

L = §3H¢’3”q>’— V(¢) ,

31

should be careful when talking about degenerate vacua. OCR Output

diction with the existence of massless states. It simply says that we

vacua by means of any local process. This is however not in contra

In other words we cannot interpolate between any pair of different

(Oialfli ···Anl0,b} = 0

to show that we can always choose a basic such that

Then for any collection of local observables A1 ·· -./4,, it is not difiicult

(0.<1lU,b)=6az»¤J>=1,·-·,NP“|0, CL) = O

have a collection of states satisfying

we have multiple vacua. This is an incorrect statement. In fact if we

In the quantum theory we have to be careful about the statement that

Remark

the classical potential by the effective potential.

the same lines except that action is replaced by the effective action and

proves the classical form of Goldstone theory. The QFT proof follows

eigenvector of M 2((<{>)) with zero eigenvalue i.e. a massless mode, thisT “ to be K A, one of the broken generators, then (KA(¢)) # 0 is anIf Ta (45) = O, this equation has no consequences. However, if we take

M (<¢>)a(T“<¢>)j

first can be written as:

The second term vanishes because (gb} is a minimum for V, and the

*··f Ta · + ·+ Ta ‘· = O .8,< )Jk<¢k> Gd), |<¢>< hzGV32V *(3¢¢)<¢>

32

`\ /’ xt z —}g(t i m) OCR Output

as shown in the figure:

L, and R is time. The light-cone wraps around the space—time cylinder

is taken to be one—dimensional with the topology of a circle of length

keeping all the right ingredients, consider an ant world: Six R. Spacesymmetries. To exhibit them in the simplest possible example but

In the last lecture we would like to study some properties of anomalous

LECTURE 5

bosons in the limit that the up and down quark masses vanish.

better known example is provided by the pions which are Goldstone

bosons are precisely the phonons. In High Energy Physics one of the

ken group is the group of lattice translations. The associated Goldstone

temperature drops the ground state becomes a crystal and the unbro

the liquid phase we have translational invariance; however, when the

the acoustic branch of the dispersion relation. While the material is in

ter physics. Perhaps the most familiar are the phonons in a crystal in

Some examples of Goldstone bosons can be taken from condensed mat

33

• Positive energy solutions pll = E > O, pl < O OCR Output

For 2,9+ we have:

¢1(<v,p) = ¢"’°*‘”‘; pi p= 0” ll l

Looking for plane wave solutions:Hence w+ (resp. z/J-) represents left (resp. right) moving wave-packets.

(6`¤+6‘i)¢- =0 rb- =¢-(w-w)ll

(@0 · @i)¢+ = U ih = 1/#(93+ $)0 1

and the free Dirac equation for massless particles becomes:

$(*6, fc) = lib-}- 1/2A two-dimensional Dirac spinor has two components

w= —vv=l °l 7 {vlwll} = 01 U O -1

The analogue of 75 is

'Y“:U`z= S 'Y`=W`y:O 1 1 O O 1 -1 O

{vl‘,v”} = 2nll”; ul U = 0,1 nlll = 1 O O -1of external electric fields. The two—dimensional Dirac algebra is:

We wish to study massless fermions in two—dimensi0ns in the presence

34

" = ¢"rl‘v5"¢ = (ww- - ww- 7 -1Dw+ —¢w-) OCR Output

V" = www = (M1/1+ + ibw- » -Mw- + vw-)

rotations of rbi. In terms of a Dirac fermion they are:

and we have two conserved currents associated to independent phase

5 = wI(60 + <9i)¤!»- + ¢@f(6¤— 6i)¢- 7

The free Lagrangian for rbi is

{<1E,¤»E} = <$EE·» {bE,b`§»} = 6EE’ 7 {GE, UE} = U

E>O E<0

= E CLEUE + Z bg’lTLE

wave equation:

operators, and creation operators to negative energy solutions of the

When we quantize we associate to positive energy solutions annihilation

pl = 27m/L, neZ. The spectrum is easily portrayed in a picture:the ant world, 1/2i(:c + L) = ’(,/Ji(.'l3) the momentum pl is quantized:(and the reverse for 2p_). Choosing periodic boundary conditions in

35

vacuum: OCR Output

V” as in four dimensions) the electric field induces currents in therespect to it (i.e. the electromagnetic Held couples to the vector currentNext we turn a constant electric field E. lf 1/ri have unit charge with

If necessary some infinite constants are subtracted so the P”|O:k >= O.

0+} 0-}

right Dirac seas can be pictured as:

ground states. Representing filled states with filled dots, the left and

purposes this gives a good semiclassical description of the fermionic

wave function associated to all the negative energy states. For many

of whip-; or in many body language, we construct the Hartree—Fock

of the Dirac sea. Imagine that we fill out all the negative energy statesto construct |Oi >. One convenient way to describe |Oi > is in terms

ators we also need to construct the vacuum state. In our case we want

ln QF']? it is not enough to present the creation and annihilation opernumber of api, fermions plus (resp. minus) the number of zb- fermions.The charge associated to the vector current V" (resp. A*‘) count the

1W*P-lb = (ww- 7 —¢f1/L)Pa Z gu i 'Y5)

¢’y"P+¢l¤ = ($1% , —¢3§¢+)

or

36

ffc = ¢Qfi2>P+c2f +¢Q1z>P.c2+ ·OCR Output

is:

1, · - · ,Nf ; a = 1, ·· - , NC. The purely fermionic part of the Lagrangianto QCD with NC colors and N f massless flavors of quarks Q; fdimensions. One of the most relevant examples is a theory analogous

With some effort we can generalize the previous argument to fourquantum effects.

A current that is conserved classically (no Dirac sea) is violated byThis is the two-dimensional analogue of the Adler-Bell-Jackiw anomaly.

8,,A" ~ E

violation:

of 2p+ minus the number of 1/1- particles we see current conservation

presence of gauge iields. On the other hand if we count the numberis clear from the picture that it is conserved, hence 3pV" = O in thecreates e+e" pairs. If we count the number of 1p+ plus 2,0- particles, itFermi surface and in the other it augments it. This means that E

on the 1/2+ vacuum it produces particles. In one case E decreases the

In the rp- vacuum E creates holes which are seen as antiparticles, while

37

coupling the axial current to two vector currents: OCR Output

This result is obtained technically by computing a triangle diagram

where e””°"’ is the four—dimensional Levi-Civitta density.

n/C6’6AM N F;F¢;6uv¤B

A" = Z QJ’v”v5Q"

corresponding current A" has an anomaly:

we are in the same situation as before, and it can be shown that the

—> QZQWSQ :. €'LOt Q-}- €—ia Qphase rotations however:

sponds to the same phase rotation for Q+,Q-. If we make opposite

One of global U (1)’s is identified with baryon number and it corre

U(N;)+ >< U(N;)- = $U(Nf)+ >< SU(N;)- >< U(1)B >< U<1)A

and right-quark Havors. Hence the global symmetry group is:The symmetry of L corresponds to arbitrary unitary rotations of left

Qi = 6%+ Qifield Q; can be written as:same as u+ (resp. u-) from the spinor point of view. In fact the Diracwhere color is ignored, Pi = (1 :l; 75)/2, and P+Q (resp. P.Q) is the

38

the answer is proportional to the trace OCR Output

theory (quarks and gluons); and apart from some kinematical factorscould be evaluated by using the high energy degrees of freedom of the

terms of quark propagators the diagram is linearly divergent, hence it

The triangle graph can be understood in two ways: if we compute it in

simplest ways of measuring that the number of colors is three.

of the rr°-lifetime, and it is an efiicient color counter. It is one of thequarks running on the loop. The results provides an accurate prediction

the third component of isospin. Further Nc is the number of colors of

where Q is the electric charge of the quarks around the loop, and T3 is

N CTT T3Q

quarks. The graph is hence proportional to:

have photons and at the other the pion, and inside the loop run the

Since 7rO = (1/ (iu — dd), we have a graph where at two vertices we

TCO =>

decay:

useful application of the triangle graph is in the computation of ¢r°fixes the ambiguities in the graph and leads to the anomaly in AH. Oneimposing current conservation and Bose symmetry on the gluon lines

V (gluon)

V (gluon)

39

to the family structure of the Standard Model. We saw in QED that OCR OutputWe should mention one final application of the anomaly which is relatedpairs in superconductors.(QQ) ~ AQSB, analogous in some respect to the formation of Cooperacquiring a vacuum expectation valve (VEV) is a fermion compositemetry SU(N_;)+ >< SU(Nf)- —+ SU(Nf)v€cm. In this case the fieldto show that in QCD we should have the breaking of the chiral symditions. In fact for Nc = 3 and N f > 2 this matching can be usedconstraint on the theory known as the ‘tHooft anomaly matching contraviolet and the infrared regimes (as in QCD) we obtain a powerfulIf the theory is described by different degrees of freedom in the ulwe compute it in terms of the low or high energy degrees of freedom.

the coefficient of the kinematical factor should be the same whether

and high-energy information about the same theory. In other wordsgram. This means (as discovered by ‘tHooft) that we can relate lowbe identified with a singularity at zero value of q of the triangle dia—ture of the graph one finds that the origin of the anomaly can alsoto solve the U (1)-problem. By looking carefully at the analytic strucThis anomaly explains not only rr"—decay but it was also used by ‘tHooft

OMA" ~ Tr T1{T2, T 3} (kinematical factor)

hence the anomaly looks like:

of the three generators of the global symmetry appearing in the vertices;

T3#·-./xr~.rv·~ p2

Tgwvww P1

TT T1{Tg,T3}

40

(3,2,1/6) (1,2,-1/2) (1,1,+2/3), (1,1 — 1/3), (1,1, -1) OCR Output

or equivalently:

Oz = 1, 2, 3

1%f, , lu, , e-.-1U Ve d +,1/6 6 +,-1/2(¤)

family (e, ue, u, cl) the representations of G involved are:and right-handed fields is quite different. For instance for the firstand for a single family of quarks and leptons the assignments of left

G = SU(3)c >< SU(2)w >< U(l)y

the gauge group is:

the Standard Model of weak strong and electromagnetic interactions,

will not reappear to any order of perturbation theory. If we now look atanomaly is removed to the first non-trivial order (the triangle graph), ittently, and a theorem due to Adler and Bardeen guarantees that if thecase the corresponding currents can be coupled to gauge fields consis—

ories: theories where the —l— and —- representations coincide. In this

Hence anomaly freedom is guaranteed for example in vector-like the

EDIT TR+{TR+, TR+} — ZW TR_{TR_, TR_}

and T R_. Then the anomaly will be proportional to:G fermions will transform under some collection of representations TR+a given theory will or will not be anomaly free. For a given gauge group

appear in the quantum theory and therefore we should verify whether

the theory is to be consistent. We have seen now that anomalies may

currents coupled to the electromagnetic field should be conserved if

41

and excellent work in making these notes readily available.

I. Canon and A. Coudert from CERN°s DTP service for their diligencethe opportunity to present these lectures. I would also like to thankto participate in the Academic Training Programme and for giving meI would like to thank A. De Rujula and E. Verlinde for inviting me

Acknowledgements

of these (and other) issue in the literature.basic concepts of QFT, and to motivate the students to pursue someThis concludes these lectures. I hope they were useful to refresh somemysteries of High Energy Physics.why this structure repeats at least three times are some of the biggestthe quarks and leptons in a family are so chirally asymmetric, and

same cancellation can be checked for other choices of T1, T2, T3. Why

So that the anomalies of quarks and leptons cancel each other. Thequarks leptons

3 3 (ul tM ed—U tr). 1/ (J.

.o =3-2+ 2—- 3—— 3~——<-1>3=5 J 6 <%><%><%><%>

—TT Yi - TT YE

if T1,T2,T3 are all Y we find:

of anomalies within a family is a very instructive exercise. For example

weak isospin, and Y the weak hypercharge). Verifying the cancellationelectric charge is given by Q = T3 + Y (T3 is the 3rd component ofweak SU (2) representation and the third the weak hypercharge. Thewhere the first entry represents the color representation, the second the

42