Erik Verlinde

Home , Feynman diagram, Probability amplitude

AN INTRODUCTION TO QUANTUM FIELD THEORY

Erik Verlinde

Theory Division, CERN, CH-1211 Geneva 23, Switzerland

Abstract

The basic concepts and ideas underlying quantum eld theory

are presented, together with a brief discussion of Lagrangian eld

theory, symmetries and gauge invariance. The use of the Feyn-

man rules to calculate scattering amplitudes and cross sections is

illustrated in the context of quantum electro dynamics.

1. INTRODUCTION

Quantum eld theory provides a successful theoretical framework for describing

elementary particles and their interactions. It combines the theory of sp ecial relativity

and quantum mechanics and leads to a set of rules that allows one to compute physical

quantities that can b e compared with high energy exp eriments. The physical quantityof

most interest in high energy exp eriments is the cross-section . It determines directly the

number of events that an exp erimentalist will b e able to see, and also is in a direct way

related to the quantum mechanical probabilities that can b e computed with the help of

eld theory.

In these notes I will present the basic concepts and ideas underlying quantum eld

theory. Starting from the quantum mechanics of relativistic particles I will intro duce

quantum eld op erators and explain how these are used to describ e the interactions among

elementary particles. These notes also contain a brief discussion of Lagrangian eld theory,

symmetries and gauge invariance. Finally, after presenting a heuristic derivation of the

Feynman rules, I'll illustrate in the context of quantum electro dynamics how to calculate

the amplitude that determines the cross section for a simple scattering pro cess. But b efore

we turn to the quantum mechanics, let us rst discuss the meaning of the cross section in

more familiar classical terms.

1.1 From Events to Cross Sections

For concreteness, let us consider an exp eriment with two colliding particle b eams in

which one b eam consists of, say, electrons and the other b eam of p ositrons. An exp erimen-

talist can now collect and count the events in which an electron and a p ositron collide and

pro duce several other particles. The number of events N that o ccur in a certain amount

of time dep ends on various exp erimental details, in particular the densities and of

1 2

the particles inside the b eams and of course the particle velo cities ~v and ~v .

1 2

To nd out how these parameters a ect the number of events, let us rst consider

the probability that an event o ccurs inside a a small volume V and within a time

interval t. In this case we can assume that the densities and velo cities are constant.

Now imagine that one of the particles, say the electron, has a nite cross-section as seen

from the p ersp ective of the other particle, the p ositron. After a time t each electron

has moved with resp ect to the p ositrons over a distance j~v ~v jt, where ~v ~v is the

1 2 1 2

relativevelo city, and in this way it has swept out a part of space with a volume equal to

j~v ~v jt. The probability that in this time the electron has collided with a p ositron

1 2

is equal to the numb er of p ositrons inside this volume, whichis j~v ~v jt. Finally,

2 1 2

to obtain the total number N of collisions wehavetomultiply this with the number

of electrons inside the volume V ; this number is V . In this waywe nd that the

number of events that o ccur in V and t is

N = j~v ~v jV t (1)

1 2 1 2

In a realistic situation the densities (~x; t) of the particles and their velo cities are not

constant in space and time. Instead of the velo city ~v it is also more appropriate to use

the current density j that describ es the ux of the particles p er unit of area. It is de ned

by j = ~v where ~v is the lo cal velo city at a certain p oint in space and time. By rep eating

the ab ove argumentation we can express the number of events in an in nitesimal volume

~ ~

dV and time interval dt as dN = j j j jd ~xdt. Then byintegrating of space and

2 1 1 2

time we obtain the following expression for the total number of events N

N = dt L(t); (2)

where

~ ~

L(t)= d xj (~x; t)j (~x; t) (~x; t)j (~x; t)j: (3)

1 2 2 1

All exp erimental details that in uence the number of events are combined in the time

integral of L(t), which is called the integrated luminosity. The integrated luminositydoes

not dep end on the kind of pro cess that one is interested in, and so it can b e measured by

lo oking at a reference pro cess for which the cross section is accurately known. With this

knowledge one can then determine the cross sections for all other pro cesses. These cross-

sections are indep endent of the exp erimental details, and thus can b e used to compare

the data that come from di erent exp eriments. They are also the relevant quantities that

one wishes to compute using quantum eld theory.

1.2 Quantum Mechanical Probabilities

Just to give an idea of how the cross section is related to a quantum mechanical

amplitude, let us assume that b efore the collision the densities and velo cities of the par-

ticles inside the b eams are accurately known. In terms of quantum mechanics this means

1) A more careful analysis shows that the integrand has an additional relativistic con-

~ ~ ~ ~

tribution jj j j. This term can b e dropp ed in a situation where j and j are

1 2 1 2

(approximately) parallel.

that one has sp eci ed the initial quantum state j i of the colliding particles. Similarly,

by measuring the energies and directions of the particles that are b eing pro duced by the

collisions one selects a certain nal wave-function j i that describ es the state of the

out-going particles. According to the rules of quantum mechanics the probablity W for

this event to happ en is given by the absolute valued squared of the overlap b etween the

nal state and the initial state

W = jh j ij

i!f i f

Quantum eld theory provides the to ols to compute the amplitudes h j i, and hence

i f

determine the quantum mechanical probabilities W . In fact, the quantity that is most

directly computed is the probability W that a collision takes place p er unit of volume

and time for a situation with constant particle densities and .For this situation W

1 2

may b e identi ed with the ratio N=V t, which according to equation (1) gives the

cross section times the pro duct of the densities and relativevelo city. All this will b e

explained in more detail in the following sections.

2. RELATIVITY AND QUANTUM MECHANICS

The elementary particles that are used in high energy exp eriments havevelo cities

close to the sp eed of light, and should therefore b e describ ed by the theory of sp ecial

relativity.For relativistic particles it is convenient to sp ecify their motion, instead by

their velo city ~v , in terms of their momentum

m~v

(4) ~p =

1 ( )

An imp ortant reason for this is that during collision the total momentum of the particles

will b e conserved. The total energy of the particles will also b e conserved. In sp ecial

relativity the energy of a particle with momentum ~p is given by

2 2 2 4

E = ~p c + m c (5)

For a particle at rest this b ecomes Einstein's famous relation E = mc .

2.1 Relativistic WaveFunctions

In quantum mechanics the energy E b ecomes identi ed with the Hamiltonian op er-

ator H . The time evolution of the wave function for a single particle is, as usual, governed

by the Schrodinger equation in which the Hamiltonian H acts as a di erential op erator.

The form of this di erential op erator can b e read of from (5) by replacing the momentum

~p by the op erator p = ih r. This gives

ih@ (~x;t)= H (~x; t) (6)

with

2 2 4

H = h r + m c : (7)

An imp ortant di erence with the non-relativistic case is, however, that the Hamiltonian

involves a square ro ot. Still the Schrodinger equation can easily b e solved by expanding

the wave-function in terms of plane waves

~ ~

ik~xi! (k )t

(~x;t)=e :

~ k

These plane waves are the eigenfunctions of the momentum op erators p with eigenvalue

~ ~

p~ =hk and are also energy eigenstates with eigenvalue E =h! (k ). The most general

solution to the Schrodinger equation can now b e written as

d k 1

~ ~

ik ~xi! (k )t

(~x;t)= e (k ); (8)

(2 )

2! (k )

where

2 2

~ ~

! (k )= k +m : (9)

To simplify the formulas we put here and in the followingh = c = 1. When the momentum

~ ~

p~ is well sp eci ed the wave-function (k ) will b e strongly p eaked near k = ~p. The p osition

of the particle will in this case not b e very precisely determined due to the uncertainty

principle. So the wave function gives us only a probability (~x; t) for nding a particle at

a p osition ~x at a certain time t. Because probabilities have to b e conserved, there must

also exist a probability current density j (~x; t) that ob eys the conservation law

@ (~x; t)= r j(~x; t) (10)

For the relativistic wave function the probability density and current density are given by

the expressions

= ( @ @ )

t t

~ ~

j = ( r r ) (11)

These densities can in fact b e identi ed with the particle densities and j inside the

! k

b eam. Notice that for a plane wave these densities are given by = and j = .To

m m

verify that these expressions indeed satisfy the conservation law (10) one has to use the

`square' of the Schrodinger equation (6)

2 2 2

(@ r + m ) (~x;t)=0 (12)

This equation is called the Klein-Gordon equation. We note, however, that the Klein-

Gordon equation allows more general solutions than the original Schrodinger equation.

~ ~

Namely,ifwechange the sign of the time frequency ! (k ) !!(k) in the expression (8)

it would still b e a solution of (12). These negative frequency solutions are not physically

acceptable as wave functions b ecause they would corresp ond to particles with negative

energy.

2.2 The Dirac Equation

For particles with spin the wave function not only carries information ab out the

momentum and energy but also sp ecify the p olarization of the particle. An electron has

spin 1=2, and so its wave function has two indep endent comp onents (~x;t), for spin up,

~ ~

2) Here the normalization factor 2! (k )ischosen so that the function (k ) has a relativistic

invariant meaning.

and and (~x; t), for spin down. Both comp onents satisfy the Schrodinger equation (6)

~ ~

and through a similar expansion as (8) lead to two indep endent functions (k ) and (k )

" #

that can b e combined in a two comp onent "spinor"

~ ~

(k )= (k) (13)

In the rest frame of the electron the direction of the spin is measured by the Pauli matrices

. These are 2 2 matrices satisfying = 1 and = = i , etc.

i 1 2 2 1 3

Dirac discovered a relativistic wave equation for a spin particle in terms of a

4-comp onentwave-function .Itisconvenient to rst decomp ose this four comp onent

spinor in twotwo-comp onent spinors and .

and intro duce the matrices

! !

0 ~ 1 0

~ = and =

~ 0 0 1

The Dirac equation can then b e given in the form of a Schrodinger equation

i@ =(i~ r+m ) (14)

One of the motivations of Dirac to write down this wave equation is that the asso ci-

ated probability density and the probability current j do not involve time or spatial

derivatives, but take the simple form

y y

= ; j = ~ : (15)

Using the Dirac equation one easily veri es that these current densities satisfy the con-

servation law (10).

It follows from (14) that the separate comp onents of satisfy the Klein-Gordon

equation. The wave functions that describ e physical particles must again have p ositive

energies ! (k ) > 0. Hence, the physical solutions to the Dirac equation can b e expanded

in plane waves as

d k 1

ik ~xi! (k )t

e u(k ) (16) (~x;t)=

(2 )

2! (k )

where

!+m

u(k )= (17)

!+m

and (k ) is the two comp onent spinor intro duced ab ove. Notice that for an electron at rest

~ ~

the momentum k = 0 and energy ! (k )=m. As a consequence, the lower two comp onents

of u vanish while the upp er two comp onents are identi ed with the spinor . This shows

that the Dirac equation indeed describ es spin particles with two p olarizations. 2

2.3 Lorentz Covariant Notation

The invariance under Lorentz transformations of the Klein-Gordon and Dirac equa-

tions can b e made more manifest byintro ducing a covariant notation for the co ordinates,

x =(~x; t), where =1;2;3;0 and the derivatives @ =(r;@ ). We also intro duce a

metric with signature diag (1; 1; 1; +1) that is used to raise, lower, and contract

indices. The Klein-Gordon equation can in this notation b e written as

(@ @ + m )'(x)= 0 (18)

Here wechanged the notation for the wave function to ' to distinguish it from the

Dirac wave-function. The conserved probability current j =(j; ) b ecomes in covariant

notation j = (' @ ' '@ ' ):

To write the Dirac equation in covariant form we de ne the Dirac gamma-matrices

=(~ ; )by~ = ~ and = . These gamma matrices satisfy the anti-commutation

0 0

relations f ; g + =2 : With these de nitions one easily veri es that the

Dirac equation (14) coincides with

(i @ m) (x)= 0 (19)

The conserved probability current (15) is in covariant notation j = ; where =

3. QUANTUM FIELDS AND THE FOCK SPACE

To describ e the pro cesses that take place in high energy collisions it is not sucient

to restrict ones attention to wave functions of single particles. These pro cesses involve

several particles which at early times b efore the collision and also after the collisions are

at di erent p ositions ~x and may carry di erent momenta k . As long as the particles are

i i

well separated in space we can represent the n-particle wave function by a pro duct of the

single particle wave-functions

(~x ;~x ;:::;~x ;t)= (~x ;t) (~x ;t)::: (~x ;t)

1 2 n 1 2 n

In this regime the total energy is approximately equal to the sum of the energies of the

individual particles. The time evolution outside the interaction region may therefore b e

describ ed by the free Hamiltonian

2 2

r + m H =

i i

i=1

In non-relativistic quantum mechanics we are used to describ e interactions among particles

byintro ducing a p otential V (~x ;~x ;:::;~x ) in the Hamiltonian. However, this prescription

1 2 n

can not b e valid in a relativistic theory b ecause it would describ e an instantaneous force

between particles that are outside each others light-cone. As we will see, the correct wayto

describ e interactions is to allow the particles to emit and absorb other particles. Through

these interactions the numb er of particles maychange in time. Indeed, we know that a

collision of a electron and p ositron may result in the pro duction of many other particles.

This is what makes quantum eld theory di erent from usual non-relativistic quantum

mechanics in which the numb er of particles is always xed. The physical explanation of

this fact is that the relation E = mc allows the center of mass energy of the colliding

particles to b e used to create new particles. Wemay conclude from this that quantum

eld theory needs to b e formulated in a Hilb ert space that contains quantum states with

arbitrary numb er of particles.

3.1 The Fock Space

The Hilb ert space that consist of all multi-particle states is called a Fock space.

We will now explain how to construct this Fock space. First, there exists one unique

state without particles which is called the vacuum state. It is denoted by jvaci and is

normalized such that hvacjvaci = 1. Then there are states jk i that describ e one particle

with momentum k .We normalize these one-particle states as

0 3 (3) 0

~ ~ ~ ~ ~

hk jk i =2!(k)(2 ) (k k ): (20)

~ ~ ~

The quantum state for n particles with momenta k , k ,:::,k is denoted by

1 2 n

~ ~ ~

jk ; k ;:::;k i

1 2 n

These states have a normalization similar to (20) including factors of 2! (k )(2 ) .

In quantum eld theory it is p ossible to have transitions b etween states with di erent

numb er of particles. These transitions are describ ed with the help creation op erators a .

When the creation op erator a acts on the vacuum state jvaci it gives the one-particle

state jk i. More generally,by acting with a on a state with n particles one adds another

particle with momentum k , and hence one obtains a state with n + 1 particles. In this

way one concludes that any n-particle state is obtained from the emptyvacuum by acting

successively with n creation op erators

y y

~ ~

jk ;:::;k i = a :::a jvaci: (21)

1 n

~ ~

k k

Here we assumed for simplicity that all momenta k are di erent. Similarly we can de ne

so-called annihilation op erators a that reduce the numb er of particles with momentum

k by one. When these annihilation op erators act on a state that contains no particle with

momentum k one gets zero. In particular, the vacuum state jvaci satis es a jvaci =0:

The creation and annihilation op erators ob ey the commutation relations

3 (3) 0

~ ~ ~

[a ;a ]= 2!(k)(2 ) (k k ); (22)

which is similar to the algebra of the creation and annihilation op erators for the harmonic

oscillator.

3.2 Quantum Fields

As its name suggests, the central ob jects in quantum eld theory are the quantum

elds. A quantum eld is an (hermitean) op erator (~x; t) that at time t destroys or creates

a particle at the p osition ~x. More precisely,by acting with the quantum eld (~x; t)on

the emptyvacuum we obtain a state with one particle

1 d k

~ ~

ik ~xi! (k )t

(~x; t)jvaci = e jk i (23)

(2 )

2! (k )

Here we notice the similarity with the expression (8) for the single particle wave function.

But it should b e clear that quantum eld (~x;t) has quite a di erentinterpretation than

the wave function (~x;t). By expanding the eld (x; t) in terms of plane waves as

d k 1

~ ~

ik ~xi! (k )t ik ~x+i! (k )t

a e + a e (24) (~x;t)=

(2 )

2! (k )

that weintro duced b efore. one obtains the creation and annihilation op erators a and a

Notice that the quantum eld (x; t) satis es the Klein-Gordon equation (12), but that

unlike the wave-function (~x;t)itcontains in its expansion plane waves with negative

time frequency ! (k ).

The canonical commutation relations b etween the creation and annihilation op er-

ators imply that the quantum eld (~x; t) do es not necessarily commute with the eld

(~y; t ) at some other p oint in space and time. However, two op erators that act at the

same time t = t but at di erent p oints in space should commute. Otherwise there would

b e a contradiction with one of the p ostulates of relativity that no signal can travel faster

than light. Indeed one can show that

(3)

[ (~y; t);(~x;t)] = i (~x ~y ) (25)

where (~y; t)= @(~y; t) is called the canonically conjugate eld of (~x;t) b ecause their

relation is similar to that of the momentum p~ and the co ordinate ~x.We remark that by

expanding the eld and in plane waves one recovers from (25) all the commutation

relations (22) of the creation and annihilation op erators.

3.3 Quantum Fields for Particles with Charge and Spin

The eld that we just describ ed is the simplest example of a relativistic quantum

eld. It describ es an uncharged particle with zero spin such as the pion .We will now

brie y describ e the quantum elds for particles with charge and spin. An imp ortant fact

ab out Nature is that all charged particle have an asso ciated anti-particle with an identical

mass and spin but with opp osite charge. For example, the anti-particle for the pion is

the negatively charged pion , and the p ositron is the anti-particle asso ciated with the

electron. The quantum eld for the charged pions is a complex scalar with expansion

1 d k

~ ~

ik ~xi! (k )t ik ~x+i! (k )t

'(~x; t)= a (26) e + b e

(2 )

2! (k )

y y

where a creates a while b annihilates a . The other op erators b and a are

~ ~

k k

contained in the hermitean conjugate eld ' . The electron and p ositron are describ ed

by a complex Dirac spinor which has four comp onents satisfying the Dirac equation. The

creation and annihilation op erators are again found by expanding the Dirac eld in Fourier

mo des

1 d k

~ ~

a ik~xi! (k )t b ik~x+i! (k )t

~ ~

(~x; t)= u (k )e + d v (k)e (27) c

k;b

k;a

(2 )

2! (k )

a b

~ ~

(k ) is de ned in (17), with a = " or #, and v (k ) is given by a similar expression where u

with the upp er and lower comp onents interchanged. The rst term gives the creation

op erators c for the electron and the last term describ es the annihilation op erators d

k;b

k;a

for the p ositron. Again the hermitean conjugate eld gives the other op erators. Because

the electron and p ositron are fermions the creation and annihilation op erators satisfy

canonical anti-commutation rules

n o

3 (3) 0

~ ~

c ;c = 2! (k )(2 ) (k k ) (28)

~0 0

k ;a

k;a

This will incorp orate the Pauli exclusion principle, and leads, as we will see, to some slight

di erences in the Feynman rules for fermions compared to b osons.

3.4 The Feynman Propagator

The forces b etween elementary particles are often describ ed by the mediation of

another particle which is created at one p oint(~x;t) in space and time and then annihi-

0 0 0

lated at another p oint(~x;t ) with t >t. The quantum mechanical amplitude for sucha

successive creation and annihilation is given by the propagator

0 0 0 0 0

(~x ~x ;t t )=hvacj(~x ;t )(~x; t)jvaci for t >t (29)

For t> t one has to change the order of the two elds, b ecause the op erator that

corresp onds to the earliest p oint in time must also acts rst on the vacuum. So in this

0 0

case the particle is rst created at the p osition (~x ;t ) and than annihilated at (~x; t).

The propagator will play an imp ortant role in the Feynman rules. But just as a

preparation, we will describ e here howtoevaluate it. First we insert the equation (23)

and then use the normalization (20). In this waywe obtain

1 d k

0 0

~ ~

0 0 ik (~x~x )i! (k )jtt j

(~x ~x ;t t )= e (30)

(2 )

2! (k )

Notice that the propagators only dep ends on the di erences in the co ordinates. We can

0 0

use this to put ~x and t equal to zero. The ab ove expression is a direct generalization of

jmjj~xj

the familiar Yukawa p otential e =j~xj to whichitwould reduce in the static limit. We

can rewrite the propagator in a manifestly Lorentz invariant form by using the identity

ik t i! jtj

e e i

dk =

2 k 2! ! + i

whichisproven by a complex contour deformation. The Feynman propagator can thus b e

written in covariant notation as

d k

ik x

(x)= e (k ) (31)

(2 )

with

(k )= (32)

2 2

k m +i

and k =(k; k ) and x =(~x; t). In a similar way one can derive the Feynman propagator

for the Dirac eld. The only di erence is that wehave a sum over p olarization states of

the electron. From the normalization of the spinor u (k ) given in section 2.2 one nds

that as a result the integrand contains an additional factor

~ ~

u (k ) u (k )=(k +m) (33)

a a

are the gamma matrices intro duced in section 2.3. where

4. AMPLITUDES AND CROSS SECTIONS

In this section we will explain how the cross section is related to quantum mechanical

amplitude, and we will describ e how these amplitudes are represented in quantum eld

theory

4.1 From Amplitudes to Cross Sections

Consider an event where two particles with momenta ~p and ~p collide and pro duce

1 2

~ ~

n particles with momenta k , :::, k . Before the collision we can represent the wave

1 n

functions of b oth particles by planes waves , and .We will denote this two-particle

~p ~p

1 2

state by j~p ;~p i . Similarly, after the collision wehave n particles with wave functions

1 2 i

~ ~

given by plane waves. The corresp onding n-particle state is jk ;:::;k i . The quantum

1 n f

mechanical probability for this pro cess is determined by the transition amplitude

~ ~

A (p ;p ;k )= h~p ;~p jk ;:::;k i (34)

i!f 1 2 i 1 2 1 n

i f

These transition amplitudes are the ob jects that one would like to compute with quantum

eld theory. But b efore we discuss that in some detail, let us rst explain in what way

these amplitudes are related to the cross section, which after all is the observable that

can b e measured.

During a collision there are various physical quantities that are conserved. In par-

ticular, the total energy and momentum b efore and after the collision must b e the same.

This tells us that the transition amplitude A will vanish unless

! (k )=! (p )+! (p )

i i 1 1 1 2

i=1

k =~p + ~p

i 1 2

i=1

Examples of other conserved quantities are charge, lepton numb er etc. Now according

to the standard rules, we can now calculate the probabilityby taking the square of the

amplitude. Notice, however, since wehave xed the momentum of the pro duced particles,

we are only calculation a partial probability for pro ducing particles with momenta b etween

~ ~ ~

k and k + dk . This transition probabilityis

i i i

d k

4 2

~ ~

dW (~p ;~p ;k )= (2) jA(~p ;~p ;k )j (35)

1 2 i 1 2 i

(2 ) 2! (k )

To obtain the full probability one has to integrate of the nal momenta k . Implicit in the

expression (35) are the delta-functions for momentum and energy conservation. To make

them explicit one should multiply the right hand side by

P P

4 (3)

~ ~

(2 ) k ~p ~p ! (k ) ! !

i 1 2 i 1 2

i i

This transition rate dW dep ends on the normalization of the wave functions of the two

incoming particles. Because these wave-functions are represented by planes waves the

! (~p )

i i

asso ciated particle densities are = and j = .Now, to go from the transition

i i

m m

i i

probability to the di erential cross-section d we can follow the same steps of section 1.1.

~ ~

The luminosity is given by the integral of jj j j = j~p ! ~p ! j=m m .Ifwe divide

1 2 2 1 1 2 2 1 1 2

the transition rate dW by this factor we obtain the cross section. Thus we obtain

m m

1 2

d = dW (p ;p ;k ) (36)

1 2 i

j~p ! ~p ! j

1 2 2 1

This expression is valid in the center of mass frame. To extend it to general Lorentz frames

2 2

(p p ) m m . we simply note that numerator can b e written in covariant form as

1 2

In the following we will not really b e concerned with calculating explicitly these cross sec-

tions. Instead we will fo cus on the quantum mechanical amplitudes A and the probability

rate dW .

4.2 The Interaction Hamiltonian

In quantum mechanics the time evolution of a quantum state is describ ed bya

Schrodinger equation. For the situation of our interest the Hamiltonian H is an op erator

that acts in the Fock space of many particles, and can b e written a sum of the free

Hamiltonian H and an additional term H that represent the interactions among the

0 int

particles. What can wesay ab out H ? Because the interactions maychange the number

int

of particles the Hamiltonian will contain op erators that can create or destruct particles.

Furthermore, the interactions should b e consistent with causality, which means that the

presence of a particle can b e felt by another particle only when they have b een in causal

contact. This is guaranteed when the interactions that are contained in the Hamiltonian

only o ccur b etween particles that are at the same p oint in space and time.

To make a long story short, the appropriate way to construct a Hamiltonian that

satis es all the required prop erties is to express the interaction Hamiltonian as an integral

over space of a density H(~x; t). This Hamiltonian density H must b e given in terms of

a pro duct of the quantum elds and and, p ossibly, the rst order spatial derivatives

H (t)= d xH ; r; (37)

I int

In fact, also the free Hamiltonian is of this form with an integrand that is quadratic in

the quantum elds

3 2 2

(38) H = d x +(r)

This is what distinguishes the free Hamiltonian from the interaction Hamiltonian, whichis

an higher order op erator. A typical example of an interaction term is H (~x;t)= (~x; t).

int

In mo dern approaches to quantum eld theory one do es not often use the Hamilto-

nian density H, but instead one formulates the theory in terms of a Lagrangian density

L(~x; t). It is related to the Hamiltonian densityby

_ _

H(; )= L(; )

@ L

= (39)

The advantage of using the Lagrangian instead of the Hamiltonian formalism is that all

symmetries, like Lorentz invariance, are manifest. This is b ecause, instead of the Hamil-

tonian H whichisanintegral over space, one works with an action S that is written as

an integral over space and time of the Lagrangian density L. In the next section we will

discuss the Lagrangian formulation in more detail.

4.3 The S -Matrix

We will now describ e how the quantum mechanical probability amplitudes

~ ~ ~

A(~p ;~p ;k )= h~p ;:::;~p jk ;:::;k i

i 2 j 1 m 1 n

i f

are represented in terms of quantum eld theory. The idea is as follows. We consider a

quantum state j (t)i which at the time t = 1 coincides with the initial state. Then

by studying its time evolution we can in principle nd out what this state lo oks likeat

t =+1. The probability for nding the n particles in the nal state j (1)i is then

determined by the overlap of the n-particle state with this nal state.

So, once we know the Hamiltonian density H in principle all wehavetodoisto

int

solve the Schrodinger equation

i@ j (t)i =(H +H ) j (t)i (40)

t 0 int

S S

Without interaction this would have b een easy, b ecause the free Hamiltonian just evolves

each of the particles according to the usual one particle Schrodinger equation. In partic-

ular, it do esn't change the numb er of particles.

The non-trivial part of the time evolution is contained in the interaction Hamilto-

nian. One can describ e the quantum states in a di erent representation which only sees

the interactions by undoing the time evolution that is generated by the free Hamiltonian.

So weintro duce a new quantum state by

itH

j (t)i : (41) j (t)i = e

S I

The time evolution of this state is generated only by the interaction Hamiltonian

i@ j i = H (t)j i (42)

t int

I I

The interaction Hamiltonian H (t) is expressed as in (37). Note, however, that in this

int

picture the elds (~x;t) and hence H (t) are time dep endent. This time dep endence

int

simply means that the elds (~x; t) satisfy the Klein-Gordon, or Dirac, or some other

linear wave equation. Formally we can write down the solution to the Schrodinger equation

(40) as

j (t )i = T exp i dt H (t) j (t )i (43)

f I i

I I

where the time ordening symbol T implies that the op erators H (t)must act successively

on the state j (t )i in the order of the time t in their argument. More explicitly,

Z Z

T exp i dt H (t) (i) dt dt :::dt H (t ):::H (t )H (t ) (44)

I 1 2 n n 2 1

I I I

1 2

n=0 f

Next we insert the expression (37) for the interaction Hamiltonian and send t !1

and t !1 to get

j (1)i = T exp i d x H (x) j (1)i (45)

int

I I

4 3

where d x = dtd x and x =(~x; t). This exp onential op erator on the right-hand-side is

called the S -matrix. Its matrix elements give the probability amplitude that for an initial

n-particle state jp ;:::;p i the outgoing state is given by the m-particle state jk ;:::;k i.

1 n 1 m

This amplitude is

A(p ; k )=hk ;:::;k jT exp i d x H (x) jp ;:::;p i (46)

i j 1 m int 1 n

From this expression it is straightforward to derive the Feynman rules for a given Hamil-

tonian densite H (x).

int

4.4 Feynman Rules

Toevaluate these amplitudes one has to use p erturbation theory. One expands the

exp onent, taking in to account the time ordening, and than one inserts the explicit expres-

sion of the interaction Hamiltonian in terms of the elds. As mentioned, the Hamiltonian

density is some p olynomial in the elds and its derivatives. These elds create or destroy

particles at a certain p osition (~x; t) in space and time. The p ower of the elds tells us how

many particles are destroyed or annihilated at one p oint.

The various contributions to the scattering amplitude can b e represented graphically

in terms of Feynman diagrams. Given a diagram there is a very precise prescription to

write down the expression for the contribution in terms of the propagators and vertices of

a given quantum eld theory. Here we will here just sketch the basic idea. EachFeynman

diagram has a numb er of external lines that represent the incoming and outgoing particles.

For the ab ove amplitude the numb er of lines would b e m incoming and n outgoing. We

lab el these lines by the asso ciated momenta k and p . According to the Feynman rules

i i

one has to write down for each of the lines the appropriate single particle wave functions

describing a particle with external momentum k or ~p . Next, the external lines come

i i

together at vertices. These vertices are in one to one corresp ondence with the various

terms in the interaction Hamiltonian. For a simple situation in which H has just one

int

term there is just one typ e of vertex where 4 lines come together. For eachvertex one

has to write down one p ower of the coupling constant times an i that comes from the i

in front of the Hamiltonian.

Further, the sum of all momenta that owtowards a vertex must vanish. It is not

necessary that all lines that come together in one vertex corresp ond to external lines. A

Feynman diagram has also internal lines that corresp ond to the propagators. Just like the

external lines, the internal lines also carry momentum.For each propagator one has to

write down a factor that dep ends on its momentum. The precise form of this propagator

dep ends on the typ e of eld. For the Klein Gordon scalar eld that we considered so far

the propagator is

(k )= (47)

2 2

k m +i

2 2 2

~ ~

where k =(k; k ) and k = k k .Wehave already explained in section 3.4 how this

expression is derived. For diagrams without closed lo ops the condition that momentum is

conserved at eachvertex determines the momenta for all the internal lines in terms of the

external momenta k . But for diagrams with ` closed lo ops there will b e ` undetermined

momenta that havetobeintegrated over. Fermion lo ops have an additional factor of

(1) b ecause of fermi statistics. Finally, one has to sum over all inequivalentways that

the external lines can b e connected through propagators and vertices. What I have just

given is only a rough sketch of the Feynman rules. The precise rules dep end of course on

the quantum eld theory that one considers and, unfortunately, also on conventions.

5. LAGRANGIAN FIELD THEORY

We will leave aside the quantum elds for while, and consider classical eld theory

in its Lagrangian formulation.

5.1 The Action Principle

A classical eld theory may b e de ned in terms of an action S that is given as an

integral over space and time of a Lagrangian density L. This Lagrangian density L is a

Lorentz invariant function of the eld and its derivative @ .Thus in general the action

S takes the form

S = d x L((x);@ (x)): (48)

According to the action principle one obtains the classical eld equations by requiring

that the action is stationary, that means it do es not change under in nitesimal variations

of the elds ! + .For general values of the eld the the action S changes under

sucha variation by a small amount S given by

@ L @L

S = d x (x)+ @ ()

@ @(@ )

!# "

@L @ L

@ ) (49) d x

@ @(@

where we dropp ed a surface terms after the partial integration. In this waywe nd that

the variation of the action vanishes provided the eld (x) satis es the Euler-Lagrange

equation

@ L @ L

@ =0 (50)

@ (@ ) @

Note that to obtain a linear Euler-Lagrange equation the Lagrangian must b e quadratic

in the elds and its derivatives. But in general one can write down Lagrangians that ,

b esides a quadratic terms, involve higher order p owers. These Lagrangians will give rise

to non-linear eld equations. It is in this way that interactions are intro duced in the

Lagrangian formalism.

Wehave seen that the elds and asso ciated with particles with spin zero and

spin 1=2 satisfy the Klein-Gordon and Dirac equation. These wave equations can in fact b e

derived from an action principle. Because the wave equations are linear it is not dicult to

construct the corresp onding Lagrangrians L.For the Klein Gordon eld the Lagrangian

2 2

1 1

L = @ @ m ; (51)

2 2

L L

One easily checks that = m and = @ and thus the Euler-Lagrange equa-

tion indeed repro duces the Klein-Gordon equation. In a similar way one can nd the

Lagrangian for the Dirac eld

L = (i @ m) (52)

. = where 0

5.2 No ether's Theorem: An Example

Charged particles with spin, such as the pions, are describ ed by a complex scalar

eld '.For this eld the Lagrangian is

L = @ ' @ ' m ' '; (53)

Notice that the Lagrangian is real, despite of the fact that ' is complex. Moreover, when

wemultiply the eld ' by a complex phase factor

' ! e ' (54)

i i

the Lagrangian L will remain the same b ecause (e ) e = 1. This has an imp ortant

consequence. Namely, supp ose that is a function of the space and time co ordinate x .

In that case the transformation (54) is not a symmetry of the Lagrangian or the action.

Yet, when the elds satisfy the equation of motion, we know that the action should b e

invariant under in nitesimal transformations ' ! ' + '.For an in nitesimal phase

rotation (54) with a small parameter the eld ' varies with ' = i '. After inserting

this variation in the action one can easily see that all terms without derivatives on

cancel. The terms that are left over are

S = ie d x@ ( )(' @ ' '@ ' )

From the action principle we know that this variation should vanish, and therefore we

conclude that the expression

j = ie(' @ ' '@ ' ) (55)

must satisfy the conservation law

@ j =0: (56)

e e

This can also b e checked directly from the equation of motion. The current j =(j ; )

represents the electric current and charge density. The fact that the presence of a conserved

current is asso ciated with a symmetry of the Lagrangian is known as No ether's theorem.

For a complex Dirac eld, which describ es the electron and p ositron, one can in a similar

way determine the expression for the electric current density.Itis

(57) j =e

This current will play an imp ortant role in describing the interactions with the electro-

magnetic eld.

5.3 Photons and the Electro Magnetic Field

The photon is a particle with spin one, and is describ ed bya vector eld A =(A;V ).

These elds A and V are in fact the vector and scalar p otential for the electric eld

~ ~ ~ ~ ~ ~

E = @ A rV and magnetic eld B = rA. Already in the previous century Maxwell

~ ~

wrote down the eld equations for E and B

~ ~ ~

rB @ E=j

t e

~ ~

rE= (58) e

where and j are the electric charge and current densities. The fact that these equations

e e

are Lorentz invariantwas discovered only later, and led to the development of sp ecial

relativity.Nowadays, we write often the Maxwell equations directly in a covariant form

in terms of the electromagnetic eld strength

F = @ A @ A (59)

which combines the electric and magnetic eld: F = E , F = B . The equation (58)

0i i ij ij k k

can b e combined as

@ F = j (60)

Notice that this equation only has solutions when the current j satis es the conservation

law (56), b ecause for the left-hand-side of this equation we automically have @ @ F =0.

The Maxwell equations can again b e derived from an action principle. The Lagrangian is

given as a function of A (x)by

(@ A @ A ) +A j (61) L =

It is not dicult to verify that the Euler-Lagrange equations give the equation (60). For

example, the linear term in A directly leads to the right hand side in terms of the electric

current.

6. QUANTUM ELECTRODYNAMICS

Quantum Electro Dynamics (QED) describ es the electro-magnetic interactions b e-

tween electrons and p ositrons. By combining the results of the previous section we can

easily construct an action for the electron and p ositron eld together with the electro-

magnetic eld. Namely,we simply take the sum of the Lagrangians(52) and (61)

L = L + L

QE D A

and identify the electric current with the No ether current (57), that is j = e .

Combining these ingredients we can write the Lagrangian of quantum electro dynamics

F L = (i D m) (62)

QE D

where

D =(@ ieA ): (63)

is called the covariant derivative.

6.1 Gauge Invariance of the Lagrangian

Avery imp ortant prop erty of this Lagrangian is that it is invariant under lo cal

gauge transformations. These tranformations act on the eld A as

@ : (64) A ! A +

and simultaneously on the fermion eld and as complex phase rotations

! e ;

! e (65)

(@ @ @ @ )= First, one easily checks that the eld strength is invariant: F ! F +

F . Next, the fact that the derivative @ on the fermion eld is replaced by the `covariant

derivative' D guarantees the invariance of the kinetic term of the fermions b ecause it

i i

follows from (63) and (64) that under a gauge transformation D ! e D e and so

D ! e D (66)

This phase factor cancels against the one of . The signi cance of the gauge invariance is

that it ensures that the photon only has twophysical p olarizations and remains massless.

6.2 The Feynman Rules of QED

Given the form of the Lagrangian one can read of the Feynman rules in a simple

manner by writing the action in terms of Fourier mo des. The quadratic terms in the elds

describ e the kinetic terms and determines the form of the propagator (p). To b e precise,

the propagator is given by the inverse of the kinetic op erator written in momentum space.

The propagators for the electron-p ositron eld is

q + k

p + m

2 2

p m + i

whichwas already derived in section 2.4. To nd the propagator for the photon is some-

what more tricky due to the gauge invariance. Because of this the kinetic term would

not giveaninvertible expression in momentum space. The right pro cedure is to add a

so-called gauge- xing term

1 2

L = (@ A ) (67)

g:f:

So that the kinetic term is invertible. This gives the following photon propagator

+(1 )q q =q

(q )= i (68)

q + i

Gauge invariance ensures that the result is indep endentof .We can use this fact to

cho ose = 1, so that the expression simpli es. We will use as photon propagator

q + i

The Lagrangian (62) contains only one term that is not quadratic in the elds. This

interaction term is

L = eA (69)

int

We conclude that QED has only one vertex, namely one with two fermion lines and one

photon line. The corresp onding factor is

The external lines for the electrons are represesented by their wave functions u(~p or v (~p)

u(p) and v (~p for outgoing lines. Similarly, the external photons for incoming lines, and by

have a factor (k ) that describ es their p olarization. The physical p olarization satisfy

k =0.Furthermore, due to the gauge invariance the Feynman amplitudes must b e

invariant under + k . This implies that longitudinal photons with = k

decouple, and hence the photon has only twophysical p olarizations. Finally, eachloop

momentum integral has a normalization factor (2 ) .

6.3 ATree Level Calculation

As an illustration of the Feynman rules let us consider the scattering amplitude

+ +

for the pro cess e e ! .At tree level there is only one Feynman diagram that

contributes. It is given by

v (k )

v (p )

u(p ) u (k )

1 1

+ +

Feynman diagram for e e !

In words what happ ens is that the electron and p ositron annihilate and pro duce a virtual

photon. This photon propagates for a while and then decays in to the muon-pair and

. The amplitude for this pro cess is given by

0 0

ab b a a b

~ ~ ~ ~

(~p ;~p ;k ;k )= ie A v (~p ) u (~p ) ie u (k ) v (k ) (70)

0 0

1 2 1 2 1 1 1 2

a b

(p + p )

1 2

The transition rate W is obtained by squaring this amplitude. For unp olarized electrons

and p ositrons one should average over the initial p olarizations a and b. Also, when one

0 0

ignores the muon-p olarization one has to sum over a and b . Combining these ingredients

one nds that the transition rate take the form

1 4e

ab 2

dW = jA (~p ; k )j = L (p ;p )L (k ;k ) (71)

0 0

i j 1 2 1 2

a b

4 s

0 0

a;b;a ;b

where

L (p ;p )= u (~p ) v (~p )v (~p ) u (~p ) (72)

1 2 a 1 b 2 2 1

b a

a;b

Here we also intro duced the Mandelstam variable s =(p +p) which is equal to the

1 2

square of the center of mass energy.Now, using identities like (33) and after evaluating

the trace of the gamma matrices one nds

p + (p p + m ) (73) +p L (p ;p )=p p

1 2 1 2

2 1

1 2

We will now assume that the energies are large so that we ignore the terms with the mass

m . Inserting the expression for L in to

2 2

t +u

dW =2e (74)

2 2

where t =(p k ) and u =(p k ) are the two other Mandelstam variables. This

1 1 1 2

result can now b e compared to exp eriments, and gives a way of determining the value of

the coupling constant e.

6.4 Lo op Corrections

At one lo op there are four kinds of Feynman diagrams that give corrections to the

tree level result. First there is the p ossibility that the electron emits a photon whichit

again re-absorbs b efore it annihilates with the p ositron. This gives the electron self-energy.

Then there is a so-called vertex correction where a photon is emitted by the electron and

absorb ed by the p ositron b efore b oth annihilate. Another p ossibility is that the emitted

photon is absorb ed only later by one of the pro duced muons.

The remain typ e of correction, whichwe will consider in much more detail, is the

so-called vacuum p olarization graph. In words it describ es a pro cess where the photon

that is created out to the electron and p ositron pro duces rst a virtual lepton (e e

or ) pair. This virtual pair again annihilates and pro duces a photon which then

in second instance pro duce the muon pair. So this correction comes from an additional

fermion lo op inside the photon propagator.

q + k

Vacuum p olarization diagram

This diagram will change the value of the electric charge e and in fact make it dep endent

on the center of mass energy of the initial electron and p ositron. Intuitively what happ ens

is that the vacuum contains uctuations of lepton pairs ` and ` . Through the electric

force the p ositively charged virtual leptons ` move somewhat towards the electron while

the virtual ` particles movetowards the p ositron. In this way the virtual uctuations

"p olarize" the vacuum and screen the electric charge e of b oth the electron and the

p ositron. This e ect b ecomes more imp ortant when the center of mass energy of the

colliding electron and p ositron is large. This may b e understo o d intuitively from the fact

that to form a virtual electron-p ositron or muon pair the vacuum has to "b orrow" part of

this center of mass energy and so the numb er of virtual pairs increases when more energy

is available .

The vacuum p olarization graph only a ects the part of the diagram that represents

the photon propagator. In fact, without much additional work one can include also higher

lo op corrections that corresp ond to rep eated vacuum p olarization diagrams. These can

b e summed up by a geometric series. Its e ect can b e included by shifting the inverse

propagator (q )by a term as follows

2 2 1

(q )= iq ! iq ( + (q )) (75)

where we again put = 1 and

Tr ( k + m) ( (q + k ) + m)

4 2 2

d k q (q )= (76)

4 2 2 2 2

(2 ) (k m )((q + k ) m )

3) Mysteriously enough, this intuition fails for non-ab elian gauge elds, where one nds

an "anti-screening" e ect that is resp onsible for asymptotic freedom.

It follows from Lorentz invariance and gauge invariance (decoupling of the longitudinal

p olarization of the photon) that this expression must b e of the form

2 2 2

(q )= (q q q )(q ) (77)

This can b e veri ed explicitely byevaluating the gamma trace and doing some manipula-

tions in the integral. Fruther, we will again assume that the energy carried by the photon

is much larger that the electron mass, and so we drop the terms dep ending on m. In this

way one nds

1 e

4 2

d k (q )= (78)

4 2 2

(2 ) k (q + k )

This integral is in nite. This is of course rather disturbing result, but fortunately there is

still a way to extract a sensible answer. First one has to regulate the integral so that it is

nite, and then apply a so-called renormalization pro cedure. Here we will not go in to o

much detail, but we just want make clear what the essential idea b ehind renormalization.

6.5 Regularization and Renormalization

There are various way to regulate the ab oveintegral expression. The simplest fashion

is to analytically continue to euclidean space and to intro duce a momentum cut o by

restricting the integral over k to values with jk j < . Again omitting some details, one

gets

2 2

e d(k )

(q )=

2 2

12 k

2 2

e q

= ln (79)

2 2

This one-lo op correction can now b e inserted in the Feynman graph for the full pro cess,

and in this way added to the tree-level result. Because it only corrects the photon propa-

gator the result is of the same form as the tree-level, but with an additional s dep endent

factor. This eventually leads to

2 2 2

t +u e s

dW =2e 1 ln (80)

2 2 2

s 12

Now there app ears to b e a problem with this expression. Namely, it dep ends on the cut-o

, whichwas chosen arbitrarily. In particular, when wewould send the cut-o !1

wewould again nd a divergent answer, at least when wekeep e xed. The solution to

this problem is that the coupling constant e that app ears in this expression should not

really b e identi ed with the physical value of the electric charge that we measure in the

lab oratory. Therefore, we are allowed to make this `bare' coupling e = e () dep endent

on the cut-o in sucha way that the physical observables like dW are indep endentof

Supp ose we rst do the exp eriment at a certain center of mass energy s = . Then

at this value of s we can now use the tree-level expression (74) to de ne the physical

coupling constant e()by

2 2

t +u

dW =2e () (81)

js= s

Comparing this to the one lo op result (80) one concludes that

2 2

e ()

2 2

(82) e ()=e () 1 ln

2 2

where we made explicit that the bare coupling e () dep ends on the cut-o . This equation

now tells us what e () is in terms of the physical coupling e(). Next we can re-express

the result for the physical observable dW at other values for s in terms of the physical

coupling e(). In this way one nds that cut-o disapp ears from the expression for dW ,

b ecause it follows from (82) that

! !!

1 1

2 2

e () e () s s

2 2

e () 1 = e () 1 (83) ln ln

2 2 2 2

12 12

and so we can write the transition probability in terms of the physical coupling constant

2 2 2

t +u e () s

dW =2e () 1 (84) ln

2 2 2

s 12

This expression is identical to (80) but now the cut-o is replaced by the mass scale ,

which is called the renormalization scale. Still there app ears to b e the problem that the

answer for dW dep ends on the arbitrary scale , while we know that the physical value

of dW must b e indep endentof .However, this condition precisely determines the way

that e() dep ends on . So nally,we conclude that the e ect of the one-lo op vacuum

p olarization can b e summarized by equation (81) with a running coupling constant e().

Given the value of e()atlow energies, sayat a few eV ,we can determine its value

at other values from the condition that value of dW remains unchanged. This gives

2 2

e ( )

2 2

e ()=e ( ) 1 ln (85)

Atlow energies we know the value of the coupling constantvery well: wehave ( )=

e ( )=4 =1=137. The value at higher energies follows from the ab ove relation which

may b e rewritten in terms of the ne structure constant ()=e ()=4 as

1 1 1

= ln( ): (86)

() ( ) 3

In fact, at high energies one also has to take in to account the contribution to the vacuum

p olarization of other charged particles, in particular the muon. This leads to additional

somewhat. This running of the ne structure contributions that change the co ecient

constant has b een con rmed by high energy exp eriments, in particular at LEP.