Introduction to Path Integrals in Field Theory

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Introduction to Path Integrals in Field Theory

U. Mosel1

Institut fuer Theoretische Physik, Universitaet GiessenD-35392 Giessen, Germany

SS 02July 6, 2002

1http://theorie.physik.uni-giessen.de



1

This manuscript on path integrals is based on lectures I have

given at the University of Giessen. If you find any conceptualor typographical errors I would like to learn about them.

In this case please send an e-mail to

[email protected]



Contents

I Non-Relativistic Quantum Theory 6

1 PATH INTEGRAL IN QUANTUM THEORY 71.1 Propagator of the Schrodinger Equation . . . . . . . . . . . . 71.2 Propagator as Path Integral . . . . . . . . . . . . . . . . . . . 101.3 Quadratic Hamiltonians . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Cartesian metric . . . . . . . . . . . . . . . . . . . . . 141.3.2 Non-Cartesian metric . . . . . . . . . . . . . . . . . . . 15

1.4 Classical Interpretation . . . . . . . . . . . . . . . . . . . . . . 17

2 PERTURBATION THEORY 20

2.1 Free propagator . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Perturbative Expansion . . . . . . . . . . . . . . . . . . . . . . 222.3 Application to Scattering . . . . . . . . . . . . . . . . . . . . . 27

3 GENERATING FUNCTIONALS 32

3.1 Groundstate-to-Groundstate Transitions . . . . . . . . . . . . 333.1.1 Generating functional. . . . . . . . . . . . . . . . . . . 37

3.2 Functional Derivativesof Transition Amplitudes . . . . . . . . . . . . . . . . . . . . . 38

II Relativistic Quantum Field Theory 43

4 CLASSICAL RELATIVISTIC FIELDS 44

4.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 444.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Symmetries and Conservation Laws . . . . . . . . . . . . . . . 504.2.1 Geometrical Space–Time Symmetries . . . . . . . . . . 514.2.2 Internal Symmetries . . . . . . . . . . . . . . . . . . . 53

2



CONTENTS 3

5 PATH INTEGRALS FOR SCALAR FIELDS 56

5.1 Generating Functional for Fields . . . . . . . . . . . . . . . . . 575.1.1 Euclidean Representation . . . . . . . . . . . . . . . . 59

6 EVALUATION OF PATH INTEGRALS 62

6.1 Free Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . 626.1.1 Generating functional . . . . . . . . . . . . . . . . . . . 626.1.2 Feynman propagator . . . . . . . . . . . . . . . . . . . 646.1.3 Gaussian Integration . . . . . . . . . . . . . . . . . . . 68

6.2 Stationary Phase Approximation . . . . . . . . . . . . . . . . 716.3 Numerical Evaluation of Path Integrals . . . . . . . . . . . . . 74

6.3.1 Imaginary time method . . . . . . . . . . . . . . . . . 746.3.2 Real time formalism . . . . . . . . . . . . . . . . . . . 75

7 S -MATRIX AND GREEN’S FUNCTIONS 78

7.1 Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 787.2 Reduction Theorem . . . . . . . . . . . . . . . . . . . . . . . . 80

7.2.1 Canonical field quantization . . . . . . . . . . . . . . . 807.2.2 Derivation of the reduction theorem . . . . . . . . . . . 82

8 GREEN’S FUNCTIONS 87

8.1 n-point Green’s Functions . . . . . . . . . . . . . . . . . . . . 87

8.1.1 Momentum representation . . . . . . . . . . . . . . . . 888.1.2 Operator Representations . . . . . . . . . . . . . . . . 89

8.2 Free Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . 918.2.1 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . 918.2.2 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . 92

8.3 Interacting Scalar Fields . . . . . . . . . . . . . . . . . . . . . 948.3.1 Perturbative expansion . . . . . . . . . . . . . . . . . . 96

9 PERTURBATIVE φ4 THEORY 99

9.1 Perturbative Expansion of the Generating Function . . . . . . 99

9.1.1 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . 1019.1.2 Vacuum contributions . . . . . . . . . . . . . . . . . . 1029.2 Two-Point Function . . . . . . . . . . . . . . . . . . . . . . . . 103

9.2.1 Terms up to O(g0) . . . . . . . . . . . . . . . . . . . . 1039.2.2 Terms up to O(g) . . . . . . . . . . . . . . . . . . . . . 1049.2.3 Terms up to O(g2) . . . . . . . . . . . . . . . . . . . . 107

9.3 Four-Point Function . . . . . . . . . . . . . . . . . . . . . . . 1099.3.1 Terms up to O(g) . . . . . . . . . . . . . . . . . . . . . 1099.3.2 Terms up to O(g2) . . . . . . . . . . . . . . . . . . . . 110



CONTENTS 4

10 DIVERGENCES IN n-POINT FUNCTIONS 113

10.1 Power Counting . . . . . . . . . . . . . . . . . . . . . . . . . . 11410.2 Dimensional Regularization

of φ4 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11610.2.1 Two-point function . . . . . . . . . . . . . . . . . . . . 11710.2.2 Four-point function . . . . . . . . . . . . . . . . . . . . 118

10.3 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . 12210.3.1 Renormalization of φ4 Theory . . . . . . . . . . . . . . 125

11 GREEN’S FUNCTIONS FOR FERMIONS 128

11.1 Grassmann Algebra . . . . . . . . . . . . . . . . . . . . . . . . 128

11.1.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 13011.1.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . 13211.2 Green’s Functions for Fermions . . . . . . . . . . . . . . . . . 138

11.2.1 Generating Functional for fermions . . . . . . . . . . . 13811.2.2 Green’s Functions . . . . . . . . . . . . . . . . . . . . . 141

12 INTERACTING FIELDS 144

12.1 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 14412.1.1 Fermion Loops . . . . . . . . . . . . . . . . . . . . . . 145

12.2 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 14812.3 Removal of Degrees of Freedom:

Yukawa Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 15012.3.1 Perturbative Expansion . . . . . . . . . . . . . . . . . 152

13 PATH INTEGRALS FOR GAUGE FIELDS 157

13.1 Gauge invariance in Abelian theories . . . . . . . . . . . . . . 15813.2 Non-abelian gauge fields . . . . . . . . . . . . . . . . . . . . . 16213.3 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

13.3.1 Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . 16913.4 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 171

13.4.1 Faddeev-Popov Determinant . . . . . . . . . . . . . . . 171

13.4.2 Ghost fields . . . . . . . . . . . . . . . . . . . . . . . . 17513.4.3 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . 176

14 EXAMPLES FOR GAUGE FIELD THEORIES 184

14.1 Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . 18414.2 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . 18414.3 Electroweak Interactions . . . . . . . . . . . . . . . . . . . . . 186



CONTENTS 5

A Units and Metric 189

A.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189A.2 Metric and Notation . . . . . . . . . . . . . . . . . . . . . . . 190

B Functionals 192

B.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192B.2 Functional Integration . . . . . . . . . . . . . . . . . . . . . . 193

B.2.1 Gaussian integrals . . . . . . . . . . . . . . . . . . . . 193B.3 Functional Derivatives . . . . . . . . . . . . . . . . . . . . . . 196

C RENORMALIZATION INTEGRALS 198

D GRASSMANN INTEGRATION FORMULA 202

E BIBLIOGRAPHY 205



Part I

Non-Relativistic Quantum

Theory

6



Chapter 1

PATH INTEGRAL IN

QUANTUM THEORY

In this starting chapter we introduce the concepts of propagators and pathintegrals in the framework of nonrelativistic quantum theory. In all theseconsiderations, and the following chapters on nonrelativistic quantum theory,we work with one coordinate only, but all the results can be easily generalizedto the case of d dimensions.

1.1 Propagator of the Schrodinger Equation

We start by considering a nonrelativistic particle in a one-dimensional po-tential V (x). The Schrodinger equation reads

Hψ(x, t) = − h2

2m

∂ 2ψ(x, t)

∂x2+ V (x)ψ(x, t) = ih

∂ψ(x, t)

∂t. (1.1)

This equation allows us to calculate the wavefunction ψ(x, t) at a later time,if we know ψ(x, t0) at the earlier time t0 < t. For further calculations werewrite this equation into the following form

ih∂

∂t− H

ψ(x, t) = 0 . (1.2)

Next, we consider the function K (x, t; xi, ti) which is defined as solutionof the equation

ih∂

∂t− H

K (x, t; xi, ti) = ihδ(x − xi)δ(t − ti) . (1.3)

7



CHAPTER 1. PATH INTEGRAL IN QUANTUM THEORY 8

K is the “Green’s function” of the Schrodinger equation (K is also often

called “propagator”) with the initial condition

K (x, ti + 0; xi, ti) = δ(x − xi) . (1.4)

The solution of the Schrodinger equation (1.2) can be written as

ψ(x, t) =

K (x, t; xi, ti) ψ(xi, ti) dxi (1.5)

for t > ti (Huygen’s principle). Relation (1.5) can be proven by inserting thelhs into the Schrodinger equation

ih

∂ ∂t

− H

K (x, t; xi, ti) ψ(xi, t) dxi

= ih

δ (t − ti) δ (x − xi) ψ (xi, t) dxi

= ihδ (t − ti) ψ(x, t) = 0 for t > ti . (1.6)

Thus the ψ defined by (1.5) is indeed a solution of the Schrodinger equationfor all times t > ti. K (x, t; xi, ti) is the probability amplitude for a transitionfrom xi, at time ti, to the position x, at the later time t. The restriction tolater times preserves causality.

We can find an explicit form for the propagator, if the solutions of thestationary Schrodinger equation, ϕn(x), and the corresponding eigenvalues,E n, are known. Since the ϕn form a complete system, K can certainly beexpanded in this basis (for t ≥ ti)

K (x, t; xi, ti) =n

anϕn(x)e−ihE ntΘ (t − ti) . (1.7)

Here the stepfunction Θ(t) = 0 for t < 0 and Θ(t) = 1 for t ≥ 0 takesexplicitly into account that we propagate the wavefunction only forward intime. The expansion coefficients obviously depend on xi, ti

an = an(xi, ti) . (1.8)

Because of the initial condition K (x, ti + 0; xi, ti) = δ (x − xi) we have

δ (x − xi) =n

an(xi, ti)ϕn(x)e−ihE nti . (1.9)

The lhs is time-independent; thus we must have

an (xi, ti) = an(xi)e+ ihE nti , (1.10)




and consequently

δ (x − xi) = n

an (xi) ϕn(x) . (1.11)

This can be fulfilled byan (xi) = ϕ∗

n (xi) (1.12)

(closure relation). Thus we have a representation of K (x, t; xi, ti) in termsof the eigenfunctions and eigenvalues of the underlying Hamiltonian

K (x, t; xi, ti) = Θ(t − ti)n

ϕ∗n (xi) ϕn(x)e−

ihE n(t−ti) . (1.13)

It is easy to show that this propagator fulfills (1.3).

In Dirac’s bra and ket notation this result can also be written asK (x, t; xi, ti) =

n

ϕ∗n (xi) ϕn(x)e−

ihE n(t−ti) Θ (t − ti)

=n

n|xie−ihE n(t−ti)x|n Θ (t − ti)

=n

n|e+ ihHti|xix|e− i

hHt |n Θ (t − ti)

= x|e− ihH (t−ti)|xi ≡ x|U (t, ti) |xi Θ (t − ti) . (1.14)

Thus the propagator is nothing else than the time development operator

U (t, ti) = e− i

hH (t−t

i)

(1.15)

for t > ti in the x representation. It is also often written as

K (x, t; xi, ti) = x|e− ihH (t−ti)|xi Θ (t − ti) ≡ xt|xiti . (1.16)

The notation here is that of the Heisenberg representation of quantummechanics. In this representation the physical state vectors are time-inde-pendent and the operators themselves carry all the time-dependence whereasthis is just the opposite for the Schrodinger representation. For example, forthe position operator x in the Schrodinger representation we obtain the time-dependent operator

xH(t) = e ih Ht xe− i

h Ht (1.17)

andxH(t)|xt = x|xt (1.18)

with|xt = e

ihHt |x . (1.19)

The state |xt is thus the eigenstate of the operator xH(t) with eigenvalue xand not the state that evolves with time out of |x; this explains the sign of the frequency in the exponent.




x

xxi

ti

t

t1

t

Figure 1.1: Possible paths from xi to x, corresponding to (1.22).

1.2 Propagator as Path Integral

We start by dividing the time-interval between ti and t by inserting the timet1. The wavefunction is first propagated until t1 and then, in a second step,until t

ψ (x1, t1) =

K (x1, t1; xi, ti) ψ (xi, ti) dxi (1.20)

ψ(x, t) =

K (x, t; x1, t1)ψ (x1, t1) dx1 .

Taking these two equations together we get

ψ(x, t) =

K (x, t; x1, t1) K (x1, t1; xi, ti) ψ(xi, ti) dxi dx1 . (1.21)

Comparing this result with (1.5) yields

K (x, t; xi, ti) =

K (x, t; x1, t1) K (x1, t1; xi, ti) dx1 . (1.22)

We can thus view the transition from (xi, ti) to (x, t) as the result of atransition first from (x, t) to all possible intermediate points (x1, t1), which isthen followed by a transition from these intermediate points to the endpoint.We could also say that the integration in (1.22) is performed over all possiblepaths between the points (xi, ti) and (x, t), which consist of two straight linesegments with a bend at t1. This is illustrated in Fig. 1.1.




We now subdivide the time interval further into (n + 1) equal parts of

length ∆t = η. We then have in direct generalization of the previous result

K (x, t; xi, ti) =

. . .

dx1 dx2 . . . dxn (1.23)

× [K (x, t; xn, tn) K (xn, tn; xn−1, tn−1) . . . K (x1, t1; xi, ti)] .

The integrals run here over all possible paths between (xi, ti) and (x, t) whichconsist of (n + 1) segments with boundaries that are determined by the timesteps ti, t1, . . . , tn, t.

We now calculate the propagator for a small time interval ∆t = η fromt j to t j+1. For this propagation we have according to (1.16)

K (x j+1, t j+1; x j, t j) = x j+1|e− ihHη |x j (1.24)

∼= x j+1|1 − i

hHη|x j

= δ (x j+1 − x j) − i

hηx j+1|H |x j

=1

2πh

eih p(xj+1−xj) d p − iη

hx j+1|H |x j

with the representation for the δ-function

δ(x − x) = 12π

eik(x−x)dk . (1.25)

We now assume that H is given by

H = T (ˆ p) + V (x) . (1.26)

Here T , ˆ p, V , x are all operators; we assume that T (ˆ p) and V (x) are Taylor-expandable. In this case, where the p- and x-dependences separate, we canalso bring the last term in (1.24) into an integral form. We have

x j+1|H |x j = x j+1|T + V |x j . (1.27)

First, we consider the first summand

x j+1|T |x j =

d p d px j+1| p p|T (ˆ p)| p p|x j (1.28)

=

d p d px j+1| pδ( p − p)T ( p) p|x j

=

d px j+1| pT ( p) p|x j .




With the normalized momentum eigenfunctions

x| p =1√2πh

eih px (1.29)

we thus obtain

x j+1|T (ˆ p)|x j =1

2πh

T ( p) e

ih p(xj+1−xj)d p . (1.30)

While there is an operator ˆ p on the lhs of this equation there are only numbers p on its rhs.

For the potential part an analogous transformation can be performed

x j+1|V |x j = V (x j) δ (x j+1 − x j) (1.31)

=1

2πh

eih p(xj+1−xj)d p V (x j)

Again, on the lhs V is an operator, while the rhs of this equation containsno operators.

In summary, we have for the propagator over a time-segment η

K (x j+1, t j+1; x j, t j) =1

2πh

d p e

ih p(xj+1−xj)

− iηh 1

2πh

d p T ( p)e ih p(xj+1−xj) + 1

2πh

d p e ih p(xj+1−xj)V (x j)

=

1

2πh

d p e

ih p(xj+1−xj)

1 − iη

hH ( p,x j)

−→η→0

1

2πh

d p j exp

i

h[ p j (x j+1 − x j) − ηH ( p j, x j)]

. (1.32)

Here H = T + V is a function of the numbers x and p and no longer anoperator! In the last step we have renamed the integration variable into p jto indicate that it may be viewed as the momentum of a classical particlemoving from x j to x j+1 between times t j and t j+1.

We now insert (1.32) into (1.23) and obtain

K (x, t; xi, ti) (1.33)

= limn→∞

nk=1

dxk

nl=0

d pl2πh

exp

i

h

n j=0

[ p j (x j+1 − x j) − ηH ( p j, x j)]

.

(with x0 = xi and xn+1 = x). The asymmetry in the range of the productsover x- and p-integrations comes about because with n intermediate stepsbetween xi and x there are n + 1 intervals and corresponding momenta.




The integrand here is, for finite n, a complex function of all the coordi-

nates x1, x2, . . . , xn and the momenta p1, p2, . . . , pn. In the limit n → ∞ itdepends on the whole trajectory x(t), p(t). Here we note that p is not themomentum canonically conjugate to the coordinate x, but instead just anintegration variable.

In the limit n → ∞ we obtain for the exponent

n j=0

[ p j (x j+1 − x j) − ηH ( p j, x j)]

=n

j=0

η

p j

x j+1 − x jη

− H ( p j, x j)

−→n→∞

t ti

dt [ p(t)x(t) − H ( p(t), x(t))] . (1.34)

With this result we rewrite (1.33) in an abbreviated, symbolic form

K (x, t; xi, ti) =

Dx

D p e

ih

t ti

dt[ p(t)x(t)−H ( p(t),x(t))]

, (1.35)

where Dx stands for

dxk and D p for

d pl/(2πh). The integrals here are

limits of n-dimensional integrals over x and p for n → ∞, they are integralsover all functions x(t) and p(t) and are defined by (1.33).Equation (1.35) represents an important result. It allows to calculate the

propagator and thus the solution of the Schrodinger equation in terms of apath integral over classical functions.

1.3 Quadratic Hamiltonians

Even though the propagator (1.35) looks like a path integral over an expo-nential function of the action, this is in general not the case, because

px − H ( p,x) = L(x, x, p) (1.36)

is not equal to the classical Lagrange function since p is not the canonical mo-mentum as already stressed above. Therefore, in general one cannot expressthe path integral (1.35) in terms of the action.

Such a simplification, however, is possible for a special p-dependence of the Hamiltonian. If H depends at most quadratically on p, then the pathintegration over the momentum p can be performed and the action appearsin the exponent. This will be discussed in the next 2 sections.




1.3.1 Cartesian metric

In the last section we have made the special ansatz H = T ( p) + V (x) inwhich the momenta and coordinates are separated. For the special case, inwhich H depends only quadratically on p with constant coefficient, e.g.

H =p2

2m+ V (x) , (1.37)

we can further simplify the path-integral (1.33)

K (x, t; xi, ti) = limn→∞

n

k=1

dxk n

l=0

d pl2πh

(1.38)

× exp

i

hηn j=0

p j

x j+1 − x jη

− p2 j

2m− V (x j)

.

Using the integral relation +∞

−∞e−ap

2+bp+c d p =

π

aeb2

4a+c (1.39)

for Gaussian integrals, discussed in more detail in App. B.2.1, we obtain byperforming the p-integration

K = limn→∞

m

i2πhη

n+12

(1.40)

× nk=1

dxk exp

i

hηn j=0

m

2

x j+1 − x j

η

2

− V (x j)

.

Thus in this special case (H = p2/2m + V ) the propagator K is given (againin abbreviated notation) by

K (x, t; xi, ti) = N Dx e

ih

t

tiL(x,x)dt

= N Dx e

ihS [x(t)]

, (1.41)

with the Lagrangian

L(x, x) =m

2x2 − V (x) (1.42)

and the action

S [x(t)] = tti

L(x(t), x(t)) dt . (1.43)

N represents the factor in front of the integral in (1.40). There is a problemwith N : the factor N is complex and becomes infinite for n → ∞, η →




0. We will see, however, later, for example in Sect. 2.1, that the whole

pathintegral leads to a well-defined expression. In addition, this problem willbe bypassed in later developments where we show that physically relevant isonly a normalized propagator in which N has been removed.

The propagator K has thus been reduced to a one-dimensional path-integral, which is only possible for Hamiltonians which are quadratic in p.This is a quite important result that we will use throughout all the follow-ing sections. Eq. (1.41) shows that the propagator is given by the phaseexp i

hS [x(t)] summed over all possible trajectories x(t) with fixed starting

and end points. This is reminiscent of the partition function in statisticalmechanics which is obtained by summing the Boltzmann factor exp (−E n/T )

over all possible states of the system.At this point we should realize that in going from (1.38) to (1.40) we have

integrated an oscillatory integrand (eif ( p)) over an infinite interval. This wasonly possible by a mathematical trick: in applying the Gaussian integrationformula (B.18) we have in effect used the quantity iη in (1.38) as if it were real.In other words: we have analytically continued expression (1.38) into thecomplex plane by setting the time interval η → −iη with η real and positive.Then (1.38) becomes a well-behaved Gaussian integral. After performingthe integration we have then gone back to the original η. This analyticalcontinuation is in general possible only if no singularities are encountered

while going to the real variable η

. Also, one has to worry about phaseambiguities connected with the appearance of the squareroot of a complexnumber.

1.3.2 Non-Cartesian metric

The momentum integration is even possible for Hamiltonians of a much moregeneral form. As an example we consider

H =1

2mf 1(x) p2 + f 2(x) p + f 3(x) . (1.44)

Here a problem arises because the canonical quantization of such a Hamilto-nian is ambiguous. This is so because the classical coordinates and momentacommute, so that H can be brought into various forms that are classicallyall equal, but differ after quantization because the operators ˆ p and x do notcommute. Even though the time development operator (1.24) could still beevaluated for any of these various forms, these would in general not lead toa path integral of the form (1.35).

We, therefore, turn the question around and ask:




Given a path integral

Dx

D p e

ih

t ti

dt ( px−H ( p,x))

(1.45)

with the classical Hamiltonian (1.52), can this still be identified as a propa-gator and, if so, for which Hamiltonian?

The answer to this question is given here without proof 1:

Dx D

p e

ih

t ti

dt [ px−H ( p,x)]

=

x

|e−

ih

(t−ti)H W

|x

. (1.46)

Here H W is the “Weyl-ordered” Hamiltonian

H W =1

8m

ˆ p2f 1(x) + 2ˆ pf 1(x)ˆ p + f 1(x)ˆ p2

(1.47)

+1

2[ˆ pf 2(x) + f 2(x)ˆ p] + f 3(x) .

In this form the momentum- and coordinate-dependent terms are symme-trized. Note that the path integral in (1.46) is well determined because allquantities on the rhs of this equation are classical, commuting quantities.

The lhs of (1.46) can be simplified by performing the integration over themomenta in (1.33)

K = limn→∞

nk=1

dxknl=0

d pl2πh

(1.48)

× exp

i

h

n j=0

p j (x j+1 − x j) − η

f 1(x j)

p2 j

2m+ f 2(x j) p j + f 3(x j)

.

Using again the Gaussian integral formula (1.39) this gives

K = limn→∞

m

i2πhη

n+12 n

k=1

dxknl=0

1 f 1(xl)

(1.49)

× exp

i

h

n j=0

η

m

2

(xj+1−xj)

η− f 2(x j)

2

f 1(x j)− ηf 3(x j)

.

1The proof can be found in [LEE], p. 475




The exponent (in round brackets) is

(. . .) =m

2

xj+1−xj

η− f 2(x j)

2

f 1(x j)− f 3(x j) (1.50)

−→η→0

m

2

x2 − 2xf 2(x) + f 22 (x)

f 1(x)− f 3(x) .

This last expression is just the Lagrangian

L =m

2g1(x)x2 + g2(x)x + g3(x) (1.51)

(a kinetic term of this form appears, for example, when rewriting the kineticenergy of a particle from cartesian into polar coordinates). For this L thecorresponding classical Hamiltonian is given by

H = px − L = (mg1x + g2) x − m

2g1x2 − g2x − g3

=m

2g1x2 − g3 =

m

2g1

p − g2

mg1

2

− g3

=1

2m

1

g1 p2 − g2

mg1 p +

1

2m

g22

g1

− g3 . (1.52)

Withf 1 =

1

g1

, f 2 = − g2

mg1

, f 3 =g2

2

2mg1

− g3 (1.53)

this is just the Hamiltonian (1.44) that we started out with.We thus have for the complete propagator in this case

K (x, t; xi, ti) (1.54)

= limn→∞

m

i2πhη

n+12 n

k=1

dxknl=0

g1 (xl) exp

i

hηn j=1

L(x j, x j)

.

Thus, in this case the path integral is changed. The square root of a functionthat determines the metric of the system appears in the integrand. ForHamiltonians even more general than (1.44) an additional potential termappears in the Lagrangian [Grosche/Steiner].

1.4 Classical Interpretation

The simple form (1.41) for the path integral allows a very physical interpre-tation of the classical limit to quantum mechanics. For the classical path the




variation of the action is, according to Hamilton’s principle, equal to zero,

i.e. the action is stationary2

δS =

t ti

L (xcl + δx, x + δxcl) dt −t ti

L (xcl, xcl) dt = 0 . (1.55)

This implies that all the paths close to the classical path give about equalcontributions to the path integral. For each path (C 1) somewhat more re-moved from the classical one there will also be another one, (C 2), whoseaction differs from that on C 1 just by πh. Then we have

ei

hS (C 1) = ei

hS (C 2)+iπ = −ei

hS (C 2) , (1.56)

so that the contributions from these two paths cancel each other. Sizeablecontributions to the path integral thus come from paths close to the classicalone. Quantum mechanics then describes the fluctuations of the action in anarrow range around the classical path.

This observation forms the basis for a semiclassical approximation. Thiscan be formulated by expanding the action functional S [x(t), x(t)] in termsof fluctuations δx around the classical path xcl(t). This gives

S [x, x] = S cl +1

2 δ2L

δx2 (δx)2

+ 2δ2L

δxδx δx δx +δ2L

δx2 (δx)2 + .....

≡ S cl + δ2S + . . . . (1.57)

Here all the derivatives have to be taken at the classical path. Because S isstationary at the classical path, there is no first derivative in this equation.The propagator (1.41) now becomes

K (x, t; xi, ti) = N

Dx eihS = N e

ihS cl

Dx exp

1

2

i

hδ2S

+ .... . (1.58)

Note that this result (without higher order terms) is exact for Lagrangiansthat depend at most quadratically on x. The second factor gives the effectsof quantum mechanical fluctuations around the classical path.

An interesting observation on the character of these fluctuations can bemade. The main contribution to the integrand in (1.40) for η → 0 comesfrom exponents

η

h

m

2

x j+1 − x j

η

2

1, (1.59)

2For an explanation of functionals and their derivatives see App. B




i.e. from velocities v j≈ 2h/(ηm) which diverge with η

→0. This implies

that the average displacement d within a time-step η is proportional to √η,so that d2 ∼ η (not d2 = v2dt2 = v2η2!), just as for a random walk. The maincontribution to the path integral, and therefore to the quantum mechanicalfluctuation around the classical trajectory, thus comes from paths that arecontinous, but have no finite derivative.



Chapter 2

PERTURBATION THEORY

In this chapter we discuss first how to calculate the propagator of a freeparticle and derive its analytic form. In most cases, however, with a potentialincluded the exact propagator cannot be calculated. Thus one has to resortto perturbation theory which will also be developed in this chapter.

2.1 Free propagator

We start with the free propagator K 0, given by

K 0 = N

Dx eihS 0

= limn→∞

m

2πihη

n+12

+∞ −∞

nk=1

dxk exp

i

hηn j=0

m

2

x j+1 − x j

η

2 . (2.1)

This path-integral can be performed exactly. With (B.19) we obtain for thefree propagator

K 0

= limn→∞ m

2πihηn+12

inπn

(n + 1) m2hηn12

(2.2)

× exp

i

n + 1

m

2hη(x − xi)

2

,

since xn+1 = x, x0 = xi. With (n + 1)η = t − ti this becomes

K 0(∆x, ∆t) =

m

2πhi (t − ti)eih

m(x−xi)2

2(t−ti) =

m

2πhi∆teihm∆x2

2∆t (2.3)

20



CHAPTER 2. PERTURBATION THEORY 21

with ∆x = x

−xi, ∆t = t

−ti. This is the propagator of a free particle for ∆t

≥0; for ∆t < 0 it has to be supplemented by the condition K = 0. Because of Galilei invariance and time homogeneity the free particle propagator dependsonly on the space- and time-distances.

The last step, from (2.2) to (2.3), shows nicely how in the limit n → ∞the infinite normalization factor combines with another equally ill definedfactor from the path integral into a well defined product in (2.2).

Since a free particle has conserved momentum, it is advantageous to trans-form K 0 into the momentum representation

K 0( p, ∆t) =1

√2πh e−ih p∆xK 0 (∆x, ∆t) d∆x

=1√2πh

m

2πhi∆t

e−

ih p∆xe

ih

m2∆t

∆x2 d∆x . (2.4)

We can now use again the integral relation (B.18) in the form

+∞ −∞

e−a∆x2+b∆xd∆x =

π

aeb2

4a (2.5)

(with a = −im/(2h∆t) and b = −i p/h) to write

K 0( p, ∆t) =1

2πh

mi∆t

π2h∆t

−ime−

ihp2

2m∆t

=1√2πh

e−ihp2

2m∆t , (2.6)

so that we obtain

K 0 (x, t; xi, ti) =1√2πh

eih p∆xK 0( p, ∆t) d p

=1

2πh eih p∆x− p2

2m∆t

d p Θ(∆t) , (2.7)

where in the last line the causality condition Θ(∆t) has been written explic-itly; it takes care of the boundary condition K = 0 for ∆t < 0. Eq. (2.7) is just the Fourier representation of the propagator (2.3).

The step function can be rewritten using the relation

Θ(∆t) =1

2πi

+∞ −∞

dωeiω∆t

ω − iε(ε > 0) , (2.8)




which follows directly from the residue theorem: for ∆t > 0 the integral can

be closed in the upper half of the complex ω plane; Cauchy’s integral theoremthen gives 2πi in the limit ε → 0 so that Θ(t) = 1. If ∆t < 0, on the otherhand, then the loop integration can only be closed in the lower half-spacethus missing the pole at ω = +iε). Multiplying (2.6) with (2.8) gives

K 0 (x, t; xi, ti) =1

(2π)2hi

+∞ −∞

d p dωeih

p∆x−

p2

2m−hω

∆t

ω − iε

. (2.9)

We now substitute

E =

p2

2m − hω (2.10)

and obtain

K 0 (x, t; xi, ti) =1

(2πh)2

d p dE e

ih

( p∆x−E ∆t) ih

E − p2

2m+ iε

(2.11)

(the “−” sign coming from the substitution is cancelled by another signobtained by inverting the integration boundaries).

Since we will need these expressions later on in three space dimensionswe give them here in a straightforward generalization of (2.7) and (2.11)

K 0 (x, t; x, t) =1

(2πh)3

eih

p·(x− x)− p2

2m (t−t)

d3 p Θ(t − t) , (2.12)

and

K 0 (x, t; xi, ti) =1

(2πh)4

d3 p dE e

ih

( p·∆x−E ∆t) ih

E − p2

2m+ iε

. (2.13)

The integrand on the right-hand side of (2.11) is the free propagatorin the energy-momentum representation. Since p and E are independent

variables in the integral, we see that propagation also takes place at energiesE = p2/2m. The classical dispersion relation does show up as a pole in thepropagator.

2.2 Perturbative Expansion

We now assume that the unperturbed particle moves freely and that theperturbing interaction is given by V (x, t). We furthermore assume that H




has the special form H = p2/2m + V (x, t), so that the propagator is given

by

K (x, t; xi, ti) = N

Dx eihS (2.14)

with

S =

t ti

L(x, x) dt =

t ti

m

2x2 − V (x, t)

dt . (2.15)

Since the integrand here is a classical function, we have

eihS = e

ih

t

tim2 x

2 dt

e− ih

t

tiV (x,t) dt

. (2.16)

The second factor can now be expanded in powers of the potential, thusyielding a perturbative expansion of the action

e

ih

t ti

V (x,t) dt ∼= 1 − i

h

t ti

V (x, t) dt − 1

2!

1

h2

t ti

V (x, t) dt

2

+ . . . . (2.17)

When we substitute this expansion into the expression (2.14) we obtain

K = N Dxe

i

h

S 0

×1 − i

h

t ti

V (x, t) dt − 1

2!

1

h2

t ti

V (x, t) dt

2

+ . . .

= K 0 + K 1 + K 2 + . . . , (2.18)

i.e. a sum ordered in powers of the interaction.

First order propagator. We next determine the first-order propagatorK 1. According to (2.18) it is given by

K 1 = − i

hN

Dx eihS 0

tf ti

V (x, t) dt (2.19)

= − i

hlimn→∞

m

2πhiη

n+12

×+∞ −∞

ni=1

dxink=1

V (xk, tk) η exp

i

m

2hη

n j=0

(x j+1 − x j)2

.




Here the time-integral has been written as a sum. In a next step we now

split the sum in the exponent into two pieces, one running from j = 0 to j = k − 1 and the other from j = k to j = n and separate the correspondingintegrals. This gives

K 1 = − i

hlimn→∞

nk=1

η

dxk

V (xk, tk) (2.20)

×

N

k2

dx1 dx2 · · · dxk−1 e

i m2hη

k−1j=0

(xj+1−xj)2

×N

n−k+12

dxk+1 · · · dxne

im2hη

nj=k

(xj+1−xj)2

withN ≡ m

2πhiη. (2.21)

The term in the first bracket is nothing else than the propagator from ti totk (K 0 (xk, tk; xi, ti)) , and that in the second bracket is that from tk to t(K 0 (x, t; xk, tk)). Thus we have

K 1 (xf , tf ; xi, ti) =

− i

h

+∞ −∞

dx

tf ti

dt K 0 (xf tf ; x, t) V (x, t)K 0 (x, t; xi, ti) . (2.22)

The time integral over the interval from ti to tf can be extended to ∞ bynoting that

K 0 (x, t; xi, ti) = 0 for t < ti (2.23)

K 0 (xf , tf ; x, t) = 0 for tf < t .

This gives

K 1 (xf , tf ; xi, ti) =

− i

h

+∞ −∞

dt

+∞ −∞

dx K 0 (xf , tf ; x, t) V (x, t)K 0 (x, t; xi, ti) . (2.24)






the higher-order terms in the interaction. In each case the prefactor (1/n!)

stemming from the expansion of the exponential in (2.17) is cancelled by thetime ordering inherent in the propagators.

Equation (2.25) is the Born series for the propagator; it can be representedgraphically as shown in Fig. 2.1. A time axis runs here from bottom totop. The straight lines denote the free propagation of the particles (K 0)and the dots stand for the interaction vertices (−iV /h) (the circles mark theinteraction range) and a space and time integration appears at each vertex.

Bethe-Salpeter equation. The expansion (2.25) can be formally summed.This can be seen as follows (in obvious shortterm notation)

K = K 0 + K 0U K 0 + K 0UK 0U K 0 + · · · (2.27)

= K 0 + K 0U (K 0 + K 0U K 0 + · · ·) with U = − ih

V .

The expression in parentheses is just again K , so that we obtain the Bethe-Salpeter equation

K = K 0 + K 0UK . (2.28)

The Bethe-Salpeter equation is an integral equation for the full interactingpropagator K as can be seen most easily from its space-time representation

K (xf , tf ; xi, ti) = K 0 (xf , tf ; xi, ti) (2.29)

− i

h

K 0 (xf , tf ; x, t) V (x, t)K (x, t; xi, ti) dx dt .

We can also represent the Bethe-Salpeter equation in a diagrammaticway. If we denote the so-called “dressed propagator” K , that includes all theeffects of the interactions, by a double line

= K (x2, t2; x1, t1) , (2.30)

then this equation can be graphically represented as shown in Fig. 2.2. Each

graph in Fig. 2.2 finds a one-to-one correspondence in (2.28): the single linesrepresent the free propagator, the double line the dressed propagator andthe dot stands for the interaction U . (2.28) shows that the factors in thesecond term of this equation have to be written from left to right against thetime-arrow in Fig. 2.2.

The Bethe-Salpeter equation can also be written in an equivalent formfor the interacting wavefunction

Ψ (xf , tf ) =

K (xf , tf ; xi, ti) Ψ (xi, ti) dxi




= +

Figure 2.2: Bethe-Salpeter equation (2.28). The arrows indicate the timedirection.

=

K 0 (xf , tf ; xi, ti) Ψ (xi, ti) dxi

− i

h

K 0 (xf , tf ; x, t) V (x, t)K (x, t; xi, ti) Ψ (xi, ti) dxi dx dt

=

K 0 (xf , tf ; xi, ti) Ψ (xi, ti) dxi (2.31)

− i

h

K 0 (xf , tf ; x, t) V (x, t)Ψ(x, t) dx dt .

This constitutes an integral equation for the unknown wavefunction Ψ(x, t).

2.3 Application to Scattering

Let us now apply the results of the last section to a scattering process. In thiscase the particle is free at t = −∞, then undergoes the scattering interactionand then, at t = +∞, is free again.

We treat this problem as usual by adiabatically switching on and off theinteraction V (x, t). The initial condition (for t → −∞) for the wavefunctionthen is

Ψin(x, t) = N e

i( ki·x−ωit)

. (2.32)We choose here a box normalization with periodic boundary conditions sothat N = 1/

√V . The scattering state that evolves from this incoming state

is denoted by Ψ(+)(x, t). The superscript (+) indicates that the state evolvesforward in time, starting from Ψ in at t = −∞; it thus fulfills the boundarycondition

Ψ(+)(x, t → −∞) = Ψin(x, t) . (2.33)

In a scattering experiment one looks at t → +∞ for a free scattered particle




with definite momentum; the corresponding final state is denoted by

Ψout(x, t) = N ei( kf ·x−ωf t) . (2.34)

The probability amplitude for the presence of Ψout in the scattered state Ψ(+)

is given by

S fi =

Ψ∗out (xf , tf ) Ψ(+) (xf , tf ) d3xf for tf → ∞ . (2.35)

This is just the transition amplitude from the initial state i to the final statef (S -matrix).

Expansion (2.29) yields the Bethe-Salpeter equation for the scattering

wavefunction

Ψ(+) (xf , tf ) =

K 0 (xf , tf ; xi, ti) Ψin (xi, ti) d3xi (2.36)

− i

h

K 0 (xf , tf ; x, t) V (x, t)K (x, t; xi, ti)Ψin(xi, ti) d3xi d3x dt .

Inserting this into (2.35) gives for the S -matrix

S fi =

Ψ∗out (xf , tf ) K 0 (xf , tf ; xi, ti) Ψin (xi, ti) d3xid

3xf (2.37)

−i

h Ψ∗out (xf , tf ) K 0 (xf , tf ; x, t) V (x, t)

× K (x, t; xi, ti)Ψin (xi, ti) d3xi d3x dt d3xf .

Since Ψin is a plane wave, we know that

φ (xf , tf ) =

K 0 (xf , tf ; xi, ti) Ψin (xi, ti) d3xi (2.38)

is also a plane wave state, since K 0 is the free propagator so that no interac-tion takes place. φ is actually the same wavefunction as Ψin, only taken at alater time and space point

φ (xf , tf ) = N e

i( ki·xf −ωitf )

. (2.39)This means that the first integral in (2.37) can be easily evaluated

Ψ∗out (xf , tf ) φ (xf , tf ) d3xf = δ3

kf − ki

. (2.40)

We thus have

S fi = δ3

kf − ki

− i

h

d3xf d

3x d3xi dt (2.41)

×[Ψ∗out (xf , tf ) K 0 (xf , tf ; x, t) V (x, t)K (x, t; xi, ti)Ψin (xi, ti) ] .




xi ti

xf tf

x t

Figure 2.3: First order scattering diagram, corresponding to the second term

in (2.41). Time runs from left to right.

The amplitude for a scattering process is given by the second term.If K on the rhs of (2.41) is now represented by the Born series expan-

sion of the Bethe-Salpeter equation (2.27) this amplitude can be graphicallyrepresented as shown in Fig. 2.3. This diagram can be translated into theamplitude just given by writing for each straight-line piece

xt t1 x2 t2

= K 0 (x2, t2; x1, t1) (2.42)

and for each interaction vertex

x t= − i

hV (x, t) + integration over x, t . (2.43)

The rules are completed by multiplying Ψ in and Ψ∗out at the corresponding

sides of the diagram and then integrating over the spatial variables of thesewavefunctions and over all intermediate times. The ordering of factors issuch that in writing the various factors from left to right one goes against the flow of time in the figure.

These rules are illustrated for a second-order scattering process in Fig. 2.4.In this figure the time runs from ti to tf from left to right. The correspondingamplitude is given by

A(2) =

d3xi d3xf d3x1 d3x2 dt1 dt2

Ψ∗

out (xf , tf ) K 0 (xf , tf ; x2, t2)− i

hV (x2, t2)

K 0 (x2, t2; x1, t1)

− i

hV (x1, t1)

K 0 (x1, t1; xi, ti) Ψin (xi, ti)

.

(2.44)




xi ti

xf tf

x1 t1

x2 t2

Figure 2.4: Second order scattering diagram, corresponding to (2.44).

We now evaluate the first order scattering amplitude (cf. (2.41))

A(1) = − i

h

Ψ∗

out (xf , tf ) K 0 (xf , tf ; x, t) V (x, t)K 0(x, t; xi, ti)

× Ψin (xi, ti)

d3x dt d3xi d3xf (2.45)

somewhat further by using expression (2.6) for the free propagator (extendedto three dimensions)

K 0 (x

, t

; x, t) =1

(2πh)3 e

ih p·(x− x)− p2

2m(t−t)d

3

p Θ(t

− t) . (2.46)

Inserting this into the expression for A(1) gives

A(1) = − i

h

1

V

1

(2πh)6

d3x dt d3xi d3xf d

3 p d3q (2.47)

e−

ih( pf ·xf −E f tf )e

ih

p·(xf −x)− p2

2m(tf −t)

V (x, t)eih

q·(x−xi)− q2

2m (t−ti)

eih

( pi· xi−E iti)

,

where the time-ordering tf > t > ti is understood and we have abbreviatedE i = p2

i

/(2m) and E f = p2

f

/(2m).We integrate first over xi and xf . This yields

d3xi −→ (2πh)3δ3 ( pi − q) (2.48) d3xf −→ (2πh)3δ3 ( pf − p) .

Performing next the integrals over p and q gives for the amplitude

A(1) = − i

hV

d3x dt




× e+ihE f tf e

ih− pf · x− p2

f 2m(tf −t)V (x, t)e

ih pi·x− p2

i2m

(t−ti)e−ihE iti

= − i

hV

eih(− pf ·x+E f t)V (x, t)e

ih

(+ pi·x−E it) d3x dt . (2.49)

The time-integration is performed next. For that we assume that V (x, t)acts only during a finite, but long time-interval ([−T, T ]), in which it is time-independent; at the boundaries of the interval it is adiabatically, i.e. withoutany significant energy-transfer, turned on and off. We then obtain

dt V (x, t)e

ih(E f −E i)t

dt =

T

−T

V (x)ei(ωf −ωi)t

dt

T →∞

−→ V (x) 2πδ (ωf − ωi) ,

(2.50)with ωf − ωi = (E f − E i)/h. Thus A(1) becomes

A(1) = − i

hV 2πδ (ωf − ωi)

eih( pi− pf )·x V (x) d3x . (2.51)

This is the well-known lowest-order Born-approximation result.



Chapter 3

GENERATING

FUNCTIONALS

In this section we consider the transition amplitude in the presence of anexternal “source” J (t), so that the Hamiltonian reads

H J (x, p) = H (x, p) − hJ (t)x . (3.1)

A classical example is that of a harmonic oscillator with externally drivenequilibrium position x0(t). Its Hamiltonian is

H x0 =p2

2m+

1

2k(x − x0(t))2

= H − kx0(t)x + O(x20) . (3.2)

With hJ (t) = kx0(t) and for small amplitudes x0 we just have the form of (3.1). It is evident that the states of this system will change as time develops,because of its changing equilibrium position.

Suppose now that the system was in its groundstate at t → −∞ andthat x0(t) acts only for a limited time period. We could then calculate thetransition probability for the system to be still in its groundstate at t

→+

∞by using the techniques developed in sections 2.2 and 2.3. There we saw(cf. (2.25) and (2.44)) that this probability is determined by matrixelementsof time-ordered products of the interaction, i.e. of the operator x in thepresent case. In this chapter we will show that these matrixelements can allbe generated once only the functional dependence of the probability for thesystem to remain in its groundstate on an external source is known.

32



CHAPTER 3. GENERATING FUNCTIONALS 33

3.1 Groundstate-to-Groundstate Transitions

Generalizing the special example of the introduction we assume that an ar-bitrary physical system is at first (at ti) stationary and then changes underthe influence of an external source J (t)x of finite duration. After the sourcehas been turned off, the groundstate of the system is still the same (up to aphase), but the system may be excited.

The propagator for this system, is quite generally, given by

xf , tf |xi, tiJ = Dx

D p e

ih

tf ti

[ px−H (x,p)+hJ (t)x] dt

. (3.3)

The value of this propagator depends obviously on the source function J (t);it is a functional of the source J (t).

In the following paragraphs we will now discuss this functional depen-dence for tf → +∞ and ti → −∞. We will also show that it determines thevacuum expectation values of time-ordered x operators.

We start by calculating the propagator (3.3) of a system under the influ-ence of a source, i.e. for a system described by (3.1). We first assume thatthe source is nonzero only for a limited time between −T and +T

J (t) = 0 for

|t

|> T . (3.4)

We can then write for the propagator (ti < −T , tf > T )

xf tf |xitiJ =

dx dx xf tf |xT xT |x −T J x −T |xiti . (3.5)

Note that the two outer propagators are taken with the sourceless Hamil-tonian (J = 0) because of condition (3.4). They are given by, e.g.,

x −T |xiti = x|e− ihH (−T −ti)|xi

=

nϕn(x)ϕ∗

n (xi) e−ihE n(−T −ti) . (3.6)

Similarly we get

xf tf |xT = xf |e− ihH (tf −T )|x

=n

ϕn(xf )ϕ∗n(x)e−

ihE n(tf −T ) . (3.7)

Here the ϕn are eigenstates of the Hamiltonian H without a source; weassume that the spectrum of H is bounded from below with eigenvaluesE n ≥ E 0. The dependence on ti and tf is now isolated.




Taking now the limits ti→ −∞

and tf →

+

∞is not straightforward

because both times appear as arguments of oscillatory functions. These func-tions oscillate the more rapidly the higher the eigenvalues E n are. We canthus expect that the dominant contribution will come from the lowest eigen-value E 0.

Wick Rotation. This can indeed be shown by by a mathematical trick,the so-called Wick rotation . In this method one looks at the propagators(3.6) and (3.7) as functions of ti and tf , respectively, and continues thesevariables analytically from the real axis into the complex plane. One thenperforms the limits and after that rotates back to real times. Mathematically

this is achieved by replacing the physical Minkowski time t by a complex timeτ

t −→ τ = te−iε . (3.8)

The direction of the rotation is mandated by the requirement that thereare no singularities of the integrand encountered in the rotation of the time-axis. This is the case for

0 ≤ ε < π . (3.9)

For larger angles the rhs of (3.6) and (3.7) develops an essential singularityfor ti/f → −/ + ∞.

This rotation corresponds to an analytic continuation in the variable ε;the real, four-dimensional Minkowski space can be viewed as a subspace of a four-dimensional complex space (or of a five-dimensional space with 3 realspace-coordinates and 1 complex time-coordinate). The substitution (3.8)corresponds to a clockwise rotation of the time-axis out of the real subspaceinto the larger complex space. For ε = +π/2 this corresponds to a rotationt = +∞ → τ = −i∞, t = −∞ → τ = +i∞; only this special case isoften called a Wick rotation. Physical results are obtained by evaluatingthe relevant expressions in the enlarged, complex space and then, at theend, rotating the result back to the real time-axis (which works only if nosingularities are encountered on this way).

We now apply this rotation to the expression

e−ihE 0tix −T |xiti =

n

ϕn(x)ϕ∗n (xi) e

ihE nT e

ih

(E n−E 0)ti (3.10)

and thus perform the limit ti → −∞ on the rotated time axis

limτ i→−∞(cos ε−isin ε)

e−ihE 0τ ix −T |xiτ i (3.11)

= limτ i→−∞(cos ε−isin ε)

n

ϕn(x)ϕ∗n (xi) e

ihE nT e+ i

h(E n−E 0)τ i .




Because we have separated out the lowest frequency the last factor vanishes

for ε > 0 , except for n = 0. Thus we get

limτ i→−∞(cos ε−isin ε)

e−ihE 0τ ix −T |xiτ i = ϕ0(x)ϕ∗

0(xi) eihE 0T . (3.12)

Analogously we also obtain

limτ f →+∞(cos ε−isin ε)

e+ ihE 0τ f xf τ f |xT = ϕ0(xf )ϕ∗

0(x) eihE 0T . (3.13)

Both expressions, (3.12) and (3.13), are the analytical continuations of thecorresponding limits on the real time axis from ε = 0 to ε > 0. Since the

right hand sides of these equations do not depend on ε they can be continuedback to the real time axis (ε = 0) without any change. By means of theWick rotation we have thus been able to make the expression (3.12) and(3.13) convergent; in this process we have found that only the groundstatecontributes to the transition probability.

We now insert these expressions into (3.5) and get

limτ i→−∞(cos ε−i sin ε)τ f →∞(cos ε−i sin ε)

eihE 0(τ f −τ i)xf τ f |xiτ iJ (3.14)

= ϕ0(xf )ϕ∗0(xi) e

ihE 02T

dx dx ϕ∗0(x) xT |x −T J ϕ0(x)

The integral in the last line can be rewritten, using ϕ0(x) = x|0. This gives dx dx ϕ∗

0(x) xT |x −T J ϕ0(x) =

dx dx0|xxT |x −T J x|0= 0T |0 −T J (3.15)

so that we obtain for (3.14)

limτ i→−∞(cos ε−i sin ε)

τ f →∞(cos ε−i sin ε)

eihE 0(τ f −τ i)xf τ f |xiτ iJ

= ϕ0(xf )ϕ∗0(xi) e

i

h

E 02T

0T |0 −T J . (3.16)

We obtain by Wick-rotating back

0T |0 −T J = limti→−∞tf →+∞

eihE 0(tf −ti)e−

ihE 02T

ϕ∗0(xi)ϕ0(xf )

xf tf |xitiJ . (3.17)

With (in the limit ti → −∞, tf → +∞)

xf tf |xitiJ =0 = ϕ∗0(xi)ϕ0(xf )e−

ihE 0(tf −ti) (3.18)




(3.17) becomes

0T |0 −T J = limti→−∞tf →+∞

xf tf |xitiJ xf tf |xiti0

e−ihE 02T . (3.19)

This immediately gives for the free (J = 0) transition amplitude

0T |0 −T 0 = e−ihE 02T , (3.20)

as it should.Equation (3.17) implies that the groundstate-to-groundstate transition

amplitude is – up to a factor – given by a path integral from arbitrary xi

to arbitrary xf and thus does not depend on these quantities, if only thecorresponding times are taken to infinity.

Gs-to-gs transition rate. The vacuum transition rates 0T |0 −T J de-serve some explanation. Formally, they are given by

0T |0 −T J = 0|e− ih

(H −hJx)2T |0 . (3.21)

The vacuum state |0 is assumed to be unique, if there is no source J present,and normalized to 1. It is the vacuum of the theory before and after theaction of the source. On the other hand, e−

ih

(H −hJx)2T

|0

is the state that

the vacuum at t = −T has evolved into at t = +T under the influence of theexternal source J (t). If this external perturbation has acted adiabatically(very gently turned on and off again) the vacuum at t > T differs from thatof the source-free theory only by a phase. The matrixelement (3.21) is thenthe probability amplitude to find the original vacuum in the time-evolvedvacuum state. The absolute value of this probability amplitude is surely 1,so that the two states can differ only by a J -dependent real phase

0T |0 −T J = eih

(S [J ]−E 02T ) (3.22)

where we have taken out the free propagation contribution. This phase isthe quantity determined by (3.17).

To conclude these considerations we note that instead of using the Wick-rotation to make the transition rates well behaved for very large times wecould also have added a small negative imaginary term −iεE n to all eigen-values E n. This would have given a damping factor to the oscillating expo-nentials that becomes larger with n and thus would have led to a suppressionof all higher excitations. This becomes apparent by looking at expressions(3.6) and (3.7). At the end of the calculation the limit ε → 0 would have tobe performed.




3.1.1 Generating functional.

This groundstate-to-groundstate transition rate is a functional of the sourceJ (t) which we denote by

W [J ] = 0 +∞|0 −∞J = limti→−∞tf →+∞

T →∞

xf tf |xitiJ xf tf |xitiJ =0

e−ihE 02T . (3.23)

W [J ] is called a generating functional for reasons that will become clear inthe next section.

In order to get rid of the phase that is produced already by a source-free

propagation (exp −i

hE 02T ) we define now a normalized generating functional

Z [J ] =W [J ]

W [0]=

0 +∞|0 −∞J 0 +∞|0 −∞J =0

= limti→−∞tf →+∞


(3.24)

with Z [0] = 1. The functional Z [J ] describes the processes relative to the un-perturbed (J = 0) time-development. The numerator in (3.24) is a transitionamplitude and can therefore be written as a path integral

xf +

∞|xi

− ∞J = Dx D p e

ih

+∞

−∞

[ px−H ( p,x)+hJ (t)x] dt

. (3.25)

If H is quadratic in p and of the form H = p2/(2m)+V (x), the propagatorcan be rewritten as (see Sect. 1.3)

xf + ∞|xi − ∞J = N

Dx e

ih

+∞ −∞

[L(x,x)+hJ (t)x] dt

. (3.26)

In the normalized functional Z [J ] the (infinite) factor N cancels out becauseit is independent of J

Z [J ] = W [J ]W [0]

= 0 +∞|0 −∞J 0 +∞|0 −∞J =0

=

Dx e

ih

+∞ −∞

[L(x,x)+hJ (t)x] dt

Dx e

ih

+∞ −∞

[L(x,x)] dt

(3.27)




3.2 Functional Derivatives

of Transition Amplitudes

In this section we will show – by using the methods outlined in App. B – thatthe groundstate expectation value of a time-ordered product of interactionoperators – in this present case of the operators x(t) – can be obtained asfunctional derivatives of the functional W [J ] with respect to J .

We start with the definition of a path integral as a limit of a finite di-mensional integral (see (1.33))

xf

tf |

xitiJ

= limn→∞

n

k=1

dxk

n

l=0

d pl

2πh(3.28)

× exp

i

h

n j=0

[ p j (x j+1 − x j) − ηH ( p j , x j) + hJ jx j]

.

and calculate its functional derivative with respect to J . In order to becomefamiliar with functional derivatives we do this in quite some detail. Using thedefinition (B.26) for the functional derivative we get (with the abbreviationF (x j, p j) = p j(x j+1 − x j) − ηH ( p j, x j))

δ

xf tf

|xiti

J

δJ (t1)

= limε→0

1

ε

limn→∞

nk=1

dxk

nl=0

d pl2πh

(3.29)

×exp

i

h

n j=0

F (x j, p j) + hx j [J j + εδ(t j − t1)]

− exp

i

h

n j=0

(F (x j , p j) + hx jJ j)

= limn→∞ nk=1 dxk

ll=0

d pl

2πh ix1 expi

h

n j=0 (F (x j, p j) + hx jJ j) ,

which can be written as

δxf tf |xitiJ δJ (t1)

= i

Dx

D p x(t1)e

ih

tf ti


. (3.30)

We now want to relate this derivative of a classical functional to quan-tum mechanical expressions and thus understand its physical significance and




meaning. In order to do so, we go back to the definition of the propagator.

There we had (in (1.23))

xf , tf |xiti (3.31)

=

dx1 . . . dxn xf tf |xn, tnxntn|xn−1tn−1 . . . x1t1|xiti .

Now, in (3.30) for J = 0, we have one factor more on the righthand side of this equation

dx1 . . . dxn xf tf |xntn . . . x1t1|xitix1

= dx1 . . . dxn xf tf |xntn . . . x1t1|x(t1)|xiti ; (3.32)

the last step is possible because |x1 is an eigenstate of the x operator witheigenvalue x1 (cf. (1.18)). The last integral is obviously equal to

xf tf |x(t1)|xiti , (3.33)

i.e. to a matrixelement of the position operator. We thus have

δxf tf |xitiJ δJ (t1) J =0

= i Dx D

p x(t1)e

ih

tf ti

[ px−H (x,p)] dt

= ixf tf |x(t1)|xiti . (3.34)

For the case that H is separable in x and p and quadratic in p, this relationreads

δxf tf |xitiJ δJ (t1)

J =0

= iN

Dx x(t1)e

ih

tf ti

L(x,x) dt

= i

xf tf

|x(t1)

|xiti

. (3.35)

The functional derivative of the propagator with respect to the source thusgives the transition matrix element of the coordinate x.

The higher order functional derivatives yield

δnxf tf |xitiJ δJ (t1)δJ (t2) . . . δ J (tn)

(3.36)

= (i)n

Dx

D p x(t1)x(t2) . . . x(tn) e

ih

tf ti


.




One might guess that

δnxf tf |xitiJ δJ (t1)δJ (t2) . . . δ J (tn)

J =0

= inxf tf |x(t1)x(t2) . . . x(tn)|xiti , (3.37)

but this equation is not quite correct. We see this by considering explicitlythe second derivative. We can proceed there exactly in the same way as forthe first. We have for the rhs of (3.36) in the case of the second derivative

δ2xf tf |xitiJ δJ (tα)δJ (tβ )

J =0

= i2

Dx

D p x(tα)x(tβ )e

ih

tf ti

[ px−H (x,p)] dt

= i2 dxi . . . dxn xf tf |xntn · · · xltl|x(tα)|xl−1tl−1· · · xktk|x(tβ )|xk−1tk−1 · · · x1t1|xiti . (3.38)

Here we have assumed that tα > tβ , since each of the infinitesimal Green’sfunctions propagates only forward in time. In this case (3.38) is indeed equalto

i2xf tf |x(tα)x(tβ )|xiti . (3.39)

However, if tα < tβ , then these two times appear in a different ordering in therhs of (3.38) and thus of the matrix element. The two cases can be combined

by introducing the time-ordering operator T

T [x(t1)x(t2)] =

x(t1)x(t2) t1 > t2

x(t2)x(t1) t2 > t1 .(3.40)

With the time-ordering operator we have

δ2xf tf |xitiJ δJ (t1)δJ (t2)

J =0

= i2

Dx

D p x(t1)x(t2)e

ih

tf ti

[ px−H (x,p)] dt

= i2xf tf |T [x(t1)x(t2)] |xiti . (3.41)

The same reasoning leads to the following result for higher-order deriva-tives

1

i

n δnxf tf |xitiJ δJ (t1)J (t2) . . . δ J (tn)

J =0

=

Dx

D p x(t1)x(t2) . . . x(tn) e

ih

tf ti

[ px−H (x,p)] dt

= xf tf |T [x(t1)x(t2) . . . x(tn)] |xiti . (3.42)




This is the generalization of (3.41). We thus have1

i

n δn

δJ (t1)δJ (t2) . . . δ J (tn)


J =0

=

1

i

n δn

δJ (t1)δJ (t2) . . . δ J (tn)Z [J ]

J =0

=

Dx D p x(t1)x(t2) . . . x(tn) e

ih

tf ti

[ px−H (x,p)] dt

Dx D p e

ih

tf

ti[ px−H (x,p)] dt

=xf tf |T [x(t1)x(t2) . . . x(tn)] |xiti

xf tf |xiti , (3.43)

where the limit ti → −∞, tf → +∞ is understood and all the times t1, . . . , tnlie in between these limits. If the Hamiltonian is quadratic in p and separatesin p and x, then we have

xf tf |T [x(t1)x(t2) . . . x(tn)] |xitixf tf |xiti =

Dx x(t1)x(t2) . . . x(tn) eihS [x(t)]

Dx e

ihS [x(t)]

,

(3.44)

where S [x(t)] is the action that depends functionally on the trajectory x(t).We now rewrite this equation. The numerator of the lhs becomes (in the

limit ti → −∞, tf → +∞, indicated by the arrow)

xf tf |T [. . .]|xitiJ =0 −→ xf tf |00|T [. . .]|00|xitiJ =0

= 0|T [. . .]|0 ϕ0(xf )e−ihE 0tf ϕ∗

0(xi)e+ ihE 0ti , (3.45)

while the denominator can be written as (c.f. (3.18))

xf tf |xiti = xf |e− ihHtf e+ i

hHti |xi −→ ϕ0(xf )e−

ihE 0tf ϕ∗

0(xi)e+ ihE 0ti (3.46)

where we have used in both cases (1.19), inserted a complete set of states and

– through the limit of infinite times – projected out the groundstate. Thegs wavefunctions and the time-dependent exponentials cancel out so that weobtain (for quadratic Hamiltonians)

0|T [x(t1)x(t2) . . . x(tn)] |0 =

Dx x(t1)x(t2) . . . x(tn) eihS [x(t)] Dx e

ihS [x(t)]

=

1

i

n δn

δJ (t1)δJ (t2) . . . δ J (tn)Z [J ]

J =0

(3.47)




with S [x(t)] = +∞−∞ L(x(t), x(t)) dt.

This is a very important result. It shows that the groundstate expectationvalue of a time-ordered product of position operators, the so-called correlation function , can be obtained as a functional derivative of the functional Z [J ]defined in (3.24). Note that |0 is the groundstate of H , as can be seenfrom (3.45). Thus the groundstate appearing on the lhs of (3.47) and thepropagator Z [J ] are linked together: if Z [J ] contains, for example, only afree Hamiltonian, then |0 is the groundstate of a free theory. If, on theother hand, Z [J ] contains interactions, then |0 is the groundstate of the fullinteracting theory.

Since all the correlation functions can be generated from Z [J ] this func-

tional is called a generating functional . For the remainder of this book wewill be concerned with these generating functionals.

Coming back to our example of the driven harmonic oscillator, discussedat the start of this section, we see that the time-ordered vacuum expecta-tion values of x are just the matrix elements that would appear in a time-dependent perturbation theory treatment of the groundstate of this system.Thus, if all the correlation functions are known, then the perturbation seriesexpansion is also known.



Part II

Relativistic Quantum Field

Theory

43



Chapter 4

CLASSICAL RELATIVISTIC

FIELDS

In this chapter a few essential facts of classical relativistic field theory aresummarized. It will first be shown how to derive the equations of motion of a field theory, for example the Maxwell equations of electrodynamics, from aLagrangian. Second, the connection between symmetries of the Lagrangianand conservation laws will be discussed1.

4.1 Equations of Motion

The equations of motion of classical mechanics can be obtained from a La-grange function by using Hamilton’s principle that the action for a givenmechanical system is stationary for the physical space–time development of the system.

The equations of motion for fields that determine their space–time depen-dence can be obtained in an analogous way by identifying the field amplitudesat a coordinate x with the dynamical variables (coordinates) of the theory.

Let the functions that describe the fields be denoted by

Φα(x) with xµ = (t, x) , (4.1)

where α labels the various fields appearing in a theory. The Lagrangian L of the system is expressed in terms of a Lagrange density L, as follows:

L =

L(Φα, ∂ µΦα) d3x (4.2)

1The units, the metric and the notation used in this and the following chapters isexplained in Appendix A.

44



CHAPTER 4. CLASSICAL RELATIVISTIC FIELDS 45

where the spatial integration is performed over the volume of the system.

The action S is then defined as usual by

S =

t1 t0

L dt = Ω

L(Φα, ∂ µΦα) d4x (4.3)

with the Lorentz-invariant four-dimensional volume element d4x = d3x dt.The, in general finite, space–time volume of the system is denoted by Ω.

As pointed out before the fields Φα(x) play the same role as the gener-alized coordinates qi in classical mechanics; the analogy here is such thatthe fields Φα correspond to the coordinates q and the points x and α to the

classical indices i. The corresponding velocities are given in a direct analogyby the time derivatives of Φα : ∂ tΦα. Lorentz covariance then requires thatalso the derivatives with respect to the first three coordinates appear; thisexplains the presence of the four-gradients ∂ µΦα in (4.2).

In order to derive the field equations from the action S by Hamilton’sprinciple, we now vary the fields and their derivatives

Φα → Φα = Φα + δΦα

∂ µΦα → (∂ µΦα) = ∂ µΦα + δ(∂ µΦα) . (4.4)

This yields

δL = L(Φα, (∂ µΦα)) − L(Φα, ∂ µΦα)

=∂ L

∂ ΦαδΦα +

∂ L∂ (∂ µΦα)

δ(∂ µΦα)

=∂ L

∂ ΦαδΦα +

∂ L∂ (∂ µΦα)

∂ µ(δΦα) . (4.5)

According to the Einstein convention a summation over µ is implicitly con-tained in this expression. In going from the second to the third line differen-tiation and variation could be commuted because both are linear operations.

The equations of motion are now obtained from the variational principle

δS =

Ω

∂ L

∂ Φα− ∂ µ

∂ L∂ (∂ µΦα)

δΦα + ∂ µ

∂ L

∂ (∂ µΦα)δΦα

d4x

= 0 (4.6)

for arbitrary variations δΦα under the constraint that

δΦα(t0) = δΦα(t1) = 0 ,




where t0 and t1 are the time-like boundaries of the four-volume Ω. The last

term in (4.6) can be converted into a surface integral by using Gauss’s law;for fields which are localized in space this surface integral vanishes if thesurface is moved out to infinity. Since the variations δΦα are arbitrary thecondition δS = 0 leads to the equations of motion

∂

∂xµ

∂ L

∂ (∂ µΦα)

− ∂ L

∂ Φα= 0 . (4.7)

The relativistic equivalence principle demands that these equations havethe same form in every inertial frame of reference, i.e. that they are Lorentzcovariant. This is only possible if

Lis a Lorentz scalar, i.e. if it has the same

functional dependence on the fields and their derivatives in each referenceframe.

In a further analogy to classical mechanics, the canonical field momentumis defined as

Πα =∂ L

∂ Φα=

∂ L∂ (∂ 0Φα)

. (4.8)

From L and Πα the Hamiltonian H is obtained as

H =

H d3x =

(ΠαΦα − L) d3x . (4.9)

The Hamiltonian H represents the energy of the field configuration.

4.1.1 Examples

The following sections contain examples of classical field theories and theirformulation within the Lagrangian formalism just introduced. We start outwith the probably best-known case of classical electrodynamics, then general-ize it to a treatment of massive vector fields and then move on to a discussionof classical Klein–Gordon and Dirac fields that will play a major role in thelater chapters of this book.

Electrodynamics

The best-known classical field theory is probably that of electrodynamics,in which the Maxwell equations are the equations of motion. The two ho-mogeneous Maxwell equations allow us to rewrite the fields in terms of afour-potential

Aµ = (A0, A) , (4.10)




defined via

B = × A ,

E = − A0 − ∂ A

∂t. (4.11)

Note that the 2 homogeneous Maxwell equations are now automatically ful-filled. The two inhomogeneous Maxwell equations2

· E = ρ ,

× B − ∂ E

∂t= j (4.12)

with external density ρ and current j can be rewritten as

∂ µF µν =∂F µν

∂xµ= jν , (4.13)

with the four-current jν = (ρ, j) (4.14)

and the antisymmetric field tensor

F µν =∂Aν

∂xµ− ∂Aµ

∂xν = ∂ µAν − ∂ ν Aµ . (4.15)

The tensor F µν is the dyadic product of two fourvectors and thus a Lorentz-tensor. Equation (4.13) is the equation of motion for the field tensor or thefour-vector field Aµ.

It is easy to show that (4.13) can be obtained from the Lagrangian

L = −1

4F µν F µν − jν Aν (4.16)

by using (4.7); the fields Aν here play the role of the fields Φα in (4.7). Wehave

∂ L∂Aν

=

− jν ,

∂ L∂ (∂ µAν )

= −1

42(+F µν − F νµ) = −F µν . (4.17)

The last step is possible because F is an antisymmetric tensor. The equationof motion is therefore

∂

∂xµ

∂ L

∂ (∂ µAν )

− ∂ L

∂Aν = −∂F µν

∂xµ+ jν = 0 , (4.18)

2Here the Heaviside units are used with c = 1 and 0 = µ0 = 1.




in agreement with (4.13). It is now easy to interpret the two terms in

L(4.16): the first one gives the Lagrangian for the free electromagnetic field,whereas the second describes the interaction of the field with charges andcurrents.

The two homogeneous Maxwell equations can also be expressed in termsof the field tensor by first introducing the dual field tensor

F µν =1

2µνρσF ρσ ; (4.19)

here µνρσ is the Levi–Civita antisymmetric tensor which assumes the values+1 or

−1 according to whether (µνρσ) is an even or odd permutation of

(0,1,2,3), and 0 otherwise. In terms of F the homogeneous Maxwell equa-tions read

∂ µF µν = 0 . (4.20)

Eqs. (4.13) and (4.20) represent the Maxwell equations in a manifestlycovariant form. The Lagrangian (4.16) is obviously Lorentz-invariant sinceit consists of invariant contractions of two Lorentz-tensors (the first term)and two Lorentz-vectors (the second term). It is also invariant under a gaugetransformation

Aµ −→ Aµ = Aµ + ∂ µφ (4.21)

because F itself is gauge-invariant by construction and the contribution of the interaction term to the action is gauge-invariant for an external conservedcurrent. The same then holds for the equation of motion (4.18). Symmetry(Lorentz-invariance), gauge-invariance and simplicity (there are no higherorder terms in (4.16)) thus determine the Lagrangian of electrodynamics.

The gauge freedom can be used to impose constraints on the four compo-nents of the vector field Aµ. In addition, for free fields this gauge freedom canbe used, for example, to set the 0th component of the four-potential equal tozero. Thus, a free electromagnetic field has only two degrees of freedom left.

Massive Vector FieldsVector fields in which – in contrast to the electromagnetic field – the fieldquanta are massive are described by the so-called Proca equation:

∂ µF µν + m2Aν = jν . (4.22)

Operating on this equation with the four-divergence ∂ ν gives, because F isantisymmetric,

m2∂ ν Aν = ∂ ν j

ν . (4.23)




For m

= 0 and a conserved current, this reduces the equation of motion

(4.22) to 2+ m2

Aν = jν ; ∂ ν A

ν = 0 . (4.24)

Thus for massive vector fields the freedom to make gauge transformationson the vector field is lost. The condition of vanishing four-divergence of thefield reduces the degrees of freedom of the field from 4 to 3. The space-likecomponents represent the physical degrees of freedom.

The Lagrangian that leads to (4.22) is given by

L = −1

4F 2 +

1

2m2A2 − j · A . (4.25)

Klein–Gordon Fields

A particularly simple example is provided by the so-called Klein–Gordon fieldφ that obeys the equation of motion

∂ µ∂ µ + m2

φ =2+ m2

φ = 0 ; (4.26)

such a field describes scalar particles, i.e. particles without intrinsic spin.The Lagrangian leading to (4.26) is given by

L =

1

2(∂ µφ) (∂

µ

φ) − m2

φ2 (4.27)

since we have∂ L

∂ (∂ µφ)= ∂ µφ (4.28)

and∂ L∂φ

= −m2φ . (4.29)

It is essential to note here that the Lagrangian density (4.27) that leads to(4.26) is not unique; unique is only the action S =

L d4x. For localized

fields for which the surface contributions vanish we can perform a partialintegration of the kinetic term in (4.26) d4x (∂ µφ) (∂ µφ) = −

d4x φ2φ (4.30)

so that we obtain

L = −1

2φ2+ m2

φ . (4.31)

The Lagrangians (4.31) and (4.27) are equivalent. Since both give the sameaction they also lead to the same equation of motion (4.26).




An interesting case occurs if we consider two independent real scalar fields,

φ1 and φ2, with the same mass m. The total Lagrangian is then simply givenby a sum over the Lagrangians describing the individual fields, i.e.

L =1

2

(∂ µφ1) (∂ µφ1) − m2φ2

1

+

1

2

(∂ µφ2) (∂ µφ2) − m2φ2

2

. (4.32)

On the other hand, we can also construct two complex fields from the tworeal fields φ1 and φ2, namely

φ =1√

2(φ1 + iφ2) (4.33)

and its complex conjugate. In terms of these the Lagrangian (4.32) can berewritten to

L = (∂ µφ)∗ (∂ µφ) − m2φ∗φ

= − φ∗2+ m2

φ . (4.34)

Dirac Fields

A particularly simple example is provided by the Dirac field Ψ for which theequation of motion is just the Dirac equation

(iγ µ∂ µ − m) Ψ = 0 , (4.35)

where the γ µ are the usual (4 × 4) matrices of Dirac theory. Ψ itself is a(4 × 1) matrix of four independent fields, a so-called spinor .

The corresponding Lagrangian is given by

L = Ψ (iγ µ∂ µ − m) Ψ . (4.36)

This can be seen by identifying the fields Φα in (4.7) with the four componentsof the Dirac spinor Ψ = Ψ†γ 0. Since

Ldoes not depend on ∂ µΨ the equation

of motion is simply given by

∂ L∂ Ψ

= (iγ µ∂ µ − m) Ψ = 0 . (4.37)

4.2 Symmetries and Conservation Laws

As in classical mechanics there is also in field theory a conservation law as-sociated with each continuous symmetry of L. The theorem which describes




the connection between the invariance of the Lagrangian under a continu-

ous symmetry transformation and the related conserved current is knownas Noether’s theorem. In the following, this will be illustrated for differenttypes of symmetries which then lead to the well-known conservation laws.

The common expression in the arguments to follow is the change of theLagrangian density under a change of the fields and their derivatives (see(4.4)). According to (4.5) and the Lagrange equations of motion (4.7) thischange is given by

δL = ∂ µ

∂ L


. (4.38)

4.2.1 Geometrical Space–Time Symmetries

In this section we investigate the consequences of translations in four-dimen-sional space–time, i.e. infinitesimal transformations of the form

xν → xν = xν + ν , (4.39)

where ν is a constant infinitesimal shift of the coordinate xν . Under suchtransformations the change of L is given by

δL = ν ∂ L∂xν

= ν ∂ ν L , (4.40)

since L is a scalar.If now L is required to be form-invariant under translations, it does not

explicitly depend on xν . In this case, δL is also given by (4.38). The changesof the fields Φα appearing there are for the space–time translation consideredhere given by

δΦα = ν ∂ Φα∂xν

= ν ∂ ν Φα . (4.41)

Inserting (4.41) into (4.38) yields

δL = ν ∂ µ ∂

L∂ (∂ µΦα) ∂

ν

Φα . (4.42)

Equating (4.42) and (4.40) finally gives

∂ µ

∂ L

∂ (∂ µΦα)∂ ν Φα − L gµν

= 0 , (4.43)

since the ν are arbitrary. By defining the tensor T µν as

T µν ≡ ∂ L∂ (∂ µΦα)

∂ ν Φα − L gµν (4.44)




this equation reads

∂ µT µν = 0 . (4.45)

Relation (4.45) has the form of a continuity equation. Spatial integrationover a finite volume yields

d

dt

V

T 0µ(x)d3x

= −

V

∂T iµ

∂xid3x = −

S

S (µ) · n dS . (4.46)

Here n is a unit vector vertical on the surface S pointing outwards and S (µ)

is a three-vector: S

(µ)

= (T 1µ

, T 2µ

, T 3µ

) . (4.47)The surface integral on the rhs of (4.46) is taken over the surface S of

volume V. For localized fields it can be made to vanish by extending thevolume towards infinity. It is then evident that the quantities

P µ =

T 0µd3x (4.48)

are conserved. These are the components of the four-momentum of the field,as can be verified for the zeroth component,

P 0 = T 00 d3x = ∂ L∂ (∂ 0Φα)

∂ 0Φα − L d3x

=

(ΠαΦα − L) d3x = H , (4.49)

according to (4.8) and (4.9). The spatial components of the field momentumare

P k =

T 0k d3x = ∂ L

∂ (∂ 0Φα)∂ kΦα d3x . (4.50)

T µν , as defined in (4.44), has no specific symmetry properties. It can,however, always be made symmetric in its Lorentz-indices because (4.45)

does not define the tensor T uniquely. We can always add a term of the form∂ λDλµν , where Dλµν is a tensor antisymmetric in the indices λ and µ, suchthat T becomes symmetric.3

3In classical mechanics the form invariance of the Lagrangian under rotations leads tothe conservation of angular momentum. Analogously, in a relativistic field theory the forminvariance of L under four-dimensional space–time rotations (Lorentz covariance) leads tothe conservation of a quantity that is identified with the angular momentum of the field.To obtain the same form for the angular momentum as in classical mechanics it is essentialthat T µν is symmetric.




Comparing (4.46) with (4.50) and assuming T to be symmetric we see that

the normal components of the vectors S (µ) in (4.47) describe the momentumflow through the surface S of the volume V and thus determine the “radiationpressure” of the field.4 These properties allow us to identify T µν as the energy-momentum tensor of the field. For the Lagrangian (4.16) of electrodynamicsT µν is just the well-known Maxwell’s stress tensor.

As already mentioned at the beginning of this chapter these conservationlaws are special cases of Noether’s theorem, which can be summarized forthe general case as follows:

Each continuous symmetry transformation that leaves the La-grangian invariant is associated with a conserved current. Thespatial integral over this current’s zeroth component yields a con-served charge.

4.2.2 Internal Symmetries

Relativistic field theories may contain conservation laws that are not conse-quences of space–time symmetries of the Lagrangian, but instead are con-nected with symmetries in the internal degrees of freedom such as, e.g.,isospin or charge.

We therefore now allow for a mixture of the different fields under the

transformationΦα(x) → Φ

α(x) = e−iεqαβΦβ , (4.51)

where is an infinitesimal parameter and the qαβ are fixed c-numbers. Wethen have

δΦα(x) = Φα(x) − Φα(x) = −iεqαβ Φβ (x) . (4.52)

The change of the Lagrangian is given by (4.38)

δL = ∂ µ

∂ L


. (4.53)

If L

is invariant under this variation δΦα, then we have

δL = ∂ µ

∂ L


= 0 . (4.54)

Equation (4.54) is in the form of a continuity equation for the “current”

jµ(x) =∂ L


1

ε. (4.55)

4More precisely, S k(µ) denotes the flux of the µth component of the field momentumin the direction xk.




Inserting the field variations δΦα yields for the current

jµ(x) = −i∂ L

∂ (∂ µΦα)qαβ Φβ . (4.56)

Equations (4.54) and (4.56) imply that the “charge”

Q =

j0(x) d3x = −i ∂ L

∂ (∂ 0Φα)qαβ Φβ d

3x (4.57)

of the system is conserved. The physical nature of these “charges” and “cur-rents” has to remain open. It depends on the specific form of the symmetrytransformation (4.63) and can be determined only by coupling the system to

external fields.

Example: Quantum Electrodynamics

To illustrate this conservation law, the theory of electromagnetic interac-tions is used as an example. However, in contrast to the considerations inSect. 4.1.1 we now consider a coupled system of a fermion field Ψ(x) and theelectromagnetic field Aµ(x) to determine the physical meaning of the con-served current. Together with a quantization procedure this theory is calledQuantum Electrodynamics (QED). The Lagrangian is given by

L = −14

F µν F µν + Ψ [iγ µ(∂ µ + ieAµ) − m] Ψ . (4.58)

L contains a part that describes the free electromagnetic field (first term).The second term describes the fermion Lagrangian; it is obtained from thefree particle Lagrangian of (4.36) by replacing the derivative ∂ µ by the co-variant derivative

Dµ = ∂ µ + ieAµ (4.59)

(minimal coupling). Here e is the electron’s charge (e = −|e|).The Lagrangian (4.58) is obviously invariant under a variation of the

fermion fields of the formΨ → Ψ = e−ieΨ . (4.60)

Comparison with (4.51) gives qαβ = e δαβ so that the conserved “current”given by (4.56) is:

jµ(x) = e Ψγ µΨ . (4.61)

Note that this conserved current is exactly the quantity that couples to theelectromagnetic field in (4.58). This property allows one to identify thecurrent (4.61) as the electromagnetic current of the electron fields.




Example: Scalar Electrodynamics

The Lagrangian for the case of a complex scalar field interacting with anelectromagnetic field is given by

L = −1

4F µν F µν + (Dµφ)∗ (Dµφ) − m2φ∗φ . (4.62)

This Lagrangian is simply the sum of the free electromagnetic Lagrangian(4.16) and the Lagrangian for a complex scalar field (4.34), where in the latteragain the derivative ∂ µ has been replaced – through minimal substitution –by the covariant derivative Dµ (4.59).

The Lagrangian (4.62) is obviously invariant under the phase transforma-tions

φ(x) −→ e−iεeφ(x)

φ∗(x) −→ e+iεeφ∗(x) (4.63)

The conserved current connected with this invariance can be obtained fromthe definition (4.56)

jµ = −i

∂ L

∂ (∂ µφ)eφ +

∂ L∂ (∂ µφ∗)

(−e)φ∗

= ie (φ∗Dµφ − φ (Dµφ)∗) . (4.64)

The conserved charge is then given by

Q =

d3x j0(x) = ie

d3x

φ∗D0φ − φ

D0φ∗

. (4.65)

It is remarkable that now the electromagnetic field Aµ appears in the con-served current (through the covariant derivative Dµ = ∂ µ+ ieAµ). Again theconserved current provides the coupling to the electromagnetic field. If thescalar field is real then it is invariant under the transformation (4.63) only for

e = 0. Eq. (4.65) then shows that in this case the conserved charge Q = 0.



Chapter 5

PATH INTEGRALS FOR

SCALAR FIELDS

In this chapter we apply the methods developed in chapter 3 to the case of scalar fields. There we showed that the vacuum expectation values of time-ordered products of x operators could be obtained as functional derivatives of a generating functional. All these results can be taken over into field theoryremembering that fields play the role of the coordinates of the theory and thespatial locations x correspond to the indices of the classical coordinates. Thisimplies that we can obtain the vacuum expectation values of time-orderedfield operators by performing the derivatives on an appropriate functional.To discuss this functional is the main purpose of this chapter.

It is easy to see that these vacuum expectation values of time-orderedproducts of field operators play an important role in quantum field the-ory. Each field operator creates or annihilates particles and a time-orderedproduct of field operators can thus describe the probability amplitude for aphysical process in which particles are created and annihilated. The quanti-tative information about such a process is contained in the S matrix whichcan be obtained from the vacuum expectation values of time-ordered fieldoperators by means of the so-called reduction theorem which we will derive

in a later chapter. The rest of this manuscript will therefore be concernedwith calculating these expectation values and with developing perturbativemethods for their determination when an exact calculation is not possible.

56



CHAPTER 5. PATH INTEGRALS FOR SCALAR FIELDS 57

5.1 Generating Functional for Fields

We assume that the system is described by a Lagrangian of the form

L (φ, ∂ µφ) =1

2

∂ µφ∂ µφ − m2φ2 − V (φ)

. (5.1)

In order to obtain the functional W [J ] for fields we note that the fields playthe role of the coordinates of the theory and that sums over the differentcoordinates have to be replaced by integrals over the space-time coordinates.In order to define more stringently what is actually meant by a path integralfor fields we write it down here in detail for a free field Lagrangian ( V = 0).

In order to do so we bring it into a form as close as possible to the classicaldefinition (1.38).

We first Fourier-expand the field

φ(x, t) =1√V

k

q k(t)ei k·x . (5.2)

For a real field φ we haveq∗− k

= q k . (5.3)

Inserting this expansion into the free Lagrangian gives1

L =

L d3x =1

2

∂ µφ∂ µφ − m2φ2

d3x (5.4)

=1

2V

kk

qkqk

k · k − m2

+ qkqk

ei( k+ k)· x d3x .

Integration over d3x gives V δ k,− k so that we have

L =1

2

k

qkq−k − ( k2 + m2)qkq−k

=

1

2k

(qkq∗

k − ωkqkq∗

k) (5.5)

with ωk =

k2 + m2. Using (5.3), writing qk = X k + iY k, and grouping thenthe coordinates X k and −iY k into a new vector xk gives

L =1

2

k

x2k − ω2

kx2k

. (5.6)

1For ease of notation we drop the vector arrows in the indices




In the general spirit of field theory we introduce a source density J which

we also expand

J (x, t) =1√V

k

j k(t)ei k·x . (5.7)

With this expansion we obtain for the source term d3x J (x)φ(x) =

k

jk(t)q−k(t) =k

J k(t)xk(t) (5.8)

where we have grouped jk and −i jk into a new vector J k.The Lagrangian (5.6) has the structure of a Lagrangian with quadratic

momentum dependence and constant coefficient, discussed in Sect. 1.3.1; alsothe souce term reads formally just the same as for the nonrelativistic systemstreated in chapter 3. We can thus again integrate the momentum dependenceout and obtain for the vacuum-to-vacuum transition amplitude (cf. (3.23))

W [J ] = limtf →+∞

ti→−∞

0tf |0tiJ

= limη→0

1

2πhiη

n+12

k

n j=1

dxkj eiη

nl=0

(Ll+J kxkl)

, (5.9)

with

Ll = L (xl, xl) = 12

k

x2kl − ω2

kx2kl

, (5.10)

where the first index (k) denotes the coordinate and the second ( j) the timeinterval. If we now identify the integration measure as

Dφ = limη→0

1

2πhiη

n+12

k

n j=1

dxkj , (5.11)

we can write the generating functional also as

W [J ] = Dφ expi 1

2∂ µφ∂ µφ − m2 − iεφ2 + Jφ d4x

=

Dφ exp

i

L (φ, ∂ µφ) + Jφ + iε

2φ2

d4x

. (5.12)

Here the volume element d4x is given by the Lorentz-invariant expression

d4x = d3x dt . (5.13)

The term iεφ2/2 with positive ε has been introduced in an ad hoc mannerto ensure the convergence of W when taking the fields to infinity with the




understanding that ultimately ε has to be taken to 0 (cf. the discussion at

the end of Sect. (3.1)).The second line of (5.12) also gives the generating functional for an in-

teracting theory with the interaction V included in L. This can be easilyproven by including an additional interaction in the Lagrangian (5.4) andFourier-expanding it. The generating functional for a scalar field theory isthus given by

W [J ] =

Dφ e−i

d4x[ 12φ(2+m2−iε)φ+V (φ)−Jφ] (5.14)

where the Lagrangian in the form (4.31) has been used.In analogy to (3.24) we define a normalized functional

Z [J ] =W [J ]

W [0]. (5.15)

Again the infinite normalization factor inherent in W [J ] cancels out in thisdefinition. Since W [J ] involves an exponential it will often be convenientin the following discussions to introduce its logarithm iS [J ] which is itself afunctional of J

W [J ] ≡ eiS [J ] =⇒ S [J ] = −i ln W [J ] = −i ln Z [J ] − i ln W [0] . (5.16)

5.1.1 Euclidean Representation

In the preceding section we have ensured convergence of the generating func-tional by introducing the ε-dependent term in the energies. In this subsectionwe go back to the alternative method that relies on the Wick rotation thatwe introduced in Sect. 3.21 and discuss the Euclidean representation of thegenerating functional. This discussion also illustrates in some more detail theremarks on integrating oscillatory functions made at the end of Sect. 1.3.1.

The real Euclidean space is obtained from Minkowski space by rotatingthe real axis in the x0 plane by δ = −π/2 into the negative imaginary axis(Wick rotation). We denote a space-time point in Euclidean space by xE; it

is related to the usual space-time point x in Minkowski space byxE = (x, x4) with x4 = ix0 = it . (5.17)

Under the Wick rotation t → −it and x4 thus becomes real. With thisdefinition we can extend the usual Minkowski-space definitions of volumeelement and space-time distance to Euclidean space

d4xE ≡ d3x dx4 = d3x idt = id4x

dx2E =

3 j=1

dx2 j + x2

4 = −dx2 . (5.18)




The d’Alembert operator is then given by

2 =∂ 2

∂t2− 2 = − ∂ 2

∂x24

− 2 = −4a=1

∂ 2

∂x2a

≡ −2E . (5.19)

With these transformations the generating functional for a free scalar fieldin Minkowski space (5.14) becomes in its Euclidean representation

W 0E[J ] =

Dφ e−

d4xE 12 [(∂ Eφ)2+m2φ2]−Jφ , (5.20)

with (∂ Eφ)2 = − (∂φ)2. Because x4 is now real, (∂ Eφ)2 is always positive

and the exponent is negative definite; the integral thus converges and is well-defined even without adding in the ε-dependent term. Since the exponentis furthermore quadratic in the fields, W 0E[J ] can be evaluated by using thetechniques for Gaussian integrals that are explained in App. B. Physicalresults are then obtained by rotating backwards after all integrations havebeen performed.

Remembering that in field theory the fields play the role of the coordinatesof a Lagrangian theory we can now directly generalize some of the results of Chapt. 3 to field theory. In particular, we have that W E[J ] of eq. (5.20) is thetransition amplitude from the vacuum state of the free theory at t → −∞to that at t

→+

∞under the influence of the external source J (cf. (3.27)

so that the normalized transition amplitude Z E is given by

Z 0E[J ] =W 0E[J ]

W 0E[0]=

0 + ∞|0 − ∞J 0 + ∞|0 − ∞0

, (5.21)

where |0 is the vacuum state of the free theory. Eq. (5.20) shows that thenormalized transition amplitude can be understood as an integration of thesource action exp (+

d4xEJφ) with the weights

w0E(φ) = e−

12

d4xE [(∂ Eφ)2+m2φ2] (5.22)

over all fields φ

Z 0E[J ] =

Dφ w0E(φ)e

d4xEJφ Dφ w0

E(φ). (5.23)

Eq. (3.47) then shows that for a function of the fields O(φ) Dφ w0E(φ)

j O j(φ(x j)) Dφ w0E(φ)

= 0|T [ j

O j(φ(x j))]|0 . (5.24)

The field operators on the rhs here are those of free fields.




In the interacting case all these relations still hold if w0E is replaced by a

weight function for the interacting theory and the vacuum state is now thatof the interacting theory which we will denote by |0. To obtain the weightfunction of the interacting theory we use a perturbative treatment of theinteraction V . By writing in analogy to (5.22)

wE(φ) = e−

d4xE 12 [(∂ Eφ)2+m2φ2]+V (φ) (5.25)

= e− V (φ) d4xE e−

12

[(∂ Eφ)2+m2φ2] d4xE≡ e

−

V (φ) d4xE w0E(φ)

we get with (5.24)

0|T [ j

O j(φ)]|0 = Dφ wE(φ)

j

O j(φ)

=

Dφ w0E(φ)e−

V (φ) d4xE

j O j(φ) Dφ w0

E(φ)e− V (φ) d4xE

=0|T [

j O j(φ0)e−

V (φ0) d4xE]|0

0|T [e− V (φ0) d4xE]|0

. (5.26)

Here |0 on the rhs is the vacuum state of the non-interacting free theory(V = 0) and all the field operators on the rhs are free field operators φ

0at all

times if they were free at t → −∞. This can be seen from the path integralwhich contains the free weight w0

E connected with free propagation.In the following chapters it is often useful to think in terms of this prob-

abilistic interpretation even when we work in Minkowski metric. This ispossible by relating Euclidean and Minkowski space by analytic continua-tion.



Chapter 6

EVALUATION OF PATH

INTEGRALS

Only a limited class of path integrals can be evaluated analytically [Grosche-Steiner] so that one is forced to use either numerical or perturbative methods.In this section we first calculate the generating functional for free scalar fields,both by a direct reduction method and, to gain familiarity with this tech-nique, with the the methods of Gaussian integration developed in App. B.2.1.We then show how more general types of path integrals can approximately bereduced to the Gaussian form. In this reduction we find a systematic methodfor a semiclassical expansion in terms of the Planck constant h. Finally, webriefly discuss methods for the numerical evaluation of path integrals.

6.1 Free Scalar Fields

The generating functional for a free scalar field theory plays a special rolein the theory of path integrals. This is so because of two reasons: first, afree-field theory is the simplest possible field theory and, second, as we have just seen in the last section the effects of the interaction term V (φ) can be

described with the help of perturbation theory in which the functional of thefull, interacting theory is expanded around that of the free theory.

6.1.1 Generating functional

The representation of the generating functional of a free field theory (V = 0 in(5.14)) as a path integral is still quite cumbersome for practical applications.For these it would be very desirable if we could factorize out the functionaldependence on J in form of a normal integral; the remaining path integral

62



CHAPTER 6. EVALUATION OF PATH INTEGRALS 63

would then disappear in the normalization.

In the following we will therefore separate the path integral (5.9) into 2factors, one depending on J and the other one being an integral over φ, i.e. just a number. For this purpose we start with the generating functional

W [J ] =

Dφ e−i

d4x[ 12φ(2+m2−iε)φ−Jφ] (6.1)

where the Lagrangian in the form (4.31) has been used. The field φ here isan integration variable; it does not fulfill a Klein-Gordon equation! We nowintroduce a field φ0 that does just that

2+ (m2

−iε)φ0(x) = J (x) . (6.2)

J thus plays the role of a source to φ0. We now take this field φ0 as a referencefield and expand φ around it, setting φ = φ0 + φ. We thus obtain for theintegrand in the exponent in (6.1)

1

2(φ0 + φ)2 (φ0 + φ) + (φ0 + φ)

m2 − iε

(φ0 + φ) − J (φ0 + φ)

=1

2φ2+

m2 − iε

φ

+1

2φ0

2+

m2 − iε

φ0 +

1

2φ0

2+

m2 − iε

φ

+ 12

φ2+

m2 − iε

φ0 − Jφ0 − Jφ . (6.3)

The third and the fourth terms on the rhs give the same contribution whenintegrated over. The second term gives, according to (6.2), 1

2Jφ0. Collecting

all terms we therefore have for the action in (6.1)

S [φ, J ] = −

d4x

1

2φ2+

m2 − iε

φ − Jφ

= −

d4x

1

2φ2+

m2 − iε

φ +

1

2Jφ0

+ φ 2+ m

2

− iεφ0 − Jφ0 − Jφ . (6.4)

In the last line of this equation we can again apply (6.2) to obtain

S [φ, J ] = −1

2

d4x

φ2+

m2 − iε

φ − Jφ0

(6.5)

We are now very close to our aim to factorize out the J dependence. Todo so we solve (6.2) by writing

φ0(x) = −

DF(x − y)J (y) d4y , (6.6)




where DF, the so-called Feynman propagator , fulfills the equation2+

m2 − iε

DF(x) = −δ4(x) (6.7)

in complete analogy to the nonrelativistic propagator K in chapter 1. Using

δ4(x) =

1

2π

4 d4k e−ikx (6.8)

we obtain

DF(x − y) =1

(2π)4

e−ik(x−y)

k2 − m2 + iεd4k . (6.9)

Substituting (6.6) into the action (6.5) gives

S [φ, J ] = − 1

2

d4x

φ2+

m2 − iε

φ + J (x)

DF(x − y)J (y) d4y

= − 1

2

d4x

φ2+

m2 − iε

φ

− 1

2

J (x)DF(x − y)J (y) d4x d4y . (6.10)

The exponential of the last term no longer depends on φ and can, therefore,be pulled out of the pathintegral (6.1). The pathintegral involving the expo-

nential of the first term appears also in the denominator of the normalizedgenerating function (5.15) and thus drops out.

We thus obtain now for the normalized generating functional

Z 0[J ] =W [J ]

W [0]= e−

i2

J (x)DF(x−y)J (y) d4xd4y . (6.11)

This is the vacuum-to-vacuum transition amplitude for a free scalar fieldtheory. Note that it no longer involves a path integral.

6.1.2 Feynman propagator

The imaginary part of the mass, originally introduced to achieve convergencefor the path integrals, appears here now in the denominator of the propagatorDF

DF(x) =1

(2π)4

e−ikx

k2 − m2 + iεd4k . (6.12)

It determines the position of the poles in DF, which are, in the k0-integration,at

k20 = k2 + m2 − iε (6.13)




– +iω δk

+ –iω δk

Re k 0

Im k 0

Figure 6.1: Location of the poles in the Feynman propagator.

or

k0 = ±

k2 + m2 iδ . (6.14)

The poles are therefore located as indicated in Fig. 6.1.The location of the poles, originally introduced only in an ad-hoc way to

achieve convergence of the path integrals, determines now the properties of the propagator of the free Klein-Gordon equation, the Feynman propagator.

This can be seen by rewriting the Feynman propagator DF in the followingform

DF(x) =1

(2π)4

d4k

e−ikx

k2 − m2 + iε(6.15)

=1

(2π)4

d3k dk0

e−ikx

k20 − k2 − m2 + iε

=1

(2π)4

d3k dk0 e−ikx 1

2ωk

1

k0 − ωk + iδ− 1

k0 + ωk − iδ

with ω2k = k2 + m2 .

We now first perform the integration over k0. Since the exponential con-tains a factor e−ik0t, the path can be completed in the upper half plane fornegative times and in the lower half for t > 0. Cauchy’s theorem then gives2πi × the sum of the residues of the enclosed poles

DF(x) =i

(2π)3

d3k

ei k·x

2ωk

−Θ(−t)e+iωkt − Θ(t)e−iωkt

= − i

(2π)3

d3k

ei k·x

2ωk

Θ(−t)e+iωkt + Θ(t)e−iωkt

(6.16)




(in the second term here an extra “

−” sign appears because of the negative

direction of the contour integral).It can now be shown that DF propagates free fields with negative fre-

quencies backwards in time and those with positive frequencies forwards. Todemonstrate this we write

DF(x) = − i

(2π)3

d3k

ei k· x

2ωk

Θ(−x0)e+iωkx0 + Θ(x0)e−iωkx0

= − Θ(−x0)i

(2π)3

d3k

1

2ωkeikx

−Θ(x0)

i

(2π)3 d3k1

2ωke−ikx . (6.17)

Here we have changed k → − k in the first integral, which does not changeits value under this substitution.

The integrands are products of orthogonal, normalized solutions of theKlein-Gordon equation

ϕ(±) k

(x) =1

(2π)32ωkeikx (6.18)

which fulfill the normalization and orthogonality conditions appropriate for

a scalar field (cf. (4.65) without an A field)

i

d3x

ϕ(±)∗ k

(x) ϕ(±) k

(x) − ϕ(±)∗ k

(x)ϕ(±) k

(x)

= ±δ3( k − k) . (6.19)

andi

d3x

ϕ(±)∗ k

(x) ϕ() k

(x) − ϕ(±)∗ k

(x)ϕ() k

(x)

= 0 . (6.20)

We can thus write

iDF(x) = Θ(−x0)1

(2π)3

d3k

1√2ωk

1√2ωk

eikx (6.21)

+ Θ(x0) 1(2π)3

d3k 1√2ωk

1√2ωk

e−ikx

= Θ(−x0)

d3k ϕ(−)∗ k

(0)ϕ(−) k

(x) + Θ(x0)

d3k ϕ(+)∗ k

(0)ϕ(+) k

(x) .

Comparison with (1.13) shows that negative-frequency solutions are prop-agated backwards in time, and positive-frequency solutions forward. Thisparticular behavior is a consequence of the location of the poles relative tothe integration path. This location is fixed by the sign of the ε term whichin turn was needed to achieve convergence for the generating functional.






6.1.3 Gaussian Integration

In Sect. 1.3 we have already used a Gaussian integral relation to integrateout the p-dependence of the path integral. In many cases the generatingfunctions appearing in field theory are of a form that contains the fields andtheir derivatives only in quadratic form so that again a Gaussian method canbe used. In order to gain familiarity with this technique we derive in thissection again the generating functional for the free scalar field theory.

We thus apply the Gaussian integration formulas of Sect. B.2.1 to thegenerating functional (6.1)

W [J ] = Dφ e−i2 φ(2+m2−iε)φd4xei Jφd4x . (6.26)

In order to make the matrix structure of the exponent more visible we usetwo-fold partial integration and write

W [J ] =

Dφ e−i2

φ(x)δ4(x−y)(2y+m2−iε)φ(y) d4xd4yei

Jφd4x

=

Dφ e−i2

φ(x)[(2y+m2−iε)δ4(x−y)]φ(y) d4xd4yei

Jφd4x

→

Dφ e− 1

2φ(xE)[(−2Ey +m2)δ(xE−yE)]φ(yE) d4yE−Jφd4xE . (6.27)

In the last step we have gone over to the Euclidean representation of thegenerating functional (cf. (5.20)), using 2E = −2 and δ4(xE−yE) = −iδ4(x−y) (cf. Sect. 5.1.1).

The integrals appearing here are now of Gaussian type and can thus beintegrated by using the expressions developed in the last section. We firstidentify the matrices A and B in the Gaussian integration formula (B.18) as

AE(x, y) = −(2E y − m2)δ4(xE − yE)

B(x) = −J T (x) . (6.28)

AE is real with positive eigenvalues (k2E+m2 > 0) as required by the derivation

in section B.2.1. It is also symmetric as can be seen by writing the d’Alembertoperator in a discretized form, e.g.

d2

dx2φ(x)

xi

= limh→0

1

h2[φ(xi + h) − 2φ(xi) + φ(xi − h)]

= Aijφ(x j) (6.29)

with φ(xi + h) = φ(xi+1) etc. and

Aij = δi,j−1 − 2δi,j + δi,j+1 .




Thus the necessary conditions for the application of (B.13) are fulfilled. Ap-

plying now (B.18) gives

W E[J ] =det(−2E + m2)δ(xE − yE)

− 12 e

12

J (xE)(AE(xE,yE))

−1J (yE) d4xEd4yE .

(6.30)The Wick rotation back to real times is easily performed by the transfor-

mation (5.19) 2E → −2− iε; it reintroduces the term +iε to guarantee theproper treatment of the poles. In this case A becomes

AE → A = (2+ m2 − iε)iδ(x − y) . (6.31)

and the volume element

d4xEd4yE → −d4xd4y (6.32)

so that we finally obtain1

W [J ] =1√

det Ae−

12

J (x)[i(2y+m2+iε)δ4(x−y)]

−1J (y) d4xd4y . (6.33)

The determinant of a matrix A is in general given by the product of itseigenvalues αi. Therefore we have

ln(det A) = lni

αi = i

ln αi = tr ln A . (6.34)

In the present case, though, A contains an operator and the notation det Adeserves some explanation. The matrix is given by

A(x, y) =2+ m2

iδ(x − y) =

2+ m2

i

1

(2π)4

d4k eik(x−y)

=i

(2π)4

d4k

−k2 + m2

eik(x−y)

= i d4k d4k1

(2π)4

eikx

−k2 + m2

δ4(k

−k)

1

(2π)4

e−iky .

Thus the momentum representation of the operator A is given by

A(k, k) =−k2 + m2

δ4(k − k) . (6.35)

We now evaluate the trace of the logarithm of this matrix where – in ac-cordance with (6.34) – the logarithm of a matrix is explained by taking the

1Note that formally we could have obtained this also by using (B.18) with A = i(2y +m2 − iε)δ4(x− y), B = −iJ,C = 0.




logarithm of each of the diagonal elements, i.e. the eigenvalues, after the

matrix has been diagonalized. This yields in the present case

trln A(x, y) =

d4xd4yd4k

(2π)4d4kδ4(x − y)ei(kx−ky) ln(−k2 + m2)δ4(k − k)

=

d4xd4k

(2π)4ln(−k2 + m2) (6.36)

Now we use that the inverse operator appearing in (6.33) is just theFeynman propagator. This can be seen by deriving the equation of motionfor the inverse operator

δ4(x − y) =

A(x, z)A−1(z, y) d4z

=

i2z + m2 − iε

δ4(x − z)

A−1(z, y) d4z

= i

δ4(x − z)2z + m2 − iε

A−1(z, y) d4z

= i2x + m2 − iε

A−1(x, y) . (6.37)

This is just the defining equation for −DF (6.7), so that we have

A−1 = iDF

. (6.38)

The propagator can thus be obtained as the inverse of the operator betweenthe two fields in the Lagrangian for the free field (4.31).

For the normalized generating functional we thus obtain

Z 0[J ] =W [J ]

W [0]= e−

i2

J (x)DF(x−y)J (y) d4xd4y . (6.39)

Equation (6.39) is the result derived earlier in section 5.1 (cf. (6.1.1)). Thepropagator that appears here is just given by the inverse of the Klein-Gordon

OperatorDF (x − y) = −

2+ m2

−1δ(x − y) , (6.40)

i.e. of that operator that appears between the two fields in the Lagrangianfor a free Klein-Gordon field

L = −1

2φ(2+ m2)φ . (6.41)




6.2 Stationary Phase Approximation

If the path integral in question is not that over a Gaussian, it can be approxi-mately brought into a Gaussian form by using the so-called stationary phaseor saddle point method. In this method one first looks for the stationarypoint of the exponent in the path integral. As explained earlier this will givea major contribution to the path integral. The remaining contributions areapproximated by expanding the exponent around the stationary point.

We illustrate this method here for the case of a scalar field with selfinter-actions. The Lagrangian is given by

L = −1

2φ 2+ m2

φ − V (φ) (6.42)

and the actionS [φ, J ] =

d4x (L + Jφ) (6.43)

is a functional of the field φ and the source J . We next determine thestationary point by looking for the zero of the functional derivative

δS [φ, J ]

δφ(x)

φ0

= −2+ m2

φ0(x) − V (φ0(x)) + J (x)

!= 0 ; (6.44)

this is the classical equation of motion corresponding to the action S [φ, J ].

The stationary field is just the classical field; the corresponding classicalaction is

S [φ0, J ] = −

d4x

1

2φ0

2+ m2

φ0 + V (φ0) − Jφ0

. (6.45)

We now expand S [φ, J ] around this stationary field (cf. (B.35),(B.37))

S [φ, J ] = S [φ0, J ] (6.46)

+1

2

d4x1 d4x2

δ2S

δφ(x1)δφ(x2)

φ0

[φ(x1) − φ0(x1)] [φ(x2) − φ0(x2)] + · · · .

The second functional derivative appearing here can be obtained by varyingthe first derivative (6.44). We thus get from the definition (B.26)

δ2S

δφ(x2)δφ(x1)

φ0

= − δ

δφ(x2)

(2+ m2)φ + V (φ) − J

1

φ0

(6.47)

Using now (B.27) we get

δ2S

δφ(x2)δφ(x1)

φ0

= −2+ m2 + V (φ0)

1

δ4(x2 − x1) (6.48)




which is an operator. The index 1 here means that the corresponding ex-

pressions are to be taken at the point x1.The action (6.45) is calculated at the fixed classical field φ0. It can,

therefore, be taken out of the path integral so that we finally obtain

W [J ] =

Dφ eih

d4x (L+Jφ) (6.49)

= eihS [φ0,hJ ]

Dφ exp

− i

2h

d4x1 d4x2

×

[φ(x1) − φ0(x1)]2+ m2 + V (φ0)

1

δ4(x2 − x1)

[φ(x2) − φ0(x2)]

+ . . . . (6.50)

In order to facilitate the following discussion we have put the unit of action,h, explicitly into this expression by setting i → i/h and J → hJ .

The path integral remaining here is now in a Gaussian form. It can beevaluated after a Wick rotation, just as in the developments leading to (6.33).After a “coordinate transformation” φ → φ = φ − φ0 and after scaling thefields by φ → √

hφ we get for the generating functional

W [J ] = eihS [φ0,hJ ]

det

i2+ m2 + V (φ0)

δ4(x2 − x1)

− 12 . (6.51)

We now perform a normalization with respect to the free case (6.33), i.e. to

W 0[0] =

deti2+ m2

δ4(x2 − x1)

− 12

= A(x1, x2)− 12 (6.52)

with A from (6.31); the index 0 on W denotes V = 0. This gives for thenormalized generating functional

W [J ] =W [J ]

W 0[0]= e

ihS [φ0,hJ ] (6.53)

×det d4z A−1(x2, z) (A(z, x1) + iV (φ0(x1))) δ4(z−

x1)− 1

2

.

With A−1 = iDF (6.38) we get

W [J ] = eihS [φ0,hJ ]

det

δ4(x1 − x2) − DF(x2 − x1)V (φ0(x1))

− 12 (6.54)

Our aim is now to write the inverse root of the determinant as a correctionterm to the classical action. For this purpose we use (6.34)

det[. . .]− 12 = e

−1

2trln[. . .]

. (6.55)




The matrix is given by

x1|1 − DFV (φ0)|x2 =δ4(x1 − x2) − DF(x2 − x1)V (φ0(x1))

. (6.56)

The trace of its logarithm is then given by

trln[. . .] =

d4x ln[1 − DF(0)V (φ0(x))] . (6.57)

We can now writeW [J ] = e

ihS [φ0,J ] (6.58)

with

S [φ0, J ] = − d4x

12

φ0

2+ m2

φ0 + V (φ0)

+ h

d4x Jφ0

+i

2h

d4x ln[1 − DF(0)V (φ0(x))] + O(h2) . (6.59)

The first line is just the classical action S [φ0, J ]. The two terms can besummed with a resulting action

S [φ0, J ] = −

d4x

1

2φ0

2+ m2

φ0 + V eff (φ0) + hJφ0

(6.60)

with the effective potential

V eff (φ(x)) = V (φ(x)) − i

2h ln[1 − DF(0)V (φ0(x))] . (6.61)

Expression (6.59) shows that the saddle point approximation amounts toan expansion of the action in powers of h. This is in accordance with thediscussion in Sect. 1.4 that quantum mechanics describes the fluctuationsof the action around the classical path. The potential V eff incorporates theeffects of these fluctuations into a classical potential.

Eq. (6.59) suggests a perturbative treatment through an expansion of thelogarithm (ln(1 + x) = x

−x2/2 + x3/3

−. . .) in terms of DFV , i.e. the

strength of the potential. The trace corresponds to an integration over xsuch that the initial and final space-time points in the individual terms inthe expansion are identical, i.e. to a closed loop integration. With higherorders in the expansion of the logarithm more and more vertices appear,but they are always located on this one closed loop. This loop expansionapproximates the quantum mechanical behavior, whereas the perturbationtreatment takes interactions into account.

If we take φ0 as a constant field, i.e. if we neglect its space-time de-pendence through the d’Alembert operator in (6.44), then the operator 1 +




DFV (φ0) becomes local in momentum space. In this case we can evaluate

the trace of its logarithm by integrating over the eigenvalues

x1| ln[1 − DFV (φ0)] |=

d4k

(2π)4e−ik(x2−x1) ln

1 − 1

k2 − m2 + iεV (φ0)

. (6.62)

This equation writes the original matrix in x-space as a result of a unitarytransformation of a diagonal matrix in k-space. The logarithm is taken of this diagonalized matrix which is then transformed back to x-space.

6.3 Numerical Evaluation of Path Integrals

An alternative method for the evaluation of path integrals is that of directnumerical computation; with rapidly increasing computer power this methodbecomes more and more important nowadays.

6.3.1 Imaginary time method

The generating functional is in general given by

W [J ] = 0|e−i(H +J )(tf −ti)

|0 (6.63)

for ti → −∞ and tf → +∞, as we have seen in chapter 3. After a Wickrotation this becomes

W [J ] = limβ →∞

0|e−β (H +J )|0 , (6.64)

where β denotes the real Euclidean time.It is immediately obvious that (6.64) also equals the groundstate expec-

tation value of the statistical operator of quantum statistics if we identify β with the inverse temperature, i.e. β = 1/T .

Inserting the explicit definition (5.9) and performing the Wick rotationgives

W [J ] = limn→0

n + 1

2πhβ

n+12

k

n j=1

dxkj e−

βn+1

nl=0

(H l+J kxkl)

, (6.65)

with

H l = H l (xl, xl) =1

2

k

x2kl + ω2

kx2kl

+ V (xkl) , (6.66)




so that W [J ] can be written as

W [J ] = limn→0

n + 1

2πhβ

k

j

dxkj ρ(xkj)e−β

n+1

lV (xkl) (6.67)

withρ(x) = e−

βn+1

l[12 (x2

kl+ω2

kx2kl

)+J kxkl] . (6.68)

The function ρ(x) is of Gaussian shape and can, therefore, analytically benormalized into

P (x) =ρ(x)

dx ρ(x)

. (6.69)

The multiple integrals appearing here can be evaluated by a Monte-Carlotechnique which samples the integrand at a large number of points, whereeach ‘point’ really corresponds to a full path x(t). Given a certain pointx = x1, . . . , xn one randomly chooses a new point x = x1, . . . , xn, often by just changing one single coordinate. One then evaluates

r =P (x)

P (x). (6.70)

If r is larger than 1, the new point is accepted. If r < 1, on the other hand,then a random number ρ between 0 and 1 is picked. If ρ < r, then the new

point is also accepted, otherwise it is rejected. This method is repeated untila large enough number of points is sampled. In this way the most importantregions in x space are sampled, thus generating finally M accepted pointsxm. The integral is then approximated by

W [J ] =1

M

M m=1

e−β

n+1

n

l=0V ((xkl)m) . (6.71)

The sampling algorithm just described is known after its inventor as theMetropolis algorithm ; it plays an important role in numerical evaluations of statistical physics expressions.

In the present case of evaluating the generating functional for gs to gstransitions one has to choose a fixed value of β , i.e. the Euclidean time.The calculations then have to be performed for several values of β with asubsequent extrapolation to β → ∞ (or T → 0).

6.3.2 Real time formalism

The major difficulty in evaluating a path integral numerically stems from theoscillatory character of the integrand. The discretized form of W [J ] (5.9) can




– in an obvious abbreviation – be written as the limit of a multidimensional

integral

W =

dx eiS (x) , (6.72)

where x is a n-dimensional vector. Importance sampling such as the one just discussed in the last section cannot directly be used because there is nopositive probability weight function in the integrand.

A weight function can, however, be inserted by a mathematical trick.Using (B.18) in the form

dx0

det(A/2π)e−12

(x−x0)T A(x−x0) = 1 (6.73)

we can write

W =

dxdx0

det(A/2π)e−

12 (x−x0)T A(x−x0)eiS (x) . (6.74)

The Gaussian factor under the integral ensures that values of x close to x0

will contribute the most to the integral. The function S (x) can, therefore beexpanded around x0

S (x) = S (x0) + S1(x)(x − x0) +1

2(x − x0)T S2(x0)(x − x0) + . . . (6.75)

with

S1(x0) =dS

dx

x0

and S2(x0) =d2S

dx2

x0

. (6.76)

After inserting this expansion we can perform the x-integration and obtainfrom (B.18)

W =

dx0 eiS (x0)

det(A)

det[A − iS2(x0)]

12

e−12ST 1 (x0)[A−iS2(x0)]−1S1(x0) . (6.77)

If we now use that (6.73) is still approximately valid even if A is a function

of x0, we can choose

A = A(x0) = iS2(x0) + c−11 (6.78)

with c > 0 and obtain

W =

dx0 eiS (x0)det(1 − iS2(x0)A−1)

− 12 e−

c2S

T 1 (x0)S1(x0) . (6.79)

The functionP (x0) = e−

c2S

T 1 (x0)S1(x0) (6.80)




thus provides a probability distribution for sampling the remaining integrand

and the expression can be evaluated with the Monte-Carlo method discussedin the last section. We obtain, therefore,

W =1

M

M i=1

eiS (xi) [det(1 + icS2(xi)]−12 , (6.81)

where the M points xi are taken randomly from the distribution P (x).For large values of c the points with S 1 ≈ 0 will contribute the most

to W . This is just the stationarity condition discussed in section 6.2 whichcorresponds to the classical solution. Since only a few points close to this

configuration will contribute significantly, a good sampling with good statis-tics can be obtained. On the other hand, for small c the probability dis-tribution becomes broad and the statistics correspondingly worse; however,quantum mechanical effects beyond the one-loop approximation now start tocontribute. Thus, by choosing the proper value of c quantum mechanics canbe switched on.



Chapter 7

S -MATRIX AND GREEN’S

FUNCTIONS

The ultimate aim of all our fieldtheoretical developments in this book isto calculate reaction and transition rates for processes involving elementaryparticles. In this chapter, we therefore, now derive a connection betweenthese transition rates and expectation values of field operators, the so-calledreduction theorem.

7.1 Scattering Matrix

The typical initial state of a scattering experiment is that of widely separatedon-shell particles at t → −∞. On-shell here means that these particlesfulfill the free energy-momentum dispersion relation. The groundstate of thesystem is the state of lowest-energy, i.e. the state with no particles present.At large times t → +∞ the final state is again that of free, non-interactingon-shell particles. The vacuum state of the theory is unique and is thereforethe same as the initial vacuum state.

The transition rate of any quantum process, be it a scattering process

m + n → m

+ n

is determined by the so-called S -matrix. We define the S-matrix as the probability amplitude for a process that leads from an ingoingstate |α, in to an outgoing state |β , out . The particles are assumed tomove freely in these asymptotic states outside the range of the interaction;both of these states can therefore be characterized by giving the momenta of all participating particles and possibly other quantum numbers as well all of which are denoted by α and β . The S -matrix is thus given by

S βα = β, out|α, in . (7.1)

78





CHAPTER 7. S -MATRIX AND GREEN’S FUNCTIONS 80

This form corresponds to (7.7).

As is obvious from its definition the S -matrix determines all the transitionrates possible within the field theory. For example, a cross-section for a1 + 1 → 1+1 collision is simply given by the absolute square of S , multipliedwith the available phase-space of the outgoing particles and normalized tothe incoming current. The Reduction Theorem to be derived in the followingsection provides a link between the S -matrix and expectation values of time-ordered products of field operators that can be calculated as derivatives of generating functionals.

7.2 Reduction TheoremSince the asymptotic states appearing in the S matrix are those of free, on-shell particles we can describe them as non-interacting quantum excitationsof the vacuum of the theory with free dispersion relations. We, therefore, firstreview the basic properties of free field creation and annihilation operatorsin the next subsection.

7.2.1 Canonical field quantization

The field operators are obtained by quantizing the free asymptotic fields φin

and φout. This is done in the usual way by imposing commutator relationsfor the fields and their momenta. We impose the canonical commutatorrelations of quantum mechanics for coordinates and corresponding canonicalmomenta. Remembering that in field theory the fields play the role of theclassical coordinates we thus impose1

[Π(x, t), φ(x, t)] = − iδ3(x − x)Π(x, t), φ(x, t)

= 0

[φ(x, t), φ(x, t)] = 0 . (7.11)

We now employ the normal mode expansion of the fields (see the discus-sion at the start of section 5.1) and momenta

φ(x, t) =1√V

k

1√2ωk

a k(t) ei k·x + a† k(t) e−i k·x

Π(x, t) = − i√V

k

ωk√2ωk

a k(t) ei k· x − a† k(t) e−i k·x

(7.12)

1From here on all the fields φ in this section are operators; only for ease of notation wedo not write the ’operator-hats’ explicitly




with ωk = k2 + m2. Since the fields φ and the momenta Π are now opera-tors, the ’expansion coefficients’ a and a† are operators as well.

The inverse Fourier transformation is given by

a† k(t) =1√

2ωkV

ei k·x (ωkφ(x, t) + iΠ(x, t)) d3x

a† k(t) =1√

2ωkV

ei k·x (ωkφ(x, t) − iΠ(x, t)) d3x (7.13)

Using the commutator relations (7.11) we obtain also the commutatorrelations for the operators a k and a† k

a k(t), a† k(t) = δ k, ka k(t), a k(t)

=

a† k(t), a† k(t)

= 0 . (7.14)

The asymptotic in and out fields are free fields so that the time-dependenceof their operators a and a† is harmonic

a k(t) = a k(0)e−iωkt

a† k(t) = a† k(0)e+iωkt (7.15)

as can be obtained from the Heisenberg equation of motion. The asymptotic

in fields φin are then given by

φin(x, t) =1√V

k

1√2ωk

a k(0)e−ikx + a† k(0)e+ikx

; (7.16)

with ikx = i(ωkt − k · x). The fields φout can be represented in the same way.Remembering that for free fields Π = Φ, the free field annihilation and

creation operators for the in and out states are given in terms of the fieldsand momenta as

a† k(0) =−i√

2ωk

V V d3x e−ikx (∂ tφ(x, t) + iωkφ(x, t))

a k(0) =i√

2ωkV

V

d3x e+ikx (∂ tφ(x, t) − iωkφ(x, t)) (7.17)

with kx = ωkt − k · x.The operators (7.17) are the same as the ones used in the well known

algebraic treatment of the harmonic oscillators for the normal field modes.The a† are the creation and the a the annihilation operators for free fieldquanta and the vacuum (groundstate) of the free field theory is given bya|0 = 0.




7.2.2 Derivation of the reduction theorem

In a realistic physics situation the scattering or decay processes that weaim to describe involve interactions between the particles. Simulating thesituation in a scattering experiment we, therefore, now assume that the in-teractions between the particles are adiabatically being switched on and off;adiabatically here means without energy transfer. The asymptotic “in” and“out” states are thus free states which can be described by the free fieldoperators (7.17). The operators a†(t) and a(t) in (7.13) have a harmonictime-dependence (7.15) only for times t → ±∞. The free field operators(7.17) acting on the vacuum of the full, interacting theory create and anni-hilate field quanta only at times t

→ ±∞while they do so at all times when

acting on the vacuum state of the noninteracting theory. They can thus beused to describe the asymptotic states.

We can thus write for the S matrix element (7.1)

S βα = β, out|α, in = β, out|a†in(k)|α − k, in , (7.18)

where we have assumed that the in-state |α, in contained a free particle withthree-momentum k ; |α−k, in is then that in state in which just this particleis missing. We can further write this as

β, out|α, in = β, out|a†

out(k)|α − k, in (7.19)+ β, out|a†in(k) − a†out(k)|α − k, in .

In the first term β, out|a†out(k) = β − k, out| (= 0, if β, out| does notcontain a particle with momentum k).

We now rewrite the in and out creation operator into a more compactform

a†in,out(k) = −i

d3x f k(x)↔

∂ t φin,out(x, t) (7.20)

with

f (t)↔

∂ t φ(x, t) = f ∂φ

∂t −∂f

∂t

φ (7.21)

and

f k(x) =1√

2ωkV e−ikx.

With this expression we can get the S -matrix element into the form

β, out |α, in = β − k, out|α − k, in (7.22)

− iβ, out|

d3x f k(x)↔

∂ t [φin(x) − φout(x)] |α − k, in .




For only 2 particles in the in and out states the first term on the rhs represents

a single particle transition matrixelement. In this case it can obviously onlycontribute if both particles do not change their energy and momentum, i.e. if |out and |in are identical. It is then just the forward scattering amplitude.

The rhs of (7.22) is time-independent. We can see this by calculating thetime derivative of its integrand

∂ t

f (x)

↔

∂ t φin(x)

= f ∂ 2t φin −

∂ 2t f

φin . (7.23)

Here f (x) = e−ikx and φin both solve the free Klein-Gordon equation withthe same mass. We therefore have

∂ tf (x) ↔∂ t φin(x) = f 2 φin − ( 2 f )φin . (7.24)

The integral over this expression vanishes after twofold partial integration of one of the terms. The same holds, of course, for the term involving φout in(7.22).

The rhs of (7.22) is thus indeed time-independent. We can, therefore,take it at any time and in particular also at t → ±∞. Then we can replacethe in and out fields at these times by the limits of the field φ(x). This givesfor the S -matrix element

β, out

|α, in

=

β

−k, out

|α

−k, in

+ lim

t→+∞iβ, out|

d3x f k(x)

↔

∂ t φ(x)|α − k, in

− limt→−∞

iβ, out|

d3x f k(x)↔

∂ t φ(x)|α − k, in . (7.25)

We now write this expression in a covariant form by usinglimt→+∞

− limt→−∞

d3x

f (x)

↔

∂ t φ

=

+∞ −∞

dt

d3x∂

∂t

f (x)

↔

∂ t φ

=

d4xf ∂ 2t φ −

∂ 2t f

φ

=

d4xf ∂ 2t −

( 2 − m2)f

φ . (7.26)

Note that here φ is an interacting field, since we integrate now over all times.Thus, in contrast to (7.23) φ does not solve the free Klein-Gordon equationand, consequently, this integral does not vanish. Twofold partial integrationin the second term on the rhs allows us now to roll the Laplace operator fromf over to φ. This gives

d4x

f ∂ 2t φ − (∂ 2t f )φ

=

d4x f (x)(2+ m2) φ(x) . (7.27)




We thus have

β, out|α, in = β − k, out|α − k, in (7.28)

+ i

d4x f k(x)(2+ m2)β, out|φ(x)|α − k, in .

In this expression we have removed one particle from the in state.We now continue by removing one particle with the momentum k from

the out state by going through exactly the same steps as before. Disregardingthe first term in (7.28), that contributes only to forward scattering, we get

β, out

|φ(x)

|α

−k, in

=

β

−k, out

|aout(k)φ(x)

|α

−k, in

= β − k, out|φ(x)ain(k)|α − k, in (7.29)

+ β − k, out|aout(k)φ(x) − φ(x)ain(k)|α − k, in .

We next replace the annihilation operators by the corresponding field oper-ators as in (7.20) (here we have to take the hermitian conjugate operator)and obtain

β, out|φ(x)|α − k, in = β − k, out|φ(x)|α − k − k, in (7.30)

+i

d3xβ − k, out|

f ∗k(x)↔

∂ t [φout(x)φ(x) − φ(x)φin(x)]|α − k, in .

Taking now the limits t → ±∞ gives, as above, for this expression

β − k, out|φ(x)|α − k − k, in+ iβ − k, out|

d4x

∂

∂t

f ∗k(x)

↔

∂ t T [φ(x)φ(x)]

|α − k, in

= . . . + iβ − k, out|

d4x

f ∗k(x)∂ 2t T [φ(x)φ(x)]

−

∂ 2t f ∗k(x)

T [φ(x)φ(x)]|α − k, in . (7.31)

We now use again (7.27) and obtain

= · · · + i β − k, out|

d4x f ∗k(x)(2 + m2)T [φ(x)φ(x)] |α − k, in .

Combining this result with (7.28) finally gives (neglecting the forward scat-tering amplitude)

β, out|α, in = i2

d4x d4x f ∗k(x)f k(x) (7.32)

×(2 + m2)(2+ m2)β − k, out|T [φ(x)φ(x)] |α − k, in .




This reduction can obviously be continued on both sides until we have

(with n particles with momenta k in the out state and m particles withmomenta k in the in state)

S βα = βn k, out|αmk, in (7.33)

= im+n mi=1

d4xi

n j=1

d4x j f ∗kj (x j)f ki(xi) ×

(2 j + m2)(2i + m2)0|T [φ(x1)φ(x2) . . . φ(xn)φ(x1)φ(x2) . . . φ(xm)] |0 .

This is the so-called Reduction Theorem that enables us to express the S -matrix in terms of the (n + m)-point Green’s function , somestimes also called

correlation function

G(x1, x2, . . . , xn, x1, x2, . . . , xm) (7.34)

= 0|T [φ(x1)φ(x2) . . . φ(xn)φ(x1)φ(x2) . . . φ(xm)] |0 .

Note that in the reduction theorem (7.33) the information about the in-teraction of the particles is contained in the (interacting) field operators φ.Also the vacuum appearing here is that of the full, interacting theory. Then+m-point function appearing there is, therefore, also that of the interactingtheory!

The physical process described by the reduction theorem is that of m on-shell particles in the initial state at the asymptotic space-time coordinatesx1, . . . , xm and n on-shell particles at the space-time coordinates x1, . . . , xnwith an interaction region in between these sets of coordinates. As we will seelater, the Klein-Gordon operators 2+ m2 when acting on G just remove thepropagators from the interaction region out to the asymptotic points, creatingso-called vertex functions Γ(x1, . . . , xn, x1, . . . , xm). The reduction theorem(7.33) then gives the transition rate S βα as the Fourier transform of thisvertex function. The Fourier transform in (7.33) contains factors of the formexp(ikx) for the outgoing particles and exp(−ikx) for the incoming ones.This could be symmetrized by changing all outgoing momenta k

→ −k.

This gives

S βα = βn − k, out|αmk, in= im+n

mi=1

d4xi

n j=1

d4x j1

2ωkjV

1 2ωkiV

e−i(kjxj+kixi)

×Γ(x1, x2, . . . , xn, x1, x2, . . . , xm) . (7.35)

The connection between the S -matrix and the Green’s function as ex-pressed by the reduction theorem has been derived here only for scalar fields,




but it is valid in general. The only formal difference is that the Klein-Gordon

operator 2+ m2 has to be replaced by the corresponding free-field operator.It is also important to note that the method of canonical quantization

used here to derive the reduction theorem has been used only for the asymp-totic states. Thus, even for fields where this method runs into difficultieswhen interactions are present, like, e.g., the gauge fields to be treated inchapter 13, the reduction theorem holds in the form given above.

In the remainder of this book we will be concerned with calculating thecorrelation functions by using path integral methods. Once these correlationfunctions are known the reduction theorem allows us to calculate any reactionrate or decay probability.



Chapter 8

GREEN’S FUNCTIONS

In chapter 7 we have found that all the S -matrix elements can be calculatedonce the correlation, or Green’s, functions are known. In this chapter wediscuss how these functions can be obtained as functional derivatives of thegenerating functionals of the theory.

8.1 n-point Green’s Functions

In chapter 7 we have seen that the correlation function, i.e. the vacuum

expectation value of time-ordered field operators,

G(x1, x2, . . . , xn) = 0|T φ(x1)φ(x2) . . . φ(xn)

|0 . (8.1)

determines the transition rate for all physical processes. Remembering thatin field theory the field operators play the role of the coordinates in classicalquantum theory we can now directly use the results obtained in Chapt. 3and Sect. 5.1.1 and write using (3.47)

G(x1, x2, . . . , xn) = 0|T φ(x1)φ(x2) . . . φ(xn)

|0

=

Dφ φ(x1)φ(x2) . . . φ(xn)eiS [φ] Dφ eiS [φ]

(8.2)

with S [φ] =+∞ −∞

L(φ, ∂ µφ)d4x. The vacuum here is that of the full, interacting

Hamiltonian.The latter expression can also be obtained as a functional derivative of

the generating functional of the theory (cf. (3.47)) so that we can also equiv-

87



CHAPTER 8. GREEN’S FUNCTIONS 88

alently define the n-point Green’s function by

G(x1, x2, . . . , xn) =

1

i

n δnZ [J ]

δJ (x1)δJ (x2) . . . δ J (xn)

J =0

. (8.3)

We note that G(x1, . . . , xn) is a symmetric function of its arguments.Therefore, according to (B.37) the following relation holds

Z [J ] =n

1

n!

dx1 . . . dxn inG(x1, x2, . . . , xn)J (x1)J (x2) . . . J (xn) . (8.4)

Connected Green’s Functions. Guided by (3.22) we define a functionalS [J ] by the relation

Z [J ] = eiS [J ] (8.5)

and introduce the so-called connected Green’s functions Gc in terms of S [J ]defined by the relation

Gc(x1, . . . , xn) =

1

i

n−1 δnS δJ (x1) . . . δ J (xn)

J =0

. (8.6)

The name of this correlation function and its physics content will becomeclear later in this section.

8.1.1 Momentum representation

Very often it is advantageous to work in momentum space because the exter-nal lines of Feynman graphs represent free particles with good momentum.In general the transformation of the Green’s function into the momentum-representation is given by

e−i( p1x1+ p2x2+...+ pnxn)G(x1, x2, . . . , xn) d4x1 d4x2 . . . d4xn

= (2π)4δ4( p1

+ p2

+ . . . + pn

) G( p1, p

2, . . . , p

n) (8.7)

The δ-function here reflects the momentum conservation due to translationalinvariance. This can be seen by performing pairwise transformations of twospace-time points to their cm. point and their relative coordinate. If wethen assume that G depends only on the latter, the integral over the c.m.coordinate can be performed and yields the δ-function. As in our discussionaround (7.35) we take all the momenta as pointing into the vertex (see Fig.8.1).




p3

p2

p1

p7

p6

p5

p4

Figure 8.1: Momentum representation of the n-point function. Note that allmomenta are pointing into the shaded interaction region.

8.1.2 Operator RepresentationsOperator representation of the generating functional. For complete-ness, we now derive an alternative expression for the generating functionalZ [J ]. We start by defining the operator functional

Z [J ] = T ei J (x)φ(x) d4x (8.8)

where φ is an operator! If we form the functional derivatives of Z [J ] we get,in analogy to (3.43),

1in δ

n

Z δJ (x1) . . . δ J (xn)

= T φ(x1) . . . φ(xn)Z [J ]

, (8.9)

so that, because of Z [0] = 1,

1

i

n δn0|Z [J ]|0δJ (x1) . . . δ J (xn)

J =0

= 0|T φ(x1) . . . φ(xn)

|0

=

1

i

n δnZ

δJ (x1) . . . δ J (xn)

J =0

. (8.10)

Thus all the functional derivatives of 0|Z [J ]|0 agree with those of Z [J ] atJ = 0. According to (8.3) and (8.4) the two expressions therefore have to beequal

Z [J ] = 0|Z [J ]|0 . (8.11)

Functional form of the scattering operator. After having seen that theGreen’s functions can be obtained as functional derivatives of a generatingfunctional in this section we show that the scattering operator S can also be




written in a functional form as

S = : ei φin(x)(2+m2) δδJ (x)d4x : Z [J ]|J =0 (8.12)

=∞k=0

ik

k!

d4x1 . . . d4xk : φin(x1) . . . φin(xk) :

δk

δJ (x1) . . . δ J (xk)Z [J ] |J =0 .

Here the : : symbol denotes the so-called normal ordered product of fieldoperators. This normal-ordered product is defined in such a way that alloperators in it are reordered so that all the annihilation operators are movedto the right. This reordering takes place without a sign change for bosonfields and with a sign-change for each pairwise exchange for fermion fields.The operator (2+ m2) δ

δJ (x)in (8.12) acts only on Z [J ].

The matrix elements of the operator (8.12) indeed agree with (7.33). Thiscan be seen by considering again a matrixelement with n particles in the out states and m particles in the in state. In the expansion of the exponential in(8.12) only that term can contribute that contains exactly m + n powers of the fields. We thus have

n, out|S |m, in = im+n mi=1

d4xi

m+n j=m+1

d4x j (8.13)

× 1

(m + n)!n| : φin(x1) . . . φin(xm+n) : |m

× (2

1 + m

2

)(2

2 + m

2

) . . . (2m+n + m

2

) i

m+n

G(x1, x2, . . . , xm+n) .Here the in field operator φin(x) can simply be replaced by operators of thefree field (7.8). The normal product reorders the expansion (8.13) such thatall the annihilation operators are on the right. Since each field contains twoindependent sums over positive and negative energy eigenstates, respectively,we have in total 2m+n operator products; of these only the term with ncreation operators and m annihilation operators can contribute. This giveswith the expansion (7.12)

n| : φin(x1) . . . φin(xm+n) : |m

=

(m + n)!

m! n! n|nk=1

f ∗

kk(xk)a†

k

n+ml=n+1

f kl(xl)al|m

=(m + n)!

m! n!

nk=1

f ∗kk(xk)n+ml=n+1

f kl(xl) . (8.14)

The degeneracy factor in front of the matrix element follows from the bi-nomial expansion of the individual terms in the normal mode expansion(f kak + f ∗ka†k)m+n. The integration in (8.13) over the xi and x j just gives anextra degeneracy factor m!n!. Taking this result together with (8.13) is thesame as (7.33).




8.2 Free Scalar Fields

We consider first the case of free fields. In this case the generating functionalcan be given analytically (6.11)

Z 0[J ] = e−i2

J (x)DF(x−y)J (y) d4xd4y . (8.15)

The first functional derivative vanishes at J = 0 because the integral (8.15)is Gaussian. For the second functional derivative we obtain

δ2Z 0[J ]

δJ (x1)J (x2)= −iDF(x1 − x2) e−

i2

J (x)DF(x−y)J (y) d4xd4y (8.16)

+ (−i)2 d4x d4y DF(x1 − x)DF(x2 − y)J (x)J (y) e−i

2 J (x)DF(x−y)J (y) d4

xd4

y ,

so that we have

G(x1, x2) = − δ2Z 0[J ]

δJ (x1)δJ (x2)

J =0

= iDF(x1 − x2) . (8.17)

The two-point function is thus just the Feynman propagator. It is there-fore also a solution of (cf. (6.7))

2+

m2 − iε

G(x1, x2) = −iδ4(x) (8.18)

8.2.1 Wick’s theorem

The higher order derivatives can be most easily obtained by expanding (8.15)

Z 0[J ] =∞n=0

1

n!

− i

2

J (x)DF(x − y)J (y) dx dy

n(8.19)

= 1 +∞n=1

1

n!

− i

2

n dx1 . . . dx2nD12D34 . . . D2n−1 2n J 1J 2 . . . J 2n

with the shorthand notation Dij = DF(xi − x j), J k = J (xk). Noting that Z always contains even powers of J , it is immediately evident that all n-pointfunctions with odd n vanish because an odd functional derivative of an evenfunction always vanishes at J = 0.

Taking now the 2k-th functional derivative of Z 0 and using (8.3) and(B.36) we obtain

G(x1, x2, . . . , x2k) =

1

i

2k δ2kZ 0δJ 1 . . . δ J 2k

|J =0

=(i)k

2kk!

P

D p1 p2 . . . D p2k−1 p2k (8.20)




where the sum runs over all permutations ( p1, p2, . . . , p2k) of the numbers

(1, 2, . . . , 2k). The factor in front of the sum removes the doublecountingbecause of the symmetry D p2 p1 = D p1 p2 (2k) and because of the randomorder of factors under the sum (k!).

Equation (8.20) states that the n-point function of a system of free bosonscan be written as a properly normalized and symmetrized product of two-point functions. This is the so-called Wick’s theorem .

As an example we consider the case n = 4. We then have

G(x1, x2, x3, x4) = −1

8

P ∈S 4

D p1 p2D p3 p4 . (8.21)

Among the 24 terms in the sum, 12 are pairwise equal because the productof the two propagators commutes. Furthermore, the propagator D p1 p2 andD p2 p1 are pairwise equal. Thus, there are only 24 : 2 : 2 : 2 = 3 essentiallydistinct terms in the sum; the factor 1/8 just takes care of all the others. Wethus have

G(x1, x2, x3, x4) = − DF(x1 − x2)DF(x3 − x4) (8.22)

− DF(x1 − x3)DF(x2 − x4)

− DF(x1 − x4)DF(x2 − x3) .

The first few n-point Green’s functions are therefore – according to (8.17),

(8.20) and (8.22) – given by

G(x1) = 0 (8.23)

G(x1, x2) = 0|T [φ(x1)φ(x2)] |0 = iDF(x1 − x2)

G(x1, x2, x3) = 0

G(x1, x2, x3, x4) = 0|T [φ(x1)φ(x2)φ(x3)φ(x4)] |0= − DF(x1 − x2)DF(x3 − x4)

− DF(x1 − x3)DF(x2 − x4)

− DF(x1 − x4)DF(x2 − x3)

etc. (8.24)

Here |0 is the vacuum state of the free Hamiltonian because it was obtainedas a functional derivative of the non-interacting functional Z 0 (8.19). Alln-point functions with odd n vanish.

8.2.2 Feynman rules

As in the classical case (cf. Sect. 2.3) we can again represent these results ina graphical way. The Feynman rules, that establish the connection between




the algebraic and the graphical representation, are for the case of free fields

still rather trivial.They are given by

1) each Feynman propagator is represented by a line:

x y= iDF (x − y) .

2) each source is represented by a cross:x

= iJ (x) .

3) There is an integration over all space-time coordinates of the currents

4) Each diagram has a factor that takes its symmetry into account.

For example, if there is an integration over the endpoints x and y, thesecould be exchanged without changing the result; the symmetry factoris, correspondingly, 1/2.

With rule 1) we get, for example, for the fourpoint function (8.22), i.e.the two-particle Green’s function

G(x1, x2, x3, x4) =1 2

3 4+

1 2

3 4

+

1 2

3 4

(8.25)

Each line connecting the two points x and y denotes the free propagator andgives a factor iDF(x − y).

We now setZ 0[J ] = eiS 0[J ] (8.26)

with

iS 0[J ] = − i

2

d4x d4y J (x)DF(x − y)J (y) (8.27)

By using rules 2), 3) and 4) we find immediately

iS 0 = (8.28)

Now we expand

Z 0[J ] = eiS 0[J ] = e

= 1 +− i

2

J (x)DF (x − y)J (y) d4x d4y

+1

2!

− i

2

J (x)DF (x − y)J (y) d4x d4y

2

+ · · ·




= 1 + iS

0[J ] +1

2!(iS

0[J ])2 +1

3!(iS

0[J ])3 + . . .

= 1 + +1

2!+

1

3!+ . . . . (8.29)

We now call all graphs that hang together connected graphs and all theothers unconnected graphs. In our simple case here S 0[J ] is representedby only one connected graph (8.28), whereas Z 0[J ] – through the powerexpansion (8.29) of the exponential function – generates unconnected graphsas well.

In the simple case of a free field discussed here there is only one connected

graph (see (8.25)). The connected Green’s function can be obtained from itsdefinition (8.6) as

Gc(x1, x2) =1

i

δ2S 0δJ (x1)δJ (x2)

|J =0 = iDF(x1 − x2) . (8.30)

All higher functional derivatives of S 0[J ] vanish. Gc is in this free case thus just given by the Feynman propagator.

8.3 Interacting Scalar Fields

In this section we consider Lagrangians of the form

L = L0 − V (φ) , (8.31)

where L0 is the free scalar Lagrangian (5.1) and V represents a selfinteractionof the field. For such a Lagrangian the generating functional for the n-pointfunctions can no longer be given in closed form. In order to obtain the n-pointfunction one has to resort to perturbative methods.

The n-point function then follows from its definition (8.2)

G(x1, x2, . . . , xn) = Dφ φ(x1)φ(x2) . . . φ(xn) eiS [φ]

Dφ eiS [φ] (8.32)

with the exponential in the generating functional of the form

eiS [φ] = ei

d4x (L0−V +i ε2φ2) . (8.33)

The action exponential can be Taylor-expanded in the interaction strength

eiS [φ] =∞N =0

1

N !

−i

d4x V

N eiS 0[φ] (8.34)




with the free action

S 0[φ] =

d4xL0 + i

ε

2φ2

. (8.35)

Inserting this into (8.32) gives the n-point function of the interacting theoryin terms of powers of the interaction and the free-field action

G(x1, x2, . . . , xn) =

Dφ φ(x1)φ(x2) . . . φ(xn) eiS [φ]

Dφ eiS [φ](8.36)

= Dφ φ(x1)φ(x2) . . . φ(xn)

∞N =0

1

N ! −i d4

x V N

eiS 0[φ]

Dφ

∞N =0

1

N !

−i

d4x V

N eiS 0[φ]

By using (8.2) and the developments in Sects. 3.2 and 5.1.1 we can rewritethis equation also in terms of normalized vacuum expectation values. The lastline of (8.36) involves the free action S 0 and thus free propagation. Therefore,if the field operators are those of free in fields at asymptotic times they remainso even during propagation. Also the vacuum appearing in the quantummechanical vacuum expectation value is then that of the non-interacting

theory so that we get

G(x1, x2, . . . , xn) = 0|T φ(x1)φ(x2) . . . φ(xn)

|0

=

0|T

φin(x1) . . . φin(xn)

∞N =0

1

N !

−i

V d4x

N |0

0|T

∞N =0

1

N !

−i

V d4x

N |0

. (8.37)

As in Sect. 5.1.1

|0

denotes here the vacuum state of the full, interacting

Hamiltonian, whereas |0 is that of the free Hamiltonian.With (8.37) we have achieved a remarkable result: (8.37) expresses the

expectation value of the time-ordered product of field operators in the vac-uum state of the full, interacting theory by a perturbative expansion over freefield expectation values. The latter can be calculated as path integrals overproducts of classical fields and powers of the interaction (8.36). This enablesus to calculate G, and ultimately also the scattering matrix S (Chapt. 7),perturbatively up to any desired order in the interaction.




8.3.1 Perturbative expansion

Eq. (8.32) allows us to calculate the perturbative expansion of the full Green’sfunction up to any desired order in V . Alternatively, these higher order termscan also be obtained as functional derivatives of the free generating functionalZ 0[J ] which is known.

According to our general considerations the functional for the Lagrangian(8.31) is given by

Z [J ] = Z 0

Dφ ei

d4x (L0−V (φ)+Jφ+i ε2φ2) (8.38)

=

Z 0 D

φ e−i

d4xV (φ)ei

d4x (L0+Jφ+i ε2φ

2) .

Here Z 0 is just the inverse of the path integral for J = 0

Z −10 =

Dφ ei

d4x (L0−V (φ)+i ε2φ

2) . (8.39)

We now use the relation

1

i

δ

δJ (y)ei

d4x (L0+Jφ+i ε2φ2) = φ(y) ei

d4x (L0+Jφ+i ε2φ

2) . (8.40)

This relation, read from right to left, will also be true for any function V (φ),as can be seen by expanding V into a series in powers of φ. We thus have

also

V [φ(y)] ei

d4x (L0+Jφ+i ε2φ2) = V

1

i

δ

δJ (y)

ei

d4x (L0+Jφ+i ε2φ2) (8.41)

and consequently, after exponentiation, also

e−i

d4y V [φ(y)] ei

d4x (L0+Jφ+i ε2φ2) (8.42)

= e−i

d4y V ( 1i

δδJ (y))ei

d4x(L0+Jφ+i ε

2φ2) .

This relation allows us to take the V -dependent factor in (8.38) out of thepath-integral

Z [J ] = Z 0e−i

d4y V ( 1i

δδJ (y))

Dφ ei

d4x (L0+Jφ+i ε

2φ2) . (8.43)

The last factor in (8.43) has been expressed in terms of the free two-particlepropagator introduced in the last section. We thus have

Z [J ] = Z 0e−i

d4z V ( 1i

δδJ (z))e−

i2

J (x)DF(x−y)J (y) d4xd4y

= Z 0e−i

d4xV ( 1i

δδJ (x))eiS 0[J ]

= Z 0e−i

d4xV ( 1i

δδJ (x))Z 0[J ] . (8.44)




Expanding the exponential that contains the interaction V then gives the

perturbative expansion for Z [J ]

Z [J ] = Z 0∞N =0

1

N !

−i

d4x V

1

i

δ

δJ (x)

N Z 0[J ] . (8.45)

Since we will later on be mostly interested in connected graphs we donot need Z[J] directly but instead its logarithm. We, therefore, now expandthe functional iS [J ] = ln Z [J ] in powers of the interaction V . We start byinserting a factor 1 = exp (+iS 0)exp(−iS 0) between Z 0 and the exponentialin (8.44) and taking the logarithm

ln Z [J ] = ln Z 0 + ln 1 · e−i d4xV ( 1i δδJ )eiS 0[J ]

= ln Z 0 + iS 0 + ln

e−iS 0e−i

d4xV eiS 0

= ln Z 0 + iS 0[J ] + ln

1 + e−iS 0[J ]

e−i

d4xV ( 1iδδJ ) − 1

eiS 0[J ]

= iS [J ] . (8.46)

A perturbation theoretical treatment is now based on a Taylor expansion of the logarithm. For that purpose we abbreviate

ε[J ] = e−iS 0[J ] e−i d4xV − 1 eiS 0[J ] (8.47)

and obtain

iS [J ] = ln Z [J ] = ln Z 0 + iS 0[J ] + ln(1 + ε[J ]) (8.48)

= ln Z 0 + iS 0[J ] +

ε[J ] − 1

2ε2[J ] + O(ε3)

.

Equation (8.48) represents an expansion of S in powers of the (for V → 0)small quantity ε. In order to obtain a perturbative expansion in terms of thepotential V we now rearrange the expansion (8.48). First we expand ε[J ] of (8.47) in terms of the strength of the interaction. This gives

ε[J ] = e−iS 0[J ]

−i

d4x V

1

i

δ

δJ

(8.49)

+1

2!

−i

d4x V

1

i

δ

δJ

2

+ · · · e+iS 0[J ] .

We now insert this expression into (8.48) and obtain

iS [J ] = ln Z 0 + iS 0[J ] +

ε[J ] − 1

2ε2[J ] + . . .




= ln Z 0 + iS 0[J ] + e

−iS 0[J ] −i d

4

x V 1

i

δ

δJ e

iS 0[J ]

+1

2!e−iS 0[J ]

−i

d4x V

1

i

δ

δJ

2

eiS 0[J ]

− 1

2

e−iS 0[J ]

−i

d4x V

1

i

δ

δJ

eiS 0[J ]

2

+ O(V 3)

= ln Z 0 + iS 0[J ] + iS 1[J ] + iS 2[J ] − 1

2(iS 1[J ])2 + O(V 3) ,

(8.50)

with

iS 0[J ] = − i

2

J (x)DF(x − y)J (y) d4x d4y

iS 1[J ] = e−iS 0[J ](−i)

d4x V

1

i

δ

δJ

e+iS 0[J ]

iS 2[J ] =1

2!e−iS 0[J ]

−i

d4x V

1

i

δ

δJ

2

e+iS 0[J ] . (8.51)

Equation (8.50) represents a perturbative expansion of S in powers of V .Note that the term of second order in the interaction V receives contributionsboth from the linear and the quadratic term in the original expansion (8.48)in ε. The term ∼ S 1 obviously just contains an iteration of the linear term.



Chapter 9

PERTURBATIVE φ4 THEORY

In this chapter we apply the formalism developed in the preceding chapterto the so-called φ4 theory whose Lagrangian is given by

L = L0 − V (φ) = L0 − g

4!φ4 . (9.1)

Here g is a coupling constant. This φ4 theory is a prototype of a field theorywith selfinteractions. It serves as a didactical example which exhibits allphenomena of more complex field theories.

9.1 Perturbative Expansion of the Generat-

ing Function

We start with the generating functional for connected Green’s functions(8.50)

iS [J ] = ln Z 0 + iS 0[J ] + iS 1[J ] + iS 2[J ] − 1

2(iS 1[J ])2 + O(V 3) . (9.2)

Inserting the interaction of φ

4

theory (9.1) into (8.51) we obtain

iS 0[J ] = − i

2

d4z d4y J (z)DF(z − y)J (y) ,

iS 1[J ] = − ig

4!e−iS 0[J ]

d4x

δ4

δJ 4(x)eiS 0[J ] (9.3)

and

iS 2[J ] =1

2!

−ig

4!

2

e−iS 0[J ]

d4x d4yδ4

δJ 4(x)

δ4

δJ 4(y)eiS 0[J ] .

99



CHAPTER 9. PERTURBATIVE φ4 THEORY 100

For notational convenience in the following we now introduce the

S i defined

by

iS 1[J ] = − ig

4!S 1[J ]

iS 2[J ] =−ig

4!

2

S 2[J ] . (9.4)

The generating function for the connected Green‘s functions in φ4 theoryreads then

iS [J ] = ln Z 0 + iS 0[J ] +−ig

4!S 1[J ] +

−ig

4!

2

S 2[J ] − 1

2

−ig

4!S 1[J ]

2

+ . . . .

(9.5)Since each of the S i[J ] contains functional derivatives of the known S 0[J ] wecan now evaluate the functional derivatives of S [J ] and obtain all the Green’sfunctions.

First, we calculate the terms linear in g

S 1[J ] = e−iS 0[J ]

d4xδ4

δJ 4(x)e+iS 0[J ] (9.6)

= e−iS 0[J ]

d4xδ4

δJ 4(x)e−

i2

J (z)DF(z−y)J (y) d4z d4y .

The fourth functional derivative is most easily obtained by going to a discreterepresentation. Noting that

∂ 4

∂J 4ke−

i2J iDijJ j = [−3DkkDkk + 6iDkk(DJ )k(DJ )k (9.7)

+ (DJ )k(DJ )k(DJ )k(DJ )k] e−i2J iDijJ j

we obtain (k=x)

S 1[J ] = −3

DF(x − x)DF(x − x) d4x

+ 6i DF(y−

x)DF(x−

x)DF(x−

z)J (y)J (z) d4x d4y d4z

+

[DF(x − y)DF(x − z)DF(x − v)DF(x − w)

× J (y)J (z)J (v)J (w) d4x d4y d4v d4w d4z

. (9.8)

The first term has no sources and will, therefore, not contribute to anyGreen’s function, the second term is quadratic in J and thus contributesto the two-point function and the last term here has the structure of a pointinteraction of four fields generated by independent sources at y, z, v, and wand thus contributes only to the four-point function.




9.1.1 Feynman rules

We can again represent these results in a graphical form by using the rulesgiven in section 8.2.2. These were

1) propagator: iDF (x − y) =x y

2) source: iJ (x) =x

3) Integration over the space-time coordinates of the sources

4) Symmetry factor for each diagram

We supplement these now for the interacting theory by the additional rules

5) Each interaction is represented by a dot:−ig

4!=

6) Integration

d4x for each loop.

If we represent the interacting connected functional iS [J ] by a doubleline

iS [J ] = (9.9)we can draw the graphs for

iS [J ] = ln Z 0 + iS 0[J ] +−ig

4!S 1[J ] + O(g2) (9.10)

as

= ln Z 0 + (9.11)

+

x +

xy z

+x

v

w

y

z

+ O(g2)

The first graph on the right-hand side is again the zeroth order term (9.3),whereas the graphs in the parentheses represent all the terms of first order inthe interaction (9.8). The first one of these describes a process without anyexternal lines; this is a vacuum process that takes place regardless if physicalparticles are present or not. It constitutes a background to all physicalprocesses. The second graph with the single loop describes a mass changedue to the selfinteraction that we will discuss in the next section. The thirdgraph, finally, describes a true interaction process.




The symmetry factors in these graphs are the factors in front of the

integrals in (9.8); they can be obtained as follows. The first graph carriesthe factor 1/2 as explained in section 8.2.2.

To construct the vacuum graph we pick one of the 4 legs of the interactionvertex and then connect with any of the other three free legs; there are always2 pairwise equal loops. Thus in total we get a weight of 4 × 3/(2 × 2) = 3.

For the second term in parentheses in (9.11) we have four legs of thevertex to connect with the external line to y; this gives 4 possibilities. Theexternal line to z can then still be connected with 3 remaining vertex legs.Since there is an exchange symmetry between y and z we get an additionalfactor 1/2 (as in S 0), so that the weight of this vertex becomes 6.

The last graph, finally, is obtained by joining one of the four legs of the vertex to one of the external points, say z. This generates 4 possibilities.Next we join any one of the 3 remaining free legs of the vertex to the externalpoint y; there are obviously 3 ways to do this. The remaining 2 legs can be joined in 2 different ways with the two external points v and w. Thus, thereare in total 4! = 24 possibilities. However, since all the external points v,w,yand z can be exchanged without changing any of the physics (v,w,y and zare integration variables), there are also 4! identical terms so that the lastgraph in the parentheses in Fig. 9.11 carries the weight 1.

These weights (symmetry factors) have to be multiplied for each graph

to the analytical expression obtained by following the rules given above forthe translation of the pictorial representation into an analytical one. Indeed,using the symmetry factors just given and following the Feynman rules forthe graphs (9.11) gives the expression (8.50) (together with (9.4) and (9.8)).

9.1.2 Vacuum contributions

We now consider the normalization term ln Z 0. Since we are working withnormalized generating functionals, Z 0 is given by W [0]−1

Z −10 = Dφ ei d

4x(L0−V +i ε2φ2) (9.12)

= e−i

d4xV ( 1i

δδJ (x))e−

i2

J (x)DF(x−y)J (y)d4xd4y|

J =0 .

We can now treat this expression in exactly the same way as we just did forS [J ]; the only change being that we have to take the final result at J = 0.This gives (see (9.10))

ln Z 0 = −iS [0] = −iS 0[0] − (−ig)

4!S 1[0] + O(g2) . (9.13)




In the graphical representation this reads, using

S 0[0] = 0 and

S 1[0] =

−3D2F(0) d4x (cf. (9.8)),

ln N = − (9.14)

The normalization constant, or – in other words – the denominator of thegenerating functional, thus contains just the vacuum graph. Inserting (9.14)into (9.11) then removes the vacuum contribution giving, finally, for Z [J ] thegraphical representation up to terms of first order in the interaction

= + + (9.15)

Although we have shown here only for first-order coupling that the de-nominator in Z [J ] (see (5.15),(5.16)) just removes the vacuum contributions,this is a general result that holds to all orders of perturbation theory.

9.2 Two-Point Function

The connected n-point function can now be obtained from its definition in(8.6)

Gc(x1, . . . , xn) =1

i

n−1 δnS δJ (x1) . . . δ J (xn)

J =0

. (9.16)

In this section we work out the connected two-point function in the lowestorders of the coupling constant.

9.2.1 Terms up to O(g0)

There is only one connected Green’s function in the free case which is justgiven by the Feynman propagator (8.30).

9.2.1.1 Momentum representation

We now evaluate the lowest order (in g) two-point function in momentumspace (cf. (8.7)). Taking the Fourier-transform of the two-point function(8.30) gives

e−i( p1x1+ p2x2)Gc(x1, x2) d4x1 d4x2 (9.17)

=

e−i( p1x1+ p2x2)

i

(2π)4

d4q

e−iq(x1−x2)

q2 − m2 + iε

d4x1d4x2 .




p − p

Figure 9.1: Two-point function of a scalar theory.

We first perform the integrations over x1 and x2 and obtain, according to thedefinition (8.1.1), for the rhs

(2π)4δ4( p1 + p2)Gc( p1, p2) = (2π)4δ4( p1 + p2)i

p21 − m2 + iε. (9.18)

From this equation we can read off the momentum representation of the prop-agator. The momenta p1 and p2 are incoming momenta that point towards avertex. Thus, the momentum representation of the free propagator is givenby

G0( p, p = − p) =

i

p2 − m2 + iε, (9.19)

pictured in Fig. 9.1. Note that here the second momentum appears with anegative sign. This is due to our notation to take all momenta as incoming(see Fig. 8.1).

9.2.2 Terms up to O(g)

Up to terms linear in the coupling strength we obtain from (9.5)

Gc(x1, x2) = −iδ2 S

δJ (x1)δJ (x2)

J =0

(9.20)

= −iδ2 S 0

δJ (x1)δJ (x2)− (−ig)

4!

δ2 S 1δJ (x1)δJ (x2)

J =0

+ O(g2) .

With S 0[J ] from (9.3) and S 1[J ] from (9.8) this gives for the two-point func-tion

Gc(x1, x2) = iDF(x1 − x2)

−− ig

4!

12i

d4xDF(x − x)DF(x − x1)DF(x − x2) + O(g2)

= iDF(x1 − x2) − g

2

d4xDF(x2 − x)DF(x − x)DF(x − x1)

+ O(g2) . (9.21)




This is the connected propagator, up to terms of

O(g), of the interacting

theory.We can represent this equation in the following graphical form, where

denotes the “dressed” propagator

x1 x2=

x1 x2+

x1 x2

(9.22)

with the rules developed above. Since the n-point functions involve deriva-tives with respect to the source current, taken at zero source, the externallines of all Feynman graphs do not contain crosses, that depict sources, any-more. They are instead given by free propagators.

The weight factors for these diagrams have been explained at the endof section 8.2.2. Since we deal here with Green’s functions with definite,fixed external points, the exchange symmetry factors must not be dividedout here. Thus, the first diagram on the rhs in (9.22) carries the weight 1and the second, the so-called tadpole diagram, the weight 12.

9.2.2.1 Momentum representation

In Sect. 8.1.1 we have introduced the momentum representation of the Green’sfunction and in Sect. 9.2.1.1 we have already evaluated it for the free case.

Here we now determine it for the φ4

theory up to terms of order O(g).Taking the Fourier-transform of the two-point function (9.21) gives e−i( p1x1+ p2x2)Gc(x1, x2) d4x1 d4x2 (9.23)

=

e−i( p1x1+ p2x2)

i

(2π)4

d4q

e−iq(x1−x2)

q2 − m2 + iε

d4x1d4x2

− g

2

e−i( p1x1+ p2x2)

d4x

1

(2π)4

3

× d4q1 d4q2 d4q3e−iq2(x2−x)e−iq1(x−x1)

(q21 − m

2

+ iε)(q22 + m

2

− iε)(q23 − m

2

+ iε) d4x1 d4x2 .

As in Sect. 9.2.1.1 we first perform the integrations over x1, x2 and x andobtain for the rhs

(2π)4δ4( p1 + p2)Gc( p1, p2) = (2π)4δ4( p1 + p2)i

p21 − m2 + iε

− g

2(2π)4δ4( p1 + p2)

1

p21 − m2 + iε

1

p22 − m2 + iε

× d4q

(2π)4

1

q2 − m2 + iε. (9.24)




This equation is used to read off the momentum representation of the propa-

gator (see (8.7)). The momenta p1 and p2 are incoming momenta that pointtowards a vertex. Thus, the momentum representation of (9.21) is given by

Gc( p, p = − p) =

i

p2 − m2 + iε(9.25)

+ S i

p2 − m2 + iε

−ig

4!

d4q

(2π)4

i

q2 − m2 + iε

i

p2 − m2 + iε

where the symmetry factor is S = 12. Equation (9.25) gives the momentumrepresentation of the two-point function up to terms of order g. The firstterm on the rhs gives the free propagator (9.19) already obtained in Sect.

9.2.1.1, whereas the second term gives the contribution of the interaction tothis two-point function.

Momentum space Feynman rules. The Feynman rules for (9.25) arenow

1) each line gives a factori

q2 − m2 + iε.

2) each vertex gives a factor−ig

4!.

3) there is four-momentum conservation for the sum of all momenta flow-ing into a vertex.

4) each internal line gives an integration d4q

(2π)4.

5) to each diagram a weight factor has to be multiplied as explained above.

Selfenergy. With the abbreviation

Σ =g

2 d4q

(2π)4

i

q2

−m2 + iε

(9.26)

and the free two-point function G0 from (9.19 we can write the two-pointfunction (9.25) as

Gc( p, − p) = G0 + G0Σ

iG0 = G0

1 +

Σ

iG0

≈ G0

1 − ΣG0

i

−1

=i

p2 − m2 + iε

1

1 − Σ 1 p2−m2+iε

=i

p2 − m2 − Σ + iε. (9.27)




h

a) b) c)

Figure 9.2: Feynman graphs for the two-point function up to O(g2).

Here we have consistently kept terms up to O(g).The quantity Σ appears like an additional mass term in the final result.

It is, therefore, called a selfenergy and the second graph on the rhs in (9.22)is called a selfenergy insertion. The appearance of this selfenergy is a firstindication that the mass m appearing in the Lagrangian is the mass of theparticle only in a classical theory. In quantum theory it gets changed by theinteractions.


As noted at the end of Sect. 8.3.1 there are two distinct contributions to thesecond order term, one being a genuine term of second order in V and theother just being an iteration of the first order term. With the help of theFeynman rules we can now construct the corresponding Feynman graphs upto terms of O(g2). For the two-point function these are given in Fig. 9.2.

Graph (a) in Fig. 9.2 obviously just represents an iteration of the firstorder tadpole graph in (9.22). Its contribution to the two-point function isgiven by

Ga( p, − p) = S ai

p2 − m2 + iε

−ig

4!

d4q

(2π)4

i

q2 − m2 + iε

(9.28)

× i

p2 − m2 + iε

−ig

4!

d4q

(2π)4

i

q2 − m2 + iε

i

p2 − m2 + iε,

S a is the symmetry factor; it is simply given by the product of the corre-sponding factors for the one-loop graphs, S a = 12 · 12 = 144.

It is evident from Fig. 9.2a, as well as from its algebraic representationin (9.28), that the graph can be cut into two parts, each representing a first




order process. This reflects the appearance of the last term in (8.50) that

is simply the square of the first order term. Such a graph that falls apartinto 2 unconnected parts, if one internal line is cut, is called one-particle-reducible; otherwise it is one-particle-irreducible (1PI). The reducible graphhere is generated by the square of the first order term ∼ S 21 in (9.5).

In order to facilitate the following discussions we introduce now the ver-tex function Γ( p1, p2, . . . , pn), sometimes also called connected proper vertex function , which describes only 1PI graphs and in which the propagators forthe external lines are missing. The n-point vertex function is, therefore, givenby

Γ( p1, p2, . . . , pn) (9.29)= G−1( p1, − p1)G−1( p2, − p2) . . . G−1( pn, − pn)Gc( p1, p2, . . . , pn) .

The free 1PI 2-point function is defined by

Γ( p, − p) = p2 − m2 . (9.30)

Note that the product of inverse two-point Green’s functions and the n-bodyfunction is just the combination that appears in the reduction theorem (cf.(7.35)).

The 1PI part of the Green’s function for the graph 9.2a reads

Γa( p, − p) = G−1( p, − p)G−1(− p,p)Ga( p, − p)

= S a−ig

4!

d4q

(2π)4

i

q2 − m2 + iε(9.31)

with S a = 144.We then get for the graph in Fig. 9.2b, which is one-particle irreducible,

Γb( p, − p) = S b

−ig

4!

2 d4q

(2π)4

d4u

(2π)4

i

q2 − m2 + iε

i(2π)4δ4(q − u)

u2 − m2 + iε

× d4r

(2π)4

i

r2

− m2

+ iε

(9.32)

with S b = 12 · 12 = 144. For the graph in Fig. 9.2c (also 1PI) we finallyobtain

Γc( p, − p) = S c

−ig

4!

2 d4q

(2π)4

d4r

(2π)4

d4s

(2π)4δ( p − (q + r + s))

× i

q2 − m2 + iε

i

r2 − m2 + iε

i

s2 − m2 + iε, (9.33)

with S c = 4 · 4! = 96.




1

3

2

4

x

Figure 9.3: Feynman graph for the four-point function.

9.3 Four-Point Function

It is easy to see that the three-point function vanishes for the model con-sidered here since the third functional derivative (see (9.16)) of the action(8.50) at J = 0 vanishes.

9.3.1 Terms up to O(g)

The four-point function up to terms of O(g) is given by

Gc(x1, x2, x3, x4) = 1

i 3 δ4 S

δJ (x1) . . . δ J (x4) J =0 (9.34)

= iδ4 S 0

δJ (x1) . . . δ J (x4)

J =0

+(−ig)

4!

δ4 S 1δJ (x1) . . . δ J (x4)

J =0

+ O(g2)

where S is given by (9.5) and Fig. 9.2. Because Gc involves the fourthderivative with respect to the source and S 0 depends on J only quadratically(see (9.3)), only the last term of S 1 in (9.8) can contribute to the Green’sfunction. Thus we get

Gc(x1, . . . , x4) = −ig d4xDF(x − x1)DF(x − x2)DF(x − x3)DF(x − x4) .(9.35)

In momentum space this is simply the product of the four propagators (9.19)times the factor −ig. The corresponding Feynman graph is given in Fig.9.3. It carries the symmetry factor 4!, corresponding to a symmetry underpermutation of all external legs.

Unconnected graphs. From the connected graphs calculated so far, wecould reconstruct also the unconnected graphs. Up to terms linear in the




coupling constant g we get in symbolic notation

Z [J ] = e = 1 + +1

2!( )2 + . . .

= 1 +

+ +

+1

2!

+ +

2

+O(g2) . (9.36)

The four-point function is generated by diagrams with four external legs

(each external leg corresponds to a factor J in the generating functional),because G is given by a fourth functional derivative. Therefore, only the lastdiagram in the second line of (9.36) and the square of the first term in thelast line can contribute to the four-point function in this order.

We thus have for the four-point function up to terms linear in the cou-pling

G(x1, x2, x3, x4) = + + (9.37)

In both of the first two diagrams the two particles just move by each other,without interaction. These unconnected graphs thus do not contribute to

any interaction processes.


.In order to become more familiar with Feynman graphs, we now construct

the connected four-point function up to terms of order g2 in a graphical way.This four-point function is shown in Fig. 9.4.

The first line in Fig. 9.4 gives the four-point function just constructed,with a symmetry factor S 1 = 4!. The four diagrams on the second line

are just the basic vertices with self-energy insertions on each of the externallegs; these insertions carry the extra symmetry factor S 2 = 12, as we haveseen in section 9.1.1. The last three diagrams in the third line are of a newtopological structure. They represent modifications of the basic interactionvertex through the insertion of internal lines.

Each one of these graphs has the same external lines. We thus have anoverall factor for all graphs

S 14k=1

i

p2k − m2

, (9.38)




1

3

2

4

=

1

3

2

4

+

1

3

2

4

+

1

3

2

4

+

1

3

2

4

+

1

3

2

4

+

1

3

2

4

+

q1

q2

1

3

2

4

+

1

3

2

4

+ O(g3)

Figure 9.4: Four-point function of φ4 theory up to terms ∼ g2.

with the external symmetry factor S 1 = 4! so that the full four-point functionis given by

Gc( p1, p2, p3, p4) = S 14k=1

i

p2k − m2

(G1 + G2 + G3) , (9.39)

where Gi denotes the contribution from the i-th line in Fig. 9.4 without theexternal line symmetry factor.The first basic vertex then just gives the contribution

G1 =−ig

4!. (9.40)

The graphs of the second line have one loop in addition. We thus have

G2 = (−ig

4!)2S 2

d4q

(2π)4

i

q2 − m2

4l=1

i

p2l − m2

(9.41)




for their contribution, with S 2 = 12. The internal momenta pl here are

those between the loop and the four-point vertex. Since the loop carries nomomentum away they are the same as the corresponding incoming momenta.

The three graphs of the third line, finally, have two internal lines, whichare, however, related by energy - and momentum conservation at the incom-ing vertex. They give

G3 = (−ig

4!)2S 3

d4q1

(2π)4

d4q2

(2π)4

i

q21 − m2

i

q22 − m2

× (2π)4

klδ4(q1 + q2 − ( pk + pl)) (9.42)

where the last sum runs only over the pairs of numbers (1,2), (1,3) and (1,4),i.e. the external legs at one of the vertices in each of the three graphs. The δ-function appears because the net momentum running into the dressed vertexhas to be zero (see (8.7)).

The symmetry factor S 3 can, for example for the middle graph of Fig.9.4, be obtained as follows. The external leg 1 can be connected with the leftvertex in 4 different ways; the same holds for the leg 2 with the right vertex.Once these connections (4 × 4 possibilities) have been done, each of the 3free legs of the left vertex can be connected with the external leg 3; the same

holds for the connections of the right vertex to the external point 4 (3 × 3possibilities). Finally, each of the remaining 2 legs of the left vertex can beconnected to the right vertex; for the remaining leg there is then no freedomleft (2 possibilities). Thus, the symmetry factor for the last 3 graphs in Fig.9.4 is

S 3 =(4 · 4)(3 · 3)2

S 1=

(4!)2

2S 1=

4!

2. (9.43)



Chapter 10

DIVERGENCES IN n-POINT

FUNCTIONS

Many of the expressions obtained in the preceding sections for two- and four-point functions are actually ill-defined because they diverge, as we will showin this section. We start with a rather general discussion of divergences inφ4 theory and then evaluate explicitly the two- and four-point functions.

To illustrate the divergence of the Green’s functions obtained we consider,as an example, the two-point function

Gc( p, − p) = i p2 − m2 + iε

(10.1)

+−i

g

2

i

p2 − m2 + iεiDF(0)

i

p2 − m2 + iε.

The loop contribution between the two Feynman propagators in the secondterm on the rhs is given by

Σ =g

2iDF(0) =

g

2

d4q

(2π)4

i

q2 − m2 + iε. (10.2)

The integral here diverges: after integrating over q0 we obtain integrals of the form (cf. section 6.1.2)

d3q√

q2 + m2. (10.3)

By introducing an upper bound Λ for the integral over | q| and then takingΛ → ∞ we see that the integral diverges as Λ2; this is called a quadraticdivergence. Because this divergence happens here for large q one also speaksof a quadratic ultraviolet divergence.

113



CHAPTER 10. DIVERGENCES IN N -POINT FUNCTIONS 114

Another way to see the degree of divergence is the so-called “power-

counting”: There are 4 powers of q in the integration measure, but onlytwo powers of q in the denominator of (10.2); the degree of divergence isthen given by the net power (2) of q.

10.1 Power Counting

The power-counting just illustrated for the case of the tadpole diagram canbe generalized to any Feynman graphs with an arbitrary number of loops.

In order to see this we consider a theory with an interaction ∼ φ p in ndimensions. Since each loop contributes according to the Feynman rules anintegral

dnq to the total expression and since each internal propagator gives

a power q−2, we have for the degree of divergence D in a diagram with Lloops and I internal lines

D = nL − 2I . (10.4)

Note that here each loop has also at least 1 internal line. For example, thetadpole diagram has L = 1 and I = 1 , giving D = 2 in four dimensions.D > 0 clearly diverges, D = 0 corresponds to a logarithmic divergence, andD < 0 seems to be convergent.

If a graph has V interaction vertices, then the total number of lines in φ p

theory is pV , since each vertex has p legs. These legs can be either externalor internal lines. If they are internal, they count twice because each internalline originates and disappears at a vertex. Thus we have

pV = E + 2I , (10.5)

where E is the number of external lines. In addition, the number of loops,L, is related to the number of vertices, V , by

L = I − V + 1 . (10.6)

Combining equations (10.5) and (10.6) with (10.4) allows us to eliminate L

and I to obtain

D = n +

n( p − 2)

2− p

V −

p

2− 1

E . (10.7)

The degree of divergence of a connected graph in this φ p theory in n dimen-sions thus depends on the number of external legs and, in general, also onthe number of vertices.

In the perturbative treatment of field theory derived in Chap. 8.3 wehave seen that the order of perturbation theory directly equals the number of




D = 4 · 1 − 2 · 1 = +2

D = 4 · 1 − 2 · 2 = 0

D = 4 · 0 − 2 · 1 = −2

D = 4 · 1 − 2 · 3 = −2

Figure 10.1: Examples for graphs with different degree of divergence. On theright equation (10.4) is illustrated for n = 4.

vertices, V , in a Feynman diagram. Thus, the degree of divergence becomeslarger and larger with increasing order of a perturbative treatment, if thefactor of V in (10.7) is positive. On the other hand, D is independent of thisorder if that factor is zero and D becomes even smaller when going to higherorders of perturbation theory if the factor is negative. Thus the perturbativetreatment leads to a finite number of divergent terms if and only if

n( p − 2)2 − p ≤ 0 . (10.8)

If this factor is zero, then the total number of divergent diagrams in a per-turbative expansion can still be infinite, but in each order of perturbationtheory only the same finite number of divergent diagrams appears. In thiscase the theory is called renormalizable. If the factor is negative, then eventhe total number of divergent terms is finite; in this case the theory is calledsuperrenormalizable. In both cases one can add a finite number of so-calledcounter terms to the Lagrangian that just remove these divergences.






These considerations show that for p = 4

[g] = n−4 = mass4−n ; (10.11)

i.e. in four dimensions the coupling constant of a φ4 theory is dimensionlessand the theory is, therefore, renormalizable; in fact, only theories with p ≤ 4are renormalizable. Therefore, according to the discussion in the precedingsection the φ4 theory contains only a finite number of divergent vertices. Inn = 4 dimensions, however, this is no longer true.

If we want to keep the coupling constant dimensionless in order to ensurerenormalizability also in n dimensions we have to modify the φ4 term inthe Lagrangian (8.51) such that an additional factor with the dimension of (mass)4−n absorbs the dimension and g becomes dimensionless

L = L0 − g

4!µ4−nφ4 . (10.12)

Note that µ here is an arbitrary mass.

10.2.1 Two-point function

The two-point function is completely determined once we know the selfenergy.We start by calculating this quantity in lowest order in the coupling by

evaluating explicitly the contribution of the tadpole diagram.To do so we first go to n dimensions so that (10.2) becomes

Σ =g

2µ4−n

dnq

(2π)ni

q2 − m2 + iε. (10.13)

This integral can be obtained analytically by going into a space of n-dimen-sional polar coordinates (see Appendix. C). The result is

Σ =g

2

µ4−n

(2π)nmn−2π

n2 Γ

1 − n

2

. (10.14)

The divergence of this expression is now manifest, since the Γ-function haspoles at 0 and the negative integers, and thus also for n = 4.

We now expand Γ around this pole. For that purpose we write

Γ

1 − n

2

= Γ

−1 +

ε

2

(10.15)

with ε = 4 − n and expand in powers of ε (cf. (C.4))

Γ−1 +

ε

2

= −2

ε− 1 + γ + O(ε) , (10.16)




where γ is the Euler-Mascheroni constant (γ

≈0.577..). We thus get for n

close to 4

Σ =g

2

µε

(2π)4−εm2−επ

4−ε2

−2

ε− 1 + γ + O(ε)

(10.17)

= gm2

32π2

4πµ2

m2

ε2

−2

ε− 1 + γ + O(ε)

.

We now usexε = eε lnx ε→0−→ 1 + ε ln x (10.18)

and obtain

Σε→0−→ gm2

32π2

1 +

ε

2ln

4πµ2

m2

−2

ε− 1 + γ + O(ε)

=gm2

32π2

−2

ε− 1 + γ − ln

4πµ2

m2

+ O(ε)

= − gm2

16π2

1

ε− gm2

32π2

1 − γ + ln

4πµ2

m2

+ O(ε) . (10.19)

We have thus split Σ into two parts. The first one is clearly diverging as napproaches 4 (ε

→0). The second term is finite and depends on the arbitrary

mass µ that was originally introduced only to keep the coupling constant freeof dimension. The appearance of the arbitrary mass µ in the finite part isrelated to the arbitrariness in separating an overall infinite expression into asum of an infinite and a finite contribution.

Note that Σ is independent of p. In next higher order the same is truefor the selfenergy contribution of Fig. 9.2b on page 107 whereas that of Fig.9.2c depends quadratically on p [Ramond].

10.2.2 Four-point function

In this section we will now evaluate the four-point function. By looking atFig. 9.4 on page 111 we expect that the four graphs in the second line justcontribute again to the selfenergy. On the other hand, we expect that thethree diagrams in the lowest line can all graphically be contracted such thatthey contain only one interaction point with possibly modified interactionstrength.

As an example, we now evaluate the middle graph in the last line of Fig.9.4. According to the Feynman rules its contribution to the 1PI vertex is




(see (9.42))

∆Γ( p1, p2, p3, p4) =−ig

4!µ4−n

2

S 3S 1

d4q

(2π)4

i

q2 − m2

i

( p − q)2 − m2

(10.20)with p = p1 + p3. The symmetry factor is S 1S 3 = 12 · 4! (9.42).

The two denominators in the integrand can be combined into one by amathematical trick due to Feynman that is very often used for the evaluationof such expressions. This trick starts from the elementary integral relation

b

adx

x2

=

−

1

x |ba =

−

1

b

+1

a

=b − a

ab

. (10.21)

By substituting nowx = az + b(1 − z) (10.22)

we obtainb a

dx

x2= (a − b)

0 1

dz

[az + b(1 − z)]2 . (10.23)

Combining (10.21) and (10.23) gives

1

ab =

1

0

dz

[az + b(1 − z)]2 . (10.24)

We now apply this to the integrand in (10.20) and obtain

1

q2 − m2

1

( p − q)2 − m2=

1 0

dz

(q2 − m2)z + [( p − q)2 − m2] (1 − z)2

=

1 0

dz

[q2 − 2 pq(1 − z) + p2(1 − z) − m2]2 .

(10.25)

The first three terms in the denominator can be combined by substituting

q = q − p(1 − z) . (10.26)

This gives for expression in the denominator

q2 − 2 pq(1 − z) + p2(1 − z) − m2

= [q − p(1 − z)]2 − m2 − p2(1 − z)2 + p2(1 − z)

= q2 − m2 − p2z(z − 1) . (10.27)




The 1PI vertex now reads in n dimensions

∆Γ( p1, p2, p3, p4) =1

2g2(µ2)4−n

dnq

(2π)n

1 0

dz1

[q2 − m2 − sz(z − 1)]2

(10.28)with s = p2 = ( p1 + p3)2.

We now interchange the order of integration and evaluate the integralover q first with the help of (C.18) in Appendix C dnq

(2π)n1

[q2 − m2 − sz(1 − z)]2

= i(2π)n

m2 + sz(1 − z)n−42 π n2 Γ 2 −n

2Γ(2)

. (10.29)

The four-point function thus reads

∆Γ( p1, p2, p3, p4) =1

2g2(µ2)4−ni

πn2

(2π)nΓ

2 − n

2

×1

0

dzm2 + sz(1 − z)

n−42 . (10.30)

We now introduce again ε = 4 − n. This gives

∆Γ( p1, p2, p3, p4) =1

2g2µεi

1

16π2Γε

2

1 0

m2 + sz(1 − z)

4πµ2

− ε2

dz . (10.31)

Now the four-point function is in a form that allows to take the limit ε → 0.Using again

xε = eε lnx → 1 + ε ln x (10.32)

and (C.2)

Γ

ε

2

=

2

ε− γ + O(ε) (10.33)

gives

∆Γ( p1, p2, p3, p4) =ig2µε

32π2

2

ε− γ + +O(ε)

×1 − ε

2

1 0

ln

m2 + sz(1 − z)

4πµ2

dz

=ig2µε

16π2

1

ε− ig2µε

32π2

γ +

1 0

ln

m2 + sz(1 − z)

4πµ2

dz

+ O(ε) with s = ( p1 + p3)2 . (10.34)




Here the four-point function has been separated into a divergent part and a

convergent term (for ε → 0). The integral appearing is a function of p, m,and µ; the latter dependence remains even when ε → 0.

So far, we have only calculated the middle graph in the last line of Fig.9.4 on page 111. It is evident, however, that the result can be directly takenover also to the other 2 graphs in the last line by taking for p the appropriatetotal momentum at one of the vertices. We, therefore, introduce now thethree Lorentz-invariant Mandelstam variables

s = ( p1 + p3)2 = p2

t = ( p1 + p2)2

u = ( p1 + p4)2 , (10.35)

with the property s + t + u = m21 + m2

2 + m23 + m2

4. These variables representthe total squared four-momentum at the vertex that involves p1 in each of the graphs in the last line of Fig. 9.4.

We thus get for the sum of all three diagrams in the last line of Fig. 9.4,the vertex correction diagrams,

Γv( p1, p2, p3, p4) =3ig2µε

16π2

1

ε(10.36)

−ig2µε

32π2[3γ + F (s,m,µ) + F (t,m,µ) + F (u,m,µ)] .

Here F (s,m,µ) denotes the integral in (10.34).Of the diagrams in Fig. 9.4 only the first and the three last ones are

1PI graphs. The four diagrams in the second line are all 1P reducible; they just differ by the propagators on the external legs that are left out when weconsider the 1PI four-point function. The complete 1PI four-point functionis, therefore, given by

Γ( p1, p2, p3, p4) = −igµε + Γv( p1, p2, p3, p4)

=−

igµε +3ig2µε

16π2

1

ε

− ig2µε

32π2[3γ + F (s,m,µ) + F (t,m,µ) + F (u,m,µ)]

= − igµε

1 − 3g

16π2

1

ε(10.37)

+g

32π2[3γ + F (s,m,µ) + F (t,m,µ) + F (u,m,µ)]

.

Equation (10.37) gives the effective, “dressed” interaction vertex. By com-parison with the free 1PI four-point function, its forms suggests to absorb all




effects of the loop graphs contained in the curly brackets into a new effective

coupling constant

g −→ g 1 + δg(s,t,u,m,ε,µ) . (10.38)

This effective coupling constant depends on s,t,u,m and µ and, as the self-energy, separates into a divergent and a finite term. It contains all the effectsof the loops.

10.3 Renormalization

In the preceding two subsections we have seen that both the 1PI two-point

and the 1PI four-point functions are (for ε = 0) regular functions of ε. The infour dimensions diverging quantities have thus been regularized . For n → 4both have divergent and finite contributions from higher order diagrams. Inthis section we now show how to handle these divergences by the renormaliza-tion technique. Any renormalization procedure requires first a regularization,either by a cut-off for the upper bounds of diverging integrals or by going ton = 4 dimensions, to be followed by a procedure in which the dependence onthese artifacts is removed.

Since the separation of a divergent quantity into a finite and an infinitecontribution is arbitrary there are various so-called renormalization schemes.

They all have in common that they either add terms to the Lagrangianor scale the fields and coupling constants such that the original form of the Lagrangian is maintained. The most obvious scheme is the so-calledminimal subtraction scheme that removes just the pole contributions in thedimensional regularization, i.e. the terms that go like powers of 1/ε. Thisscheme is straighforward and well-suited for the dimensional regularization,but it leads to expressions in which the parameters m (mass) and g (coupling)have no direct relation to measurable quantities.

Here we discuss another scheme, in which we require that the parame-ters of the Lagrangian assume their physical, measured values, i.e. m is the

physical mass and g the physical coupling constant.We start by looking at the structure of the most general two-point func-tion. When we go to terms that depend on g2 and higher orders of theinteractions we always encounter 1P reducible graphs, such as the one inFig. 9.2a. The general structure of the (reducible) two-point function is of the form

Gc( p, − p) = G0( p, − p) + G0( p, − p)Σ

iG0( p, − p)

+ G0( p, − p)Σ

iG0( p, − p)

Σ

iG0( p, − p) + . . . , (10.39)




where Σ is the so-called proper self-energy. Note that (10.39) is an expansion

in terms of irreducible diagrams. Equation (10.39) can be summed and gives

Gc( p, − p) = G0( p, − p)

1 +Σ

iG0( p, − p) +

Σ

iG0( p, − p)

Σ

iG0( p, − p) + . . .

= G0( p, − p)

1 − Σ

iG0( p, − p)

−1

=

G−10 ( p, − p) − Σ

i

−1

=i

p2

−m2

−Σ + iε

. (10.40)

The self-energy Σ is in general a function of the momentum p of theparticle. We can therefore expand Σ( p2) around the on-shell point p2 = m2,where m is the physical, observable mass

Σ( p2) = Σ(m2) + ( p2 − m2)Σ1 + Σ2( p2) . (10.41)

Here

Σ1 =∂ Σ

∂p2| p2=m2 and Σ2(m2) = 0 . (10.42)

In (10.41) only the first two terms of the expansion of Σ have been written

out explicitly; Σ2( p2) denotes the whole remainder of the expansion. Σ, beingthe selfenergy insertion of a two point function, is quadratically divergent.Consequently, Σ1 as the first derivative of Σ with respect to p2 is logarith-mically divergent and Σ2 as a second derivative is convergent since takingthe derivate always adds one more power of q2 in the denominator of theFeynman propagators.

Inserting this expansion into (10.40) gives for the propagator

Gc( p, − p) =i

p2 − m2 − Σ(m2) − ( p2 − m2)Σ1 − Σ2( p2) + iε

= 11 − Σ1i

p2 − m2 − Σ(m2)+Σ2( p2)1−Σ1

+ iε. (10.43)

The pole of this propagator should be at the physical mass and its residuumshould be i. However, this is in general not the case, if we start with theobservable, physical mass in the Lagrangian, because the selfinteractions con-tribute to the self-energy.




Counter terms. In order to get the pole to the correct, physical location

we therefore have to change the mass in the Lagrangian by adding a so-calledcounterterm to it

Lcm = −1

2δm2Zφ2 with Z =

1

1 − Σ1

. (10.44)

Z is usually called “field renormalization constant” for reasons that will be-come obvious a little later (see (10.56)). Note that this counterterm hasexactly the same form as the mass term in the original Lagrangian. It willthus also add a term −Zδm2 in the denominator of the dressed propagator(10.43).

We determine the unknown δm2 by the requirement

δm2 + Σ(m2) = 0 (10.45)

so that the new term just cancels the selfenergy contribution at the on-shellpoint. With the counterterm added, the propagator then becomes

G( p, − p) =1

1 − Σ1

i

p2 − m2 − Z Σ2( p2) + iε. (10.46)

Since Σ2(m2) = 0 by definition, this propagator has the correct pole at thephysical mass m.

Its residue, however, is – instead of being simply i –

i1 − Σ1

= iZ . (10.47)

This deficiency can be cured by adding another, additional counterterm

Lcφ =1

2(Z − 1)


(10.48)

to the Lagrangian. Then the propagator becomes

G( p, − p) = Z i

p2 − m2 − Z Σ2( p2) + (Z − 1)( p2 − m2) + iε

=i

p2 − m2 − Σ2( p2) + iε. (10.49)

This propagator has the pole at the correct, physical mass m (because of Σ2(m2) = 0) and the correct residue i.

By adding the given counterterms we have thus removed the divergentquantities Σ(m2) and Σ1 from the propagator. The one remaining quantityΣ2( p2) involves a second derivative of the selfenergy with respect to p2 andis convergent; it vanishes at the on-shell point. Note that the countertermsall have the structure of terms already present in the original Lagrangian. If this is the case, in general a theory is called renormalizable.




10.3.1 Renormalization of φ4 Theory

We now specify all of these general considerations to the example of φ4 the-ory. As we have discussed in Sect. 10.2 the φ4 theory is renormalizable,i.e. only a finite number of elementary vertices diverges. These are just theterms we have calculated in the last two subsections, namely the selfenergycontribution (10.19) and the vertex function (10.37).

For the selfenergy we have, up to one-loop diagrams, (cf. (10.19)),

Σ( p2) = − gm2

16π2

1

ε− gm2

32π2

1 − γ + ln

4πµ2

m2

+ O(ε) , (10.50)

i.e. Σ1 = Σ2 = 0 and Z = 1. Equation (10.45) reduces to

δm2 = −Σ(0) = −Σ =⇒ Z = 1 . (10.51)

Thus in this one-loop approximation, there is no field renormalization, butalready in the two-loop approximation we would get Σ 1 = 0 because thegraph Fig. 9.2c is p-dependent. To be general we, therefore, keep the factorZ in the following expressions.

So far we have not taken the change of the coupling constant due to higherorder loop diagrams into account. In (10.37) we have indeed already seen thatalso the interaction vertex gets modified due to higher order loop corrections.

The structure there was such that the coupling constant g was replaced byan effective coupling constant g(1 + δg(s,t,u,m,ε,µ)). If we again want tohave the physical, observable coupling constant in our Lagrangian we mustget rid of the modification by a proper counter term. We, therefore, define avertex renormalization constant Z g by

Z g = (1 + δg(s,t,u,m,ε,µ))−1 |r , (10.52)

where r denotes an in principle arbitrary renormalization point. Since wewant to relate the coupling to a measurable quantity we use the so-calledsymmetric point

pα · pβ = m24

3δij − 1

3

(10.53)

for i, j = 1, . . . , 4. At this point we have s = t = u = −4m2/3. We can thenintroduce the additional counter term

Lcv = −gµε

4!(Z g − 1) φ4 . (10.54)

In this way we can ensure that the coupling constant g in the Lagrangianhas its physical, observable value at r.




The Lagrangian with all the counterterms now reads

L =(∂ µφ)2 − m2φ2

− gµε

4!

+1

2(Z − 1)


− 1

2δm2Zφ2 − gµε

4!(Z g − 1) φ4

=Z

2

(∂ µφ)2 − m2φ2

− 1

2δm2Zφ2 − gµε

4!Z gφ4 . (10.55)

This Lagrangian leads to the correct physical behaviour for the 2-pointGreen’s function with the physical mass m and the proper residue. Thefields φ, in terms of which the propagator is defined, are thus the physical

fields, which include already the effects of selfinteractions.By introducing the “bare field” φ0 and the “bare mass” m0 by

φ0 =√

Zφ

m20 = m2 + δm2 (10.56)

and a “bare coupling constant” by

g0 = gµεZ gZ 2

, (10.57)

then the whole Lagrangian can be expressed in terms of bare quantities only

L =1

2

(∂ µφ0)2 − m2

0φ20

− g0

4!φ4

0 . (10.58)

This bare Lagrangian has the same form as the original one, becauseall the counter terms had the same form as terms already appearing in theoriginal Lagrangian. As a consequence it leads to finite physical quantitiesin all orders of perturbation theory. If this is the case, then the theory is welldefined, i.e. it is said to be renormalizable. The bare Lagrangian is reallyconsidered to be the ‘true’ Lagrangian of the theory because it leads only tofinite physical quantities.

In the preceding considerations we have chosen 2 arbitrary renormaliza-tion points; we have required the propagator to have a pole at p2 = m2,where m is the physical mass (cf. (10.49)), and for the vertex we have chosenthe symmetric point r (s = t = u = −4m2/3) (cf. (10.52)). Choosing otherrenormalization points will lead to other values for the mass and the cou-pling. This arbitrariness in the renormalization point reflects the fact thatthe divergent expressions for the selfenergy and the vertex all separate intoa divergent and a finite term. Any scheme, that removes the infinite terms,will lead to finite expressions for the physical observables, independent of




what it does with the finite contributions. It will also lead to the same n-

point function, and – with the help of the reduction theorem – to the sameobservable transition rates.

In the dimensional renormalization discussed here all the counter termsand renormalization factors also depend on the arbitrary mass µ, even afterε → 0 and after the infinite terms have been removed. This µ was originallyintroduced to keep the coupling constant g dimensionless also for n = 4.It is obvious from our considerations above that – for specified renormal-ization points – choosing different values for µ will lead to different valuesfor the wavefunction renormalization Z and the coupling constant g. Thisarbitrariness, however, is nothing else than the arbitrariness we have already

encountered in the choice of the renormalization point. The physics must beindependent of µ. This observation is the starting point for the developmentof the renormalization group method.

It is essential to realize that even in a massless theory, which containsno dimensional scale parameters, µ introduces a scale that determines themomentum dependence of the coupling constant. Thus, at the quantum level,a bare Lagrangian is not enough to specify a theory, but a renormalizationscheme must be added that introduces necessarily a scale into the theory.



Chapter 11

GREEN’S FUNCTIONS FOR

FERMIONS

For simplicity of notation we have so far in this book discussed only thepath integrals and generating functions for scalar fields. All this formalismcan be easily generalized also to vector fields which obey the Proca equation(4.24) and thus fulfill component by component the Klein-Gordon equation.For fermion fields, however, there is a problem. The main idea in using pathintegrals is to express quantum mechanical transition amplitudes by integralsover classical fields; the values of these fields at the discrete coordinate-sites were taken to be commuting numbers. Such a formalism can, however,not “know” about the Pauli principle. For example, with the formalismdeveloped so far, a fermion could be propagated to a point in configurationspace which is already occupied. In nature, however, this propagation isPauli-forbidden.

For the description of fermions it is, therefore, necessary to extend thetheory developed so far such that the Pauli principle is taken into account.This can be achieved by using an anticommuting algebra for the classicalfields, the so-called Grassmann algebra.

11.1 Grassmann Algebra

In this section we outline the basic mathematical properties of the Grassmanalgebra as far as we will need them in the later developments.

We define the n generators i, . . . , n of an n-dimensional Grassmann al-gebra by the anticommutation relations

i, j ≡ i j + ji = 0 . (11.1)

128



CHAPTER 11. GREEN’S FUNCTIONS FOR FERMIONS 129

Let us now consider series expansions in these variables. Since

21 = 2

2 = . . . = 2n = 0 , (11.2)

because of (11.1), any series in i must have the form

φ() = φ0 +i

φ1(i)i +i<j

φ2(i, j)i j (11.3)

+i<j<k

φ3(i,j,k)i jk + . . . + φn(1, 2, . . . n)12 . . . n .

Here the φi are ordinary, commuting c-numbers. Note that the expansion

actually terminates because of (11.2). Every element of the algebra can bewritten in the form (11.3); thus, the various powers of in this expansionconstitute a basis of the algebra.

If we choose the function φi to be totally antisymmetric in all their vari-ables, then we can also write

φ() = φ0 +i

φ1(i)i +1

2!

ij

φ2(i, j)i j

+1

3!

ijk

φ3(i,j,k)i jk + . . . . (11.4)

From now on we can assume that the φi are indeed antisymmetric since anysymmetric part of φ would not contribute anyway.

As a consequence of this relation any analytical function in 1 dimensionhas the form

φ(1)() = φ0 + φ1 , (11.5)

and in 2 dimensions

φ(2)() = φ0 + φ1(1)1 + φ1(2)2 + φ2(1, 2)12 . (11.6)

These relations imply for the Taylor expansion of a Gaussian

e−2

= 1 (11.7)

ande−ij = 1 − i j . (11.8)




11.1.1 Derivatives

Since functions of Grassmann variables can, because of (11.2), be at mostlinear in ε, the derivative operator is defined as an algebraic operation that,when applied to a Grassmann variable, simply replaces that variable by a 1;when applied to a product the variable first has to be commuted to a positionnext to the derivative before it can be removed. We then get, for example,

∂

∂ j, i

1 =

∂

∂ j(i1) + i

∂1

∂ j

=∂i∂ j

1 − ∂1

∂ ji + i

∂1

∂ j= δij1 . (11.9)

The “−” sign in the second line appears, because the factor 1 must befirst brought to the left before the derivative can act. This result is validin general. We thus can define the derivative also by the anticommutationrelation

∂

∂ j, i

= δij . (11.10)

This relation can be used to obtain the derivative of a function φ().Applying the lhs of (11.10) to φ, which can always be written as polynomial(see (11.4)), yields

∂

∂ j, i

φ() =

∂

∂ j[iφ()] + i

∂

∂ jφ() (11.11)

= δijφ() +

∂

∂ j

φ0 −

k

φ1(k)k +1

2!

kl

φ2(k, l)kl

− 1

3!

klm

φ3(k, l , m)klm + . . .

i + i

∂

∂ jφ() .

Here all the terms in φ() with an odd number of s have changed sign whenthey were brought to the left so that the derivative can act on them. With

∂ ∂ j

kl = δkjl − δljk (11.12)

and∂

∂ jklm = δkjlm − δljkm + δmjkl (11.13)

we get∂

∂ j, i

φ() = δijφ() +

−φ1( j) +

1

2!

l

φ2( j,l)l − 1

2!

k

φ2(k, j)k




−1

3! lm φ3( j,l,m)lm +1

3! km φ3(k,j,m)km

− 1

3!

kl

φ3(k, l , j)kl + . . .

i + i

∂

∂ jφ()

= δijφ()

−

φ1( j) − k

φ2( j,k)k +1

2!

kl

φ3( j,k,l)kl − . . .

i

+ i∂

∂ jφ() . (11.14)

Because of (11.10) this has to be equal to δij φ(). Commuting i throughto the left side of the square parentheses changes again the signs of all oddterms. The last two terms in (11.14) then cancel each other if the expressionin the square parentheses equals the derivative of φ

∂

∂ jφ() = φ1( j) +

k

φ2( j,k)k +1

2!

kl

φ3( j,k,l)kl + . . . . (11.15)

This gives the derivative of a general function φ(). Eq. (11.15) could alsohave been obtained by differentiating the expansion (11.4) directly.

The second derivatives can also be defined

∂

∂i

∂

∂ jφ() = φ2( j,i)

+1

2!

l

φ3( j,i,l)l − 1

2!

k

φ3( j,k,i)k + . . .

= φ2( j,i) +l

φ3( j,i,l)l + . . . . (11.16)

Because the φi are antisymmetric this gives immediately

∂ 2

∂i∂ j φ() +

∂ 2

∂ j∂iφ() = 0 , (11.17)

or ∂

∂i,

∂

∂ j

= 0 . (11.18)

Thus we have, in particular,∂ 2

∂2i

= 0 . (11.19)




This relation implies that there is no inverse to differentiation. This can

be seen by multiplying the defining equation for the inverse

∂

∂ε

∂

∂ε

−1

φ() = φ() (11.20)

from the left by ∂ ∂ε

. This gives

∂

∂ε

∂

∂ε

∂

∂ε

−1

φ() = 0 ·

∂

∂ε

−1

φ() =∂

∂εφ() (11.21)

and thus the inverse does not exist.

11.1.2 Integration

Integration in the space of Grassmann variables can, therefore, be definedonly in an operational sense. This is achieved by the following relations

di = 0 di i = 1 . (11.22)

The symbols di obey the commutation relations

di, dk = di, k = 0 , (11.23)

but note that d is not an infinitesimal interval and, in particular, not amember of the Grassmann algebra of the i.

Integration over Grassmann variables has thus the same effect as dif-ferentiation. The integral in (11.22) is the only nonvanishing integral overfunctions of i. It has the property of translational invariance

d φ(1)() =

d φ(1)( + α) = φ1 (11.24)

with the definition of φ(1) in (11.5); α is also a Grassmann number.For Grassmann variables we can also define a δ-function. In 1 dimension

we have

−

d( − )φ() = −

d( − )(φ0 + φ1)

= φ0 + φ1 = φ() . (11.25)

We can thus identify the δ function as

δ( − ) = −( − ) ; (11.26)




note that δ(

−) is an odd function. There is also a Fourier-representation

for the δ-function

δ( − ) = −( − ) =

dζ [1 − ζ ( − )] =

dζ e−ζ (−) ; (11.27)

here (11.8) has been used.

11.1.2.1 Multiple integrals

In multiple integrals the integration variable first has to be commuted to aposition next to the integration measure. For example, we get

di d j e−ij =

di d j(1 − i j)

= 0 −

di d j i j = +

di

d j j

i

=

di i = 1 . (11.28)

We now determine the Jacobian J that appears when we perform a lineartransformation

= O (11.29)

of the integration variables in n dimensions; O is a general matrix. J isdefined by

d1 . . . dn 1 . . . n =

d1 . . . dn J (O)1(O)2 . . . (O)n (11.30)

=

d1 . . . dn J O1αO2β . . . Onν αβ . . . ν .

Note that this equality holds for general matrices O. The product of theGrassmann variables α, . . . , ν vanishes, if any of the indices appears twice.Thus only the permutations of 1, . . . , n survive. We can therefore bring eachof them into the ordered form by writing

αβ . . . ν = (−)P 1 . . . n , (11.31)

where (−)P is the sign of the permutation. We thus have d1 . . . dn 1 . . . n =

d1 . . . dn J

P

(−)P O1α . . . Onν 1 . . . n

=

d1 . . . dn J 1 . . . n det(O) . (11.32)




On the other hand, we can also evaluate the integrals over the ’s directly.

This gives d1 . . . dn 1 . . . n =

d1 . . . dn 1 . . . n , (11.33)

because of (11.22). Comparison with (11.33) yields

J = [det(O)]−1 . (11.34)

Note that this is the inverse of the ordinary result!

11.1.2.2 Gaussian integrals

We next consider the Gaussian integral

I (n) =

d1 . . . dn e−T M (11.35)

where M is an antisymmetric (n×n) matrix with ordinary c-number elementsand is a vector with n elements. Expanding the exponential function weobtain

e−T M = 1 − T M +

1

2!

T M

2 − · · · . (11.36)

For simplicity let us consider the case n = 2 first. We have for I (2)

I (2) =

d1d2e−T M =

d1 d2

1 − T M +

1

2

T M

2 − . . .

= 0 −

d1 d2 T M +1

2

d1 d2

T M

2 − . . .

≡ −I 1(2) + I 2(2) − . . . (11.37)

For the first integral we obtain

I 1(2) =

d1 d2 T M =

d1 d2 iM ij j

= d1 d2 (1M 111 + 1M 122 + 2M 211 + 2M 222)

=

d1 d2 (1M 122 + 2M 211) , (11.38)

because the integrands of the other two terms contain squares of Grassmannvariables which vanish. We therefore have (because M is antisymmetric)

I 1(2) = −M 12 + M 21 = −2M 12 = −2

det(M ) . (11.39)




The second integral in (11.37) is

I 2(2) =

d1 d2

T M

2(11.40)

=

d1 d2 iM ij j kM kll .

The indices i,j,k and l all have to be different, because otherwise the productof the s vanishes. This, however, is impossible because we have only 2different variables; the integral I 2(2) thus vanishes. The same holds for allhigher order terms in (11.37).

In summary we obtain for the integral (11.37)

I (2) = −I 1(2) = +2

det(M ) = 222

det(M ) . (11.41)

For 3 variables we obtain immediately d1 d2 d3 e−

T M = 0 , (11.42)

because the exponent is quadratic in and, therefore, the series expansioninvolves only even powers. The term of second order in gives 0, becauseof

d = 0; the term of fourth order also vanishes, because, with only three

variables, it has to contain a square of one of them, which vanishes.

A similar reasoning leads one to the conclusion that with four variablesonly the term of fourth order in can contribute. Thus we have

I (4) =

d1 d2 d3 d4 e−T M

=1

2

d1 . . . d4

T M

2

=1

2

d1 . . . d4 iM ij j kM kll

=1

2

4

ijkl=1

M ijM kl d1 . . . d4 i jkl . (11.43)

Here only the terms with all four indices different can contribute. We thusget 4! = 24 nonvanishing terms. Of these 24 terms many are equal to eachother: the antisymmetry M ij = −M ji gives M ijM kl = M jiM lk and M ijM lk =M jiM kl = −M ijM kl and thus a factor 22, the symmetry M ijM kl = M klM ijanother factor 2, so that only 3 essentially different terms remain. We thenhave

I (4) = 4M 12M 34

d1 . . . d4 1234






All the κi anticommute with each other.

With this definition we evaluate a Gaussian integral containing also conjugate Grassmann numbers

Z ∼

d∗1d1d∗2d2 . . . d∗ndn e−†M . (11.50)

The matrix M in (11.50) can now be a general, complex matrix withoutany specific symmetries. In contrast to the c-number integrals discussed insection B.2.1 it does not have to have special properties since convergence of the integrals is guaranteed here by the Grassmann algebra. In order to beable to apply the results obtained above (see (11.45)) to this integral we nowenlarge the general (n

×n)-dimensional matrix M into an antisymmetric

(2n × 2n)-dimensional matrix N , so that the exponent in (11.50) can berewritten

†M =n

i,j=1

∗iM ij j =1

2

2nk,l=1

κkN klκl =1

2κT Nκ . (11.51)

For example, for the case n = 2 the matrix N reads

N =

0 M 11 0 M 12

−M 11 0 −M 21 00 M 21 0 M 22

−M 12 0 −M 22 0

. (11.52)

One then gets in general

det(N ) = [det(M )]2 . (11.53)

Using now (11.45) gives

Z ∼

dκ1 . . . dκ2n e−12κ

T Nκ =

det(N ) , (11.54)

and with (11.53) we finally get

Z ∼

d∗1d1 . . . d∗ndn e−†M = det(M ) . (11.55)

This result can also easily be obtained for a diagonal matrix M . Even though

the integral is a 2n-dimensional one, the matrix M on the rhs is only n-dimensional. Note again that the matrix M here is a general complex matrix.

A slightly more general form follows when we add linear terms in theexponent and write an expressively imaginary M . Then we obtain

n

d∗ndn e−i†M+iη†+i†η = det(iM ) eiη†M −1η . (11.56)

In all these formulas for Grassmann integrals the determinant appears inthe numerator in contrast to the Gaussian integrals over ordinary numbers(B.22).






η(x), η(x)

=

η(x), η(x)

=

η(x), η(x)

= 0

η(x), ψ(x) = η(x), ψ(x) = η(x), ψ(x) = 0 . (11.61)

While (11.60) is an obvious generalization of our earlier results obtainedin Chap. 5 for scalar fields, its actual (numerical) evaluation may not seem tobe straightforward, because of the Grassmann nature of the fields. In orderto define more stringently what is actually meant by the path integral in(11.60) we can proceed as in chapter 5.1 for scalar fields and Fourier-expandthe fields. This is carried through in detail in [LEE]. For our present purposeit is enough to mention that the fields Ψ and Ψ can be expanded in terms of Grassmann numbers α and α such that

Ψ(x) =α

φα(x)α(t)

Ψ(x) =α

φα(x)α(t) , (11.62)

where the φα are x-dependent Dirac spinors (u( pα, sα)ei pα· x or v( pα, sα)e−i pα·x,for positive and negative energies, respectively, see [LEE], p. 517-521).

Equation (11.62) just amounts to an x-dependent change of the basis fromα, α to the Grassmann numbers Ψ and Ψ. The fermionic action, e.g. forfree fields, then reads in terms of the Grassmann numbers (with k ≡ γ µkµ)

S =

Ψ (i ∂ − m) Ψ d4x

=αβ

φαα (i ∂ − m) φβ β d

4x , (11.63)

and the path-integral integration measure is really

DΨ DΨ =k

j

dΨ(xk, t j)dΨ(xk, t j)

∼α j

dα(t j) dα(t j) . (11.64)

Here the first index (α) denotes the Grassmann generator, the second ( j) thetime interval. The last step here was possible because of the orthonormalityof the functions φα and φα.

11.2.1.1 Feynman propagator for fermions

In the case of free boson fields we succeeded in writing the generating func-tional as a term involving only space-time integrals over the sources (see




(6.11) and (6.39)). This had the advantage that the two-point function could

be directly read off from that expression.We achieve here the same for fermions by using (11.56). This gives

Z 0[η, η] = Z 0

DΨDΨ ei

d4x(L+ηΨ+Ψη)

= Z 0 det(iM ) eiηM −1η , (11.65)

where the matrix M is obtained by identifying

−i†M ←→ i

d4xΨ(x)(i ∂ − m)Ψ(x)

(11.66)

= − i d4x d4y Ψ(x) (i ∂ y + m)δ4(x − y)Ψ(y)

after partial integration. Since Z 0[0, 0] = 1, we have Z 0 = [det(iM )]−1. Wethus get

Z 0[η, η] = ei η(x)M −1(x,y)η(y)d4xd4y . (11.67)

The inverse operator appearing here can be obtained as at the end of section (6.1.3)

M (x, z)M −1(z, y) d4z =

(i ∂ z + m) δ4(x − z)

M −1(z, y) d4z

= − δ4(x − z) (i ∂ z − m) M −1(z, y) d4z

= − (i ∂ x − m) M −1(x, y)!

= δ4(x − y) .

(11.68)

The last equation is fulfilled by

M −1(x, y) = − (i ∂ + m) DF(x − y) , (11.69)

since

−(i ∂ − m)(i ∂ + m)DF(x − y) = (2+ m2)DF(x − y)

= − δ4(x − y) , (11.70)

because of (6.7). Here we have used γ µ, γ ν = 2gµν .We therefore have now the desired result. For free fields the normalized

generating functional is

Z 0[η, η] = e−i η(x)S F(x−y)η(y) d4xd4y , (11.71)




with the Feynman propagator for fermions

S F(x − y) = (i ∂ + m)DF(x − y) (11.72)

being the propagator of the Dirac equation

(i ∂ − m) S F(x − y) = δ4(x − y) . (11.73)

Note that S F is a (4 × 4) matrix because of the Dirac matrix structure of γ µ.For S F we can also find an integral representation

S F(x − y) = (i ∂ + m)1

(2π)4

d4k

e−ik(x−y)

k2 − (m − i)2

= 1(2π)4

d4k e−ik(x−y) k + m

k2 − (m − i)2

=1

(2π)4

d4k e−ik(x−y) 1

k − (m − i)(11.74)

because of (k + m)(k − m) = k2 − m2.Note that S F(x−y) is not symmetric under exchange x ↔ y, while DF(x−

y) is. However, it is antisymmetric under an operation that corresponds inDirac theory to simultaneous x ↔ y exchange and hermitean conjugationsince

γ 0S F(y−

x)†γ 0 = S F(x−

y) , (11.75)

which can easily be derived with the help of the relation

γ 0γ †µγ 0 = γ µ . (11.76)

As we have discussed in Sect. 6.1.2 for scalar fields the Feynman propaga-tor moves positive-energy solutions forward in time and those with negativeenergy backwards. The same is true for the propagator for fermions as canbe shown following steps very similar to those leading to (6.21).

11.2.2 Green’s Functions

The n-point functions for fermions can now be obtained, as in the case of bosons, as functional derivatives of the functional Z [J ]. For this purpose wemodify the definition (8.3) such that the presence of two fermion fields, Ψand Ψ, is taken into account.

We define the 2n-point function for fermions by

Gα,β,...,2ν (x1, x2, . . . , x2n) (11.77)

=

1

i

2n δ2nZ

δη2ν (x2n) · · · δην +1(xn+1)δην (xn) · · · δηα(x1)

η=η=0

,




where Z [η, η] is given by (11.60)

Z [η, η] = Z 0

DΨ DΨ ei

d4x [L+η(x)Ψ(x)+Ψ(x)η(x)] . (11.78)

Here the order of the functional derivatives with respect to η and η, respec-tively, is fixed by convention (see discussion below). Note that the Green’sfunction now depends also on the Dirac spinor-indices, in addition to thespace-time coordinates.

As in (8.2) for bosons, the Green’s function can be written as a vacuumexpectation value of a time-ordered product of field operators

G(x1, x2, . . . , x2n) (11.79)

=

DΨ DΨ Ψ(xn) · · · Ψ(x1)Ψ(x2n) · · · Ψ(xn+1) eiS DΨ DΨ eiS

= 0|T Ψ(xn)Ψ(xn−1) · · · Ψ(x1) ˆΨ(x2n) · · · ˆΨ(xn+1)

|0 ,

where S denotes the action. The special ordering of fields in (11.79) is aconsequence of our ordering of the derivatives in (11.77); it ensures that thereis no additional phase present. This can be seen as follows: In calculating thederivatives with respect to η no phase appears, because first the derivative

is anticommuted through all field operators already pulled down from theexponential, then acts without a further sign change on the exponential andthen is anticommuted back, thus creating the same phase backwards as onthe way forward. The extra phase (−) that appears from the derivative withrespect to η due to the necessary reordering in the exponent is cancelled bythe same phase caused by anticommuting the derivative operator δ/δη withall the Grassmann fields Ψ and Ψ, that were already pulled down, and byreordering them.

Thus, the n-point Green’s function, originally introduced as a functionalderivative of a generating functional, can in general also be defined as the

vacuum expectation value of a time-ordered product of field operatorsˆΦ(x)that can describe either boson or fermion fields

G(x1, x2, . . . , x2n) = 0|T Φ(xn)Φ(xn−1) · · · Φ(x1) Φ(x2n) · · · Φ(xn+1)

|0 .

(11.80)The only difference between boson and fermion fields appears in the reorder-ing caused by the T operator; here a ”(−)” sign appears whenever a pairof fermions changes its order whereas there is no such sign when the bosonoperators are reordered.




2-point function. The two-point function is given by

Gαβ (x1, x2) = − δ2 Z [η, η]

δηβ (x2)δηα(x1)

η=η=0

. (11.81)

Using (11.65) for the generating functional of the free theory we obtain

Gαβ (x1, x2) =

DΨDΨ Ψα(x1)Ψβ (x2)ei Ld4x DΨDΨ ei

Ld4x

(11.82)

Here α and β are Dirac-Spinor indices.This expression can, as for bosons, be written as an expectation value

over a time-ordered product of field operators

G(x1, x2) = 0|T Ψ(x1) ˆΨ(x2)

|0 (11.83)

where, now for fermions, we have

T Ψ(x1) ˆΨ(x2)

=

Ψ(x1) ˆΨ(x2)

− ˆΨ(x2)Ψ(x1)for

t1 > t2

t2 > t1. (11.84)

Note the extra minus sign compared to the boson case. This sign appearsbecause of the Grassmann nature of the fermion fields when we reorder thefields as in the developments leading to (3.47).

We could have obtained the two-point function also from (11.71). Thisgives

Gαβ (x1, x2) = i[S F(x1 − x2)]αβ . (11.85)

Again, as in scalar field theory, the 2-point function is – up to a phase – justequal to the propagator of the relevant equation of motion.

The propagator of our fermionic theory is thus just the inverse of theoperator appearing between the fields Ψ and Ψ in the Lagrangian

L = Ψ(i ∂ − m)Ψ

G(x, y) = i(i ∂ − m)−1 = iS F(x − y) . (11.86)

For boson fields this was also the case (see the discussion at the end of section6.1.3)

L = −1

2φ(2+ m2)φ

G(x, y) = − i(2+ m2)−1 = iDF(x − y) . (11.87)

We can thus read off the propagator directly from the Lagrangian 1.

1The factor 1/2 in the Lagrangian for the bosons is a special feature of the unchargedfield and would not be there if we were working with charged fields.



Chapter 12

INTERACTING FIELDS

So far we have treated only the case of free fermion fields. In the case of aninteracting theory, expression (8.44) can directly be taken over for the caseof fermions so that we have

Z [η, η] = Z 0 e−i

d4xV ( 1iδδη, 1i

δδη ) e−i

η(x)S F(x−y)η(y)d4xd4y . (12.1)

Here the interaction V is defined by the Lagrangian of the interacting theory

L = L0 − V (Ψ, Ψ) . (12.2)

If the theory involves both boson and fermion fields and their couplings,i.e. if the Lagrangian is given by

L = LΨ0 + Lφ0 − V 1(Ψ, Ψ) − V 2(φ) − V int(Ψ, Ψ, φ) , (12.3)

then Z [J,η, η] is given simply by (cf. (8.44))

Z [J,η, η] = Z 0 e−i

d4xV int( 1iδδη, 1i

δδη, 1i

δδJ )

×e−i

d4xV 1( 1iδδη, 1i

δδη)e−i

d4xV 2( 1

iδδJ )

×e−i η(x)S F(x−y)η(y)d4xd4y

×e−

i2 J (x)DF(x−y)J (y)d4xd4y . (12.4)

This functional can generate all the n-point functions of the interacting the-ory and these can again be graphically represented in the form of Feynmandiagrams.

12.1 Feynman Rules

The Feynman rules of Sect. 9.1 can be extended by the following rules (inmomentum space)

144



CHAPTER 12. INTERACTING FIELDS 145

Figure 12.1: Fermion-boson vertex (see (12.6)). The solid line denotes thefermion, the dashed one the boson. The dot denotes the interaction vertex.

1) each internal fermion line gives a factor

ik − (m − i)

. (12.5)

The fermion lines carry now an arrow that points into the directioninto which the fermion’s charge flows. This is a consequence of the factthat the fermion propagator is no longer symmetric in its arguments(cf. (11.75)) so that we have to give the direction of motion explicitly.For fermions we deal with two fields, ψ and ψ, that carry differentcharges. In a Feynman graph, in which time runs from left to right,particle lines are always rightwards directed whereas antiparticle linesmove leftwards. In a loose way of speaking the fermion lines move fromψ to ψ, because ψ creates particles and ψ annihilates them. Since theopposite holds for antiparticles their lines run leftwards, reflecting thefact that the Feynman propagator propagates them backwards in time.

In a coupled theory a new class of diagrams can appear because of thecoupling of fermion and boson fields. For example, a coupling term of theform

V int = gΨΨφ = gΨαΨαφ , (12.6)

the so-called Yukawa coupling, will generate vertices of the form shown inFig. 12.1. We, therefore, have a new Feynman rule dealing with fermion-

boson vertices:

2) each boson-fermion vertex gives a factor −ig. If the coupling terminvolves other bilinear forms of the Dirac spinors, e.g. Ψγ µΨ, then anadditional factor γ µ would appear at each vertex.

12.1.1 Fermion Loops

A change in the rules appears when we consider fermion loop insertions inthe boson propagator, e.g. the graph shown in Fig. 12.2, where the solid




(x α) (y β )

Figure 12.2: Fermion loop contribution to the boson propagator.

line denotes the fermions and the dashed line the bosons. Such a term isobviously of second order in the fermion-boson coupling constant (indicated

by the two vertices) and contributes to the bosonic 2-point function. Itcan thus be generated by the second order term in the expansion of (12.4),properly generalized to fermion fields. Because the graph in Fig. 12.2 is one-particle irreducible only the true second order term (∼ S 2 in (8.50),(8.51))can contribute. This term has the structure

iS [η, η, J ] = . . .

+1

2e−iS 0[η,η,J ]

(−i)

d4x V int

1

i

δ

δη(x),

1

i

δ

δη(x),

1

i

δ

δJ (x)

2

e+iS 0[η,η,J ]

+ . . . . (12.7)

Here S 0[η, η, J ] is given by

iS 0[η, η, J ] = − i

2

d4v d4w J (v)DF(v − w)J (w)

− i

d4x d4y η(x)S F(x − y)η(y) . (12.8)

We now remember that only those terms in the generating functional cancontribute to the boson propagator in Fig. 12.2 that are of second order inthe bosonic current J and of zeroth order in the fermionic currents η and η.Therefore, the only contributing term in (12.7) with the interaction (12.6) is

iS loop2 = − (−ig)2

2e−iS 0 (12.9)

×

d4x d4yδ6

δηα(x)δηα(x)δJ (x)δηβ (y)δηβ (y)δJ (y)e+iS 0

=(−ig)2

2e−iS 0

d4x d4y

d4v d4w

×

DF(x − v)J (v)DF(y − w)J (w)δ4

δηα(x)δηα(x)δηβ (y)δηβ (y)

e+iS 0 ,




after performing the derivative with respect to J .

We now concentrate on the fermionic derivative (for notational conve-nience we drop here the index F for the propagators)

δ3

δηα(x)δηα(x)δηβ (y)

δ

δηβ (y)e−i

η(x)S (x−y)η(y) d4x d4y

=δ2

δηα(x)δηα(x)

δ

δηβ (y)(−i)

S βγ (y − y)ηγ (y) d4y e−i

···

=δ

δηα(x)

δ

δηα(x)

− iS ββ (y − y)

− S βγ (y − y

)ηγ (y

) d4

y ηδ(x

)S δβ (x

− y) d4

x e

−i ···=

δ

δηα(x)

− S ββ (y − y)

S αγ (x − y)ηγ (y) d4y

⊕

S βγ (y − y)ηγ (y) d4y S αβ (x − y) + O(ηη)

e−i ···

. (12.10)

Here the symbol ⊕ denotes a ‘plus’ sign that is due to the Grassmann natureof the source functions and would have been opposite, if we had workedwith bosons instead of fermions. The terms denoted by O(ηη) stand forexpressions that involve products of the two source functions and would thus

be linear in η or η after the last functional derivative has been taken. Sincethe n-point function is given by a functional derivative at vanishing source,these terms do not contribute to the 2-point function we are after.

Performing now the last derivative gives for the expression (12.10)

= [−S ββ (y − y)S αα(x − x) ⊕ S βα(y − x)S αβ (x − y) + O(ηη)]

× e−i η(x)S F(x−y)η(y) d4x d4y . (12.11)

The sign of the second term would have been opposite, if we had workedwith bosons.

We now combine this result with (12.9). Since we are interested in partic-ular in the fermion loop insertion to the boson propagator shown above, weneed to construct a bosonic 2-point function with no external fermion lines.Therefore, only those terms of the fermionic part (12.11) can contribute tothis graph that contain no sources η or η, except in the exponential. Allother terms ∼ ηη would vanish after setting η = η = 0.

Disregarding the vacuum contributions (first term in (12.11)) we thushave as the only term of (12.11) that contributes to the loop diagram

⊕ tr [S F(y − x)S F(x − y)] . (12.12)




Inserting this result into (12.9) gives for the relevant part of S 2

iS loop2 = ⊕ (−ig)2

2

d4x d4y d4v d4w tr [S F(y − x)S F(x − y)]

× DF(x − v)DF(y − w)J (v)J (w) (12.13)

and, correspondingly, for the 2-point function

Gloopboson(x1, x2) = − δ2(iS loop

2 )

δJ (x1)δJ (x2)(12.14)

=

(−

ig)2 d4x d4y DF(x1

−x)tr[S F(x

−y)S F(y

−x)] DF(y

−x2) .

When we retrace this detailed calculation we find that the positive sign in(12.12) and thus the negative sign in (12.14) is due to the fact that η and ηare Grassmann fields.

Thus, we get the additional Feynman rule, which holds in general,

3) Any fermionic loop graph is associated with an additional (−) sign anda trace over the Dirac indices.

12.2 Wick’s TheoremIn Sect. 8.2.1 we had expressed the n-point function for bosons G(x1, . . . , xn)in terms of a symmetrized product of n/2 2-point functions (Wick’s theorem).The same can now be done also for the general theory that involves bosonsand fermions.

Mixed n-point functions. First, it is clear that boson and fermion fieldscommute; the corresponding field operators act in different Hilbert spaces sothat the groundstate expectation value separates into a product of vacuumexpectation values over fermion and boson operators separately, e.g.

0|Ψ(1)φ(2)φ(3) ˆΨ(4)|0 = 0|Ψ(1) ˆΨ(4)|00|φ(2)φ(3)|0 . (12.15)

We, therefore, need to formulate here Wick’s theorem only for the fermions.

Wick’s theorem for fermions. We now derive a simple method, knownas Wick’s Theorem that we encountered for bosons already in Sect. 8.2.1, forthe evaluation of a vacuum expectation value of a time-ordered product of free field operators. In that case we can use the form (11.67) of the generating




functional for the evaluation of the functional derivative. Performing first the

functional derivatives with respect to η on that expression gives1

δ2n

δη(x2n) · · · δη(xn+1)δη(xn) · · · δη(x1)e−i

η(x)S F(x−y)η(y) d4xd4y |η=η=0

=δn

δη(x2n) · · · δη(xn+1)

(−i)n

S F(xn − y)η(y) d4y

×

S F(xn−1 − y)η(y) d4y · · ·

S F(x1 − y)η(y) d4y

× e

−i η(x)S F(x−y)η(y) d4xd4y n=η=0 . (12.16)

Now the differentiation with respect to η is performed. Since the wholeexpression has to be taken at η = η = 0, only the prefactors of the exponentialcan contribute. This gives

= (−i)nP

(−)P S F(x1 − x p2n)S F(x2 − x p2n−1) . . . S F(xn − x pn+1) , (12.17)

where the sum over P runs over all permutations of the indices n + 1, . . . , 2n.The sign of the permutations is such that (−)P = +, if always the outermost

operators in (11.77) are combined into the 2-point function, i.e. if (1, 2n),(2, 2n − 1) · · · (n, n + 1) are combined. The Dirac indices, that have not beenwritten down here, have to be combined in the same way.

Inserting this result into the definition of the Green’s function in (11.77)gives

G(x1, x2, . . . , x2n) =

1

i

2n δ2nZ

δη(x2n) . . . δ η(xn+1)δη(xn) . . . δη(x1)

= inP

(−)P S F(x1 − x p2n) · · · S F(xn − x pn+1) .

(12.18)

Equation (12.18) is Wick’s theorem for fermions. Written as a vacuum ex-pectation value of time-ordered field operators it reads

0|T Ψ(xn) · · · Ψ(x1) ˆΨ(x2n) · · · ˆΨ(xn+1

|0 (12.19)

= inP

(−)P 0|T Ψ(x1) ˆΨ(x p2n)

|0 · · · 0|T

Ψ(xn) ˆΨ(x pn+1)

|0 .

1To facilitate the notation the Dirac indices are not written down here.




Wick’s theorem for fermions (12.18) has a form that is analogous to that for

bosons (8.20). The essential difference is the appearance of the sign of thepermutation that reflects the antisymmetric of fermionic states. In additionthe degeneracy factor 1/2n is missing here because for fermions we have nolonger a symmetry under exchange: S F(x − y) = S F(y − x) (see (11.75)).

This form also exhibits clearly the important consequence of using anti-commuting Grassmann fields for the fermions. The exchange of any two of the coordinates appearing in the fields Ψ or of any 2 in the fields Ψ givesa “−” sign to the Green’s function. In particular, the 2-particle, 4-pointGreen’s function has the property

Gαβγδ(x1, x2, y1, y2) =−

Gβαγδ(x2, x1, y1, y2) =−

Gαβδγ (x1, x2, y2, y1)

= Gβαδγ (x2, x1, y2, y1) . (12.20)

This implies that for G not to vanish we must have x1 = x2 and y1 = y2, if all the Dirac-indices are the same, thus reflecting the Pauli principle.

12.3 Removal of Degrees of Freedom:

Yukawa Theory

In the beginning of Chap. 12 we have first met a coupled fermion-boson theory

with a simple Yukawa coupling. We now consider a theory that containsfree fermions coupled to a self-interacting boson field by means of a Yukawainteraction. Its Lagrangian reads

L = −1

2φ2φ − V (φ) + ψ (i ∂ − mF) ψ − gψψφ

= − 1

2φ2φ − V (φ) + ψ (i ∂ − mF − gφ) ψ . (12.21)

The term V (φ) may contain both the mass term and further selfinteractionterms. Since the coupling constant g here is dimensionless we expect that thistheory is renormalizable (cf. the discussions in Sect. 10.2). The second line of

(12.21) shows that the Yukawa coupling contributes the fermion mass and,for mF = 0, can even generate the whole mass of a fermion. This mechanismplays an important role in the theory of electroweak interactions.

In the preceding sections we have discussed the perturbative treatmentof such a Lagrangian. Here we consider now an alternative treatment that isbased on the fact that the generating functional is given by

Z [η, η, J ] = N

Dψ Dψ Dφ exp

i

L[ψ , ψ , φ] + Jφ + ηψ + ψη

d4x

.

(12.22)




It is thus quadratic in the fermion fields and, therefore, of Gaussian form in

the fermion fields. The path integral over the fermionic degrees of freedomcan thus be performed. According to (11.56) and (11.65) this gives

Z [η, η, J ] = N

Dφ det(M ) eiηM −1ηe−i

[ 12φ2φ−V (φ)−Jφ] d4x (12.23)

with (see (11.66))

M = i ∂ − m∗ with m∗ = mF + gφ ; (12.24)

the factor ‘i’ before M i/n the determinant has been absorbed into the nor-malization constant N . Note that the fermion determinant M is a function

of φ. Equation (12.23) is completely equivalent to (12.22).We now concentrate on processes that contain only bosons on the externallegs and thus require only bosonic n-point functions for the calculation of transition rates etc. We can then set η and η to zero. In doing so we giveup the possibility to calculate any n-point function with fermionic externallegs because these are obtained as functional derivatives with respect to η.Our procedure thus amounts to keeping only fermionic loops in our theoryfor the scalar field φ. More specifically, as we will see below, our procedurekeeps fermionic effects only on the one-loop level.

Using now (6.55)det(M (φ)) = etrlnM (φ) (12.25)

gives

Z [J ] = N

Dφ e−i

[12φ2φ+V (φ)−Jφ] d4x+trlnM (φ) . (12.26)

Equation (12.26) shows that now formally all the fermion degrees of freedomhave been removed. Alternatively, we could also say that we are now workingwith a new, effective action for the bosonic sector

S B = −

φ1

22φ + V (φ)

d4x − itrln(i ∂ − m∗(φ)) . (12.27)

The trace here has to be performed over Dirac indices and over space-time

coordinates (or, equivalently, in momentum space). The corresponding effec-tive Lagrangian density reads

LB = −1

2φ2φ − V (φ) − iTrln(i ∂ − m∗(φ)) , (12.28)

where we have usedtr = Tr

d4x ; (12.29)

Tr is a trace over the Dirac (and possibly other internal) indices. In this form,which is still exact, the original Lagrangian (12.21) is said to be bosonized .




12.3.1 Perturbative Expansion

We now develop a perturbative treatment of this bosonized Lagrangian thatuses an expansion around the groundstate of the bosonic sector.

The true groundstate (vacuum) of the bosonic sector is, of course, deter-mined by the potential V (φ), but it may not coincide with the minimum of this potential. Nevertheless, we may take here the constant, translationallyinvariant field φ0 which minimizes V as our reference field. In order to nor-malize the action, and thus the physics, to this state we subtract the actionbelonging to φ0. This gives (with m∗

0 = mF + gφ0)

S B =− 1

2φ2φ

−V (φ)d4x

−itrln(i

∂

−m∗) + i trln (i

∂

−m∗

0

) ,

(12.30)where a constant term has been dropped. Next we use the identity

trln(i ∂ − m∗) = trln(i ∂ − m∗)† = tr ln (−i ∂ − m∗) (12.31)

which holds because of the hermiticity of the Dirac operator i ∂ − m∗; herewe remember from our discussion in Sect. 6.1.3 that the determinant of anoperator is just the product of its eigenvalues. We thus get

trln(i ∂ − m∗) =1

2trln[(i ∂ − m∗) (−i ∂ − m∗)]

= 12

trln2+ m∗2 − ig ∂φ

. (12.32)

Thus the effective bosonic action becomes

S B = −

1

2φ2φ + V (φ)

d4x (12.33)

− i

2trln

2+ m∗2 − ig (∂φ)

+

i

2trln

2+ m∗

02

= − 1

2φ2φ + V (φ)

d4x − i

2trln

1 +

m∗2 − m∗0

2 − ig ∂φ

2+ m∗0

2

and the Lagrangian density now reads

LB = −1

2φ2φ − V (φ) − i

2Tr ln

1 − m∗2 − m∗

02 − ig ∂φ

−2− m∗0

2

= − 1

2φ2φ − V (φ) − i

2Trln(1 − D(0)W (x)) (12.34)

with the operator D and the field-dependent quantity W

D = − 1

2+ m∗0

2 and W = m∗2 − m∗0

2 − ig ∂φ . (12.35)




Note that D is nothing else than the Feynman propagator (cf. (6.7)). So far,

no approximations have been made and the theory is still exact; note thatthe reference field is in principle arbitrary.

The action is now in a form that is suitable for a perturbative treatment.We thus expand the logarithm (− ln(1 − x) = x + x2

2+ . . .) in (12.33) and

obtain

−tr ln(. . .) = tr∞n=1

1

n(DW )n (12.36)

The trace in (12.36) indicates that each term in the expansion contains oneclosed fermion loop (with n vertices W coupling to the field φ). Equation(12.36) is therefore called the one-loop expansion , it represents a perturbative

expansion of the effective action. Integrating out the fermion degrees of freedom has thus led to an effective theory that takes all fermionic one-loopdiagrams into account. The expansion is expected to converge the morerapidly the closer the actual field φ is to the chosen field φ0 and the smallerthe gradients of φ are.

Our aim in the following developments is to write the trace term as anintegral over a space-time density which could then be interpreted as a cor-rection term to the Lagrangian density, thus yielding an effective Lagrangianwhich contains the effects of the fermions on the one-loop level. In order tobe able to do so we have to separate the terms under the trace into a product

of operators that are local in x and p. In this case the trace can be separated

tr [F (ˆ p)G(x)] = Tr d4 p

(2π)4 p|F (ˆ p)G(x)| p

= Tr d4 p

(2π)4

d4 p

(2π)4

d4x

d4x

× p|F (ˆ p)| p p|xx|G(x)|xx| p= Tr

d4 p

(2π)4F ( p)

d4x G(x) . (12.37)

The expressions in (12.36) are, however, not in a form that allows directapplication of (12.37), because the field φ, and thus also W , is dependent onx and the propagator D acts on it. We thus first reorder each term such thatall propagators are moved to the left. To do so we start from the identity

W D = DW + DD−1, W

D (12.38)

and iterate this equality by applying it again to the quantity W = [D−1, W ]

W D = DW + DD−1, W

D . (12.39)




Combining now (12.38) and (12.39) gives

W D = DW + D2D−1, W

+ D2

D−1,

D−1, W

D . (12.40)

This procedure can be continued and yields the operator identity

W D = DW + D2D−1, W

+ D3

D−1,

D−1, W

+ . . . . (12.41)

The basic commutator appearing here is given byD−1, W

= [−2, W ] = −2W − 2 (∂ µW ) ∂ µ . (12.42)

Since W contains a derivative of the field this term is of order O((∂ µφ)3) andthe next higher order term in (12.41) would be of order O((∂ µφ)4). Equation(12.41) thus represents an expansion in terms of gradients of the field. Wealso see that it contains increasing powers of D so that we expect to encounteronly a finite number of ultraviolet, i.e. for large momenta, divergent terms.

We now use the identity (12.41) to rewrite the expansion (12.36). Up toterms of second order in W the expansion reads

−trln(. . .) = tr(DW ) +1

2tr(DWDW ) + O(W 3) (12.43)

= tr(DW ) +1

2tr(D2

W 2

) +1

2tr D3 D−1

, W W + O(W 3

) .

Note that all the terms on the rhs have the momentum dependence and thex-dependence separated so that – according to (12.37) – they can be writtenas an integral over the momentum times a spatial integral over a Lagrangiandensity. Eq. (12.43) is an expansion in terms of powers of W .

We now discuss the terms in the expansion (12.43) and start with theterm tr(DW ). This is explicitly given by

tr(DW ) = tr

D

m∗2 − m∗

02 − ig ∂φ

. (12.44)

Sincetr γ µ = 0 (12.45)

andm∗2 − m∗

02 = g

2mF (φ − φ0) + g

φ2 − φ2

0

(12.46)

we get

tr(DW ) = g tr

D2mF (φ − φ0) + g

φ2 − φ2

0

. (12.47)




The trace operation can now be performed. This gives

tr(DW ) = 4g d4k

(2π)4

1

k2 − m∗0

2

d4x

2mF (φ(x) − φ0) + g

φ2(x) − φ2

0

,

(12.48)where we have used (12.37) with the trace over the Dirac indices Tr = 4. Thisterm is now in the desired form, i.e. it is given by a space-time integral overfields. However, the momentum-integral appearing here actually divergesquadratically as power counting shows us.

Also the next term in (12.43) diverges. It is given by

tr(D2W 2) = g2Tr d4k

(2π)4

1k2 − m∗

022 (12.49)

×

d4x

2mF (φ(x) − φ0) + g

φ2(x) − φ20

− i ∂φ(x)

2

and diverges logarithmically.The next term in (12.43)

tr

D3D−1, W

W

= tr(D)3 (−2W − 2(∂ µW )∂ µ) W

(12.50)

and all higher order terms are clearly ultraviolet convergent because they

contain higher and higher powers of the propagator D.This means that we have only two divergent terms ((12.48) and (12.49)) in

the expansion (12.36). These can be removed by introducing proper counterterms. We can thus define a finite, renormalized effective Lagrangian densityin the boson sector of our theory by just adding the two divergent terms(12.48) and (12.49) to the Lagrangian

LB(x) = −1

2φ(x)2φ(x) − V (φ(x))

−

i

2

Tr ln 1 −D m∗2(x)

−m∗

02

−ig

∂φ(x)

− i2g d4k

(2π)4

2mF (φ(x) − φ0) + g (φ2(x) − φ20)

k2 − m∗0

2

− ig2

4Tr

d4k

(2π)4

1k2 − m∗

022 (12.51)

×

2mF (φ(x) − φ0) − i ∂φ(x) + g

φ2(x) − φ20

2

.




Writing this expression in a somewhat more compact form gives

LB = − 1

2φ(x)2φ(x) − V (φ(x)) − i

2Tr(1 − DW )

− i

2Tr

DW +

1

2(DW )2

(12.52)

= − 1

2φ(x)2φ(x) − V (φ(x)) +

1

2Tr

D3

D−1, W

W

,

where Tr denotes a trace over the Dirac indices only and where it is un-derstood that we have to use the expansion (??) for the Tr ln term. Thetwo counter terms have the structure of terms already present in the original

boson Lagrangian; they just remove the first two terms in the Taylor expan-sion of the logarithmic term in (12.34). The Yukawa theory is thus indeedrenormalizable, provided the potential V (φ) is ‘well-behaved’ and containsat most terms of fourth order in φ.



Chapter 13

PATH INTEGRALS FOR

GAUGE FIELDS

The fundamental interactions of nature like, e.g. the electroweak and thestrong interactions are described by so-called gauge field theories. In thesetheories the action is invariant under certain space-time dependent unitary“gauge” transformations. As a consequence for each field configuration thereare infinitely many others that can all be obtained by a gauge transformationfrom the original one. The physics remains unchanged under such transfor-mations.

Special problems arise when such gauge theories are quantized. The rea-son is quite easy to see in the framework of path integrals. If we consider afield Aµ, the path integral would be naively written down as

DA eiS [Aµ] (13.1)

where the action S is a functional of Aµ. The integration measure stands fora straightforward generalization of the measure used for scalar fields

DA ∼ x

n j=1

4µ=1

dAµ(x, t j) (13.2)

in the limit n → ∞ (cf. (5.11)); the measure so defined is gauge invariantas we will see later in this chapter. For a fixed field Aµ there are infinitelymany other fields, connected to Aµ by gauge transformations, that leave theaction S invariant and thus all give the same contribution to the integrand.Thus the path integral as written down here cannot converge.

In this chapter we develop methods to deal with this difficulty. After adiscussion of gauge-invariance in electrodynamics we summarize the proper-ties of a more general class of gauge-field theories and then develop the pathintegral description for them.

157



CHAPTER 13. PATH INTEGRALS FOR GAUGE FIELDS 158

13.1 Gauge invariance in Abelian theories

We start this section with a discussion of gauge invariance in electrodynamicsby considering a free field configuration without currents.

For the electromagnetic field the free Lagrangian is given by

L = −1

4F µν F µν = −1

4(∂ µAν − ∂ ν Aµ) (∂ µAν − ∂ ν Aµ) (13.3)

and is clearly invariant under a gauge transformation

Aµ(x) −→ Aµ(x) = Aµ(x) + ∂ µ(x) . (13.4)

Here (x) is an arbitrary differentiable function of space-time. The corre-sponding equation of motion for the fields Aµ is

∂ ν F νµ = ∂ ν (∂ ν Aµ − ∂ µAν ) = (δµν 2− ∂ ν ∂ µ)Aν = 0

−→ (gµν 2− ∂ µ∂ ν ) Aν = 0 , (13.5)

which is also gauge invariant. If we now fix a gauge, e.g. the Lorentz gauge∂ µAµ = 0, we obtain the free wave equation for Aµ

2Aµ = 0 , (13.6)

which is no longer gauge invariant.

We now proceed to obtain the propagator for the electromagnetic field.As mentioned at the end of section 11.2.2 the propagator of the theory couldbe read off from the term in the Lagrangian that is quadratic in the field.The Lagrangian (13.3) can indeed also be converted into a form quadratic inthe fields by writing

L d4x = −1

4

d4x (∂ µAν − ∂ ν Aµ)(∂ µAν − ∂ ν Aµ) (13.7)

= − 1

4

d4x [ (∂ µAν )(∂ µAν ) − (∂ µAν )(∂ ν Aµ)

−(∂ ν Aµ)(∂ µAν ) + (∂ ν Aµ)(∂ ν Aµ)] .

By partial integration we obtain L d4x =

1

4

d4x (Aµ2Aµ − 2Aµ∂ ν ∂ µAν + Aµ2Aµ) (13.8)

=1

2

d4x Aµ (gµν 2− ∂ µ∂ ν ) Aν ,

so that L can also be written as

L =1

2Aµ (gµν 2− ∂ µ∂ ν ) Aν . (13.9)




This Lagrangian is very similar to that of a scalar field theory (4.31) so

that we expect the generating functional to have the form

Z [0] = ei

d4xL . (13.10)

Since the Lagrangian is quadratic in the fields it can be treated in analogyto the developments in Sect. 6.1.3. According to our normal procedure (cf.(11.87)) we thus expect to obtain the propagator of the electromagnetic fieldas

D−1µν ∼ gµν 2− ∂ µ∂ ν . (13.11)

The problem arises when we realize that this operator D does not exist.

We see this by applying its inverse (13.11) to an arbitrary four-gradient ∂ ν G

D−1µν ∂ ν G = (2∂ µ − ∂ µ2) G = 0 . (13.12)

Thus, D−1 has a zero eigenvalue and, loosely speaking, its inverse D is in-finite. This infinity is linked to that in the path integral over gauge fieldsdiscussed at the start of this chapter. This can be seen by a straightforwardapplication of (B.18) to (13.10) which gives the equivalent of (6.33) withdet(A) = 0 in the denominator1. In other words: we are integrating over toomany degrees of freedom when the gauge condition is not taken into account.

Therefore, we expect that one way to overcome this difficulty is to fix agauge, for example by imposing the Lorentz gauge condition

∂ ν Aν = 0 . (13.13)

If we then consider only potentials that fulfill this condition and integrateonly over them, we expect the path integral to be well-behaved.

This expectation is based on the fact that we can find a Lagrangian thatincorporates the gauge condition (13.13) at the price of loosing its manifestgauge invariance

L = −1

4F µν F µν

+ LGF

= −1

4F µν F µν − 1

2λ(∂ µAµ)2 . (13.14)

This Lagrangian, that leads for λ = 1 to the wave equation (13.6) as we willshow below, contains a so-called gauge-fixing term as a quadratic constraint,coupled in by means of a Lagrange-multiplier ( 1

2λ). This additional term

1Remember here that the determinant of an operator is given by a product of itseigenvalues.






gives the so-called Feynman gauge, whereas λ = 0 describes the Landau-

gauge. The Feynman gauge is the one that we are used to from classicalelectrodynamics; in it the equations of motion for the electromagnetic field(13.16) read

2Aν = 0 . (13.20)

This is the equation of motion that we obtained earlier for the Lorentz gauge.

Generating functional. Now that we have constructed a propagator forthe electromagnetic field, we can write down the generating functional as

Z [J ] = DA ei

(L+LGF +J µA

µ)d4x (13.21)

where DA =4µ=1 DAµ and LGF is the gauge-fixing term

LGF = −1/(2λ)(∂ µAµ)2 . (13.22)

The gauge-fixing term suppresses the contributions of fields that do not fulfillthe Lorentz gauge condition to the path integral.

We will see a little later, in section 13.3, that this functional can actuallybe rewritten into a form that exhibits explicitly the integration over the gaugedegree of freedom. For a free field theory the generating functional can thenbe obtained as usual (cf. (B.18))

Z [J ] = e−i2 J µ(x)Dµν(x−y)J ν(y) d4xd4y , (13.23)

and the two-point function is, as usual (see (11.86), (11.87)), found to be

Gµν (x, y) = iDµν (x − y) (13.24)

The path integral (13.21) does not suffer from the problems mentioned atthe start of this section. Now the matrix in the quadratic term of L (13.9)can be inverted and the propagator exists. Furthermore, all fields that differfrom a specific field just by a gauge transformation that leads out of thecovariant Lorentz gauge have a higher action and thus contribute less to thepath integral.

The action itself is gauge-independent, whereas the integral over thegauge-fixing term and the source terms do depend on the gauge. Thus, thegenerating functional itself is gauge dependent and, consequently, also theGreen’s functions obtained from it by functional derivatives. All of the gaugescorresponding to the various values of λ, however, lead to the same physicsbecause in calculating a transition amplitude Dµν will couple to conservedcurrents only which fulfill ∂ µ j

µ = 0 → kµ jµ(k) = 0. As a consequence the

λ-dependent term in (13.19) always drops out and the physically observabletransition amplitudes are all gauge independent.




13.2 Non-abelian gauge fields

In more general gauge field theories one demands invariance of the Lagrangianunder the transformation of the particle fields (see [MOSEL], chapter 7)

ψ(x) → ψ(x) = U (x)ψ(x) . (13.25)

The particle fields are described by N -component spinors ψ where the com-ponents represent the internal degrees of freedom. Here U is a unitary, localgauge transformation

U (x) = e−ik(x)T k (13.26)

where the T k

are the generators of a Lie group. They are Hermitian sothat the Lie group is unitary and have vanishing trace so that the generatedLie algebra is special , i.e. all group elements are represented by (N × N )matrices that have unit determinant. The latter implies that the matrixrepresentations of the generators T k have vanishing trace. The generatorsform a Lie algebra with the defining commutation relations

[T l, T m] = if lmnT n (l = 1, 2, . . . N 2 − 1) . (13.27)

The generators are normalized such that

tr T lT m = 12 δlm (13.28)

in the fundamental representation; the f lmn in (13.27) are the completelyantisymmetric structure constants of the group. Another important matrixrepresentation of the generators is the so-called regular representation

T jkl

= −if jkl . (13.29)

The special unitary groups in N dimensions with these properties arecalled SU (N ). Since the generators do not commute the group SU (N ) is a

non-Abelian one. Only for the case N = 1 there is only one generator andthe group becomes Abelian, called U(1).Invariance under the transformation (13.25) can be achieved only if the

derivative ∂ µ is replaced by the so-called covariant derivative

Dµ = ∂ µ + igT lAlµ (13.30)

where g is the coupling constant and Alµ is a vector (gauge) field; the super-script l labels the internal degrees of freedom. This “minimal substitution”fixes the interaction between particle and gauge fields.




The gauge transformation of these fields must then be given by

Aµ → AU µ = UAµU −1 − i

gU∂ µU −1 , (13.31)

if the Lagrangian in the particle sector is required to be invariant. Here wehave introduced the fields Aµ in (13.31) as contractions of the generatorswith the fields Akµ

Aµ = T kAkµ . (13.32)

The change of A under an infinitesimal gauge transformation is given by

δA jµ = − lf ljkAkµ +1

g ∂ µ j = il T kAkµ jl +1

g ∂ µ j

=1

g(Dµ) j (13.33)

where we have used the regular representation of SU (N ), i.e. (T j)kl

= −if jkl.Also for non-Abelian gauge fields the free Lagrangian is given by

L = −1

4F lµν F lµν , (13.34)

but now the field tensor also carries the quantum numbers of the internalgauge group and must have a more complicated structure

F lµν = ∂ µAlν − ∂ ν Alµ − gf lmnAmµ Anν (13.35)

in order to be invariant under the gauge transformation (13.31). Gaugeinvariance also requires that the gauge fields are massless, since a mass termof the form m2A · A would clearly violate gauge invariance.

The field tensor F lµν can also be contracted with the generators to forma scalar in the intrinsic space (F µν = F lµν T l). These scalars change in thegauge-transformation just as under any unitary transformation

F µν → F U µν = U F µν U −1 . (13.36)

Equation (13.34) together with (13.35) show that the free gauge field isselfinteracting, with terms ∼ A3 and ∼ A4 in L . These terms are generatedby the non-Abelian piece of F µν and are absent for an Abelian theory (f lmn =0). Notice also that the coupling constant g, determining the interactionstrength, is the same for gauge field–gauge field (13.35) and gauge field–particle (13.30) interactions.




The complete ‘generic’ Lagrangian of a non-Abelian gauge field theory

with many distinct fermion fields then reads

L = −1

4F lµν F l µν +

f

ψf (iγ µDµ − mf )ψf

. (13.37)

The sum over f runs over all fermions. The fact that the coupling constantappears both in the fermion–gauge field coupling and in the gauge field–gaugefield coupling has an important consequence: since there is a common gaugefield for all fermions, all these different fermions must couple with the sameg to the gauge fields. This universality is a special property of non-Abeliangauge field theories.

An example for the relevance of a non-Abelian gauge field theory is pro-vided by Quantum Chromodynamics (QCD), the theory of the strong in-teractions between quarks. Here the relevant internal degree of freedom of the quarks is color and the symmetry group is SU (3). Another example isprovided by the theory of the electroweak interaction where the gauge groupis U (1)×SU (2); in this case the relevant degrees of freedom are the electricalcharge (U (1)) and the weak isospin (SU (2)). Both of them will be discussedin some more detail in Chapt. 14.

Abelian gauge field theories. For the case of QED U (x) = e−i(x), T k =

1 and f lmn

= 0; thus U is Abelian. Indeed (13.31) then reduces to the wellknown gauge transformation of electrodynamics

Aµ → Aµ = Aµ +

1

g∂ µ (13.38)

and also the field tensor assumes its well known form. The case of QED isthus contained as a special case in the developments in this section. Since inthis case there is no coupling constant g in the field tensor F , universalitydoes not hold for non-Abelian gauge field theories.

13.3 Path integralsWe now want to develop a path integral formulation also for non-Abelianfields. We start again with the naive expression for a path integral overgauge fields

Z [0] =

DA eiS [A] . (13.39)

Here the action is that of the original theory without any gauge-fixing term

S =

d4x L (13.40)




and the integration measure

DA stands for

DA =x

j

N 2−1l=1

4µ=1

d Alµ(x, t j) (13.41)

(cf. (5.11)). This integral, of course, diverges as in the case of the Abeliantheory because, as discussed in the last section, the path integral runs overall field configurations, i.e. both over essentially distinct ones and those thatdiffer only by gauge transformations from each other. It is then tempting to just add a gauge-fixing term to L as we have done in (13.21).

While this problem of the gauge-equivalent potentials appears for both

Abelian and non-Abelian gauge field theories, there is another difficulty thatis specific for non-Abelian theories. The path integral in the form givenabove, in which an integration only over the fields appears, was shown inchapter 1 to be valid only for quadratic Hamiltonians with constant coeffi-cients in front of the kinetic term. This, however, is no longer the case fora non-Abelian gauge theory, where the field tensor F depends not only onthe derivatives of the fields, but also on the fields themselves. Effectivelythis leads to a kinetic energy term in which the coefficient of the momentum-dependent terms depends on the fields themselves. When considering thenonrelativistic analogon of this case in section 1.3.2 we have started from the

Hamiltonian formulation of the path integral. There we have found that anew factor involving the squareroot of the coordinate dependent coefficientappears under the path integral (see (1.54)). It is therefore natural that inthe treatment of non-Abelian gauge field theories we would also have to startfrom the Hamiltonian representation of the path integral (1.32) (see [Lee] fora detailed treatment). At the expense of some mathematical rigor, however,the concept of a path integral over the fields only can be maintained and thisis what we are going to do in this chapter.

In the following we will now develop a method to split the path integralup into two factors, one containing the divergent part and the other one beingconvergent. The divergent part will be seen to be an infinite constant thatdrops out when we work with the normalized functional. The divergence ishere – as in the Abelian case – connected with the presence of zero eigenvaluesof the matrix in the quadratic term of the gauge field action. It is just thosezero eigenvalues that we have to factor out by integrating only over thosefields that correspond to nonzero eigenvalues. We start by dividing up thetotal configuration space of all possible Aµ(x) into equivalence classes. Eachof these equivalence classes contains all the possible fields AU µ that can beobtained from an initial field Aµ(x) by gauge transformations. The integrandof (13.39) is constant along the fields AU µ , i.e. in an equivalence class. This




means that the path integral (13.39) is proportional to an infinite constant

which is just the “volume” of the gauge group U . The physically relevantpart of the path integral then just runs over the essentially distinct fieldsAµ(x). This restriction can be achieved by requiring that the fields thatcontribute to the integration all fulfill a gauge condition which we write inthe form

F l(Akµ(x)) = 0 . (13.42)

Examples of such gauge conditions are given by the

Lorentz gauge F k = ∂ µAkµ = 0

Coulomb gauge F k =

· Ak = 0

Axial gauge F k = Ak3 = 0Temporal gauge F k = Ak0 = 0 .

(13.43)

The challenge is now to include these conditions in the path integrals suchthat gauge-invariance of the theory is maintained.

With this aim in mind we now define a functional integration over theelements of the symmetry group SU (N ), determined by the transformation

U (x) = e−ikT k , (13.44)

and denote the integration measure by DU ; in Fig. 13.1 this integration over

the gauge group corresponds to an integration along the solid lines. Thisintegration measure is gauge invariant

DU = D(U U ) = DU (13.45)

because of the group-property of the transformations U ; this can also be seenby noting that we integrate over all gauge transformations U . To factor outan integral over DU is the aim in the following paragraphs.

With this integration measure we now consider the integral

M−1[Alµ] =

DU δ [F l(Akµ

U )] (13.46)

Here F l(Akµ) = 0 is a gauge-fixing condition (13.42). The superscripts l, kstand for internal degrees of freedom of the symmetry group SU (N ); fornotational convenience, we will write them out only when necessary. Theδ-functional in (13.46) is really a product of Dirac δ-functions at each space-time point (x, t j). AU is the gauge-transformed field. Since the δ-functionalin (13.46) restricts the gauge transformations to U ≈ 1 it suffices to consideronly infinitesimal gauge transformations

U (x) ≈ 1 − il(x)T l with 1 . (13.47)




In this case the measure can be written as

DU = x,j

N 2−1l=1

dl(x, t j) (13.48)

in a discrete space-time representation. For the measure over infinitesimaltransformations in (13.48) we have

U = U U ≈ (1 − ilT l)(1 − ilT l) ≈ 1 − i(

l+ l)T l

= 1 − ilT l , (13.49)

so that the Jacobian from to = + is

det

δl

δk

= 1 . (13.50)

Thus the measure (13.48) is indeed gauge-invariant.

A xµ

( )

Aµ

AµA

U

U

Figure 13.1: Lines of constant integrands. The horizontal axis contains thephysically distinct fields, the vertical one the gauge-transformation degree of freedom. The solid lines depict the fields within a given equivalence class;the dashed line represents the gauge condition F (AU µ ) = 0 for an Abeliantheory, the dotted line for a non-Abelian theory exhibits the appearance of Gribov copies for large fields.

The condition F (AU µ ) = 0 then defines a surface that crosses that of eachequivalence class (dashed line in Fig. 13.1); we assume that for an initial field




Aµ this is the case for one group element. This is certainly the case for an

Abelian gauge field theory, like e.g. QED. There, the gauge transformationU reads (cf. (13.38))

Aµ(x) −→ AU µ (x) = Aµ(x) +1

g∂ µ(x) (13.51)

and the gauge condition becomes

F (AU µ ) = ∂ µAµ(x) +1

g∂ µ∂ µ(x) = 0 (13.52)

where we have chosen as an example the covariant gauge ∂ µ

Aµ = 0. Eq.(13.52) represents an inhomogeneous wave equation for . With the boundarycondition Aµ(x) → 0 as |x| → ∞ it specifies a unique (x). As Gribov haspointed out this is no longer the case if we are dealing with a non-Abelianfield theory. In this case there can be several equivalent fields and all fulfillthe same gauge condition (open points in Fig. 13.1). Thus, in this case onealso has to remove these so-called Gribov copies from the path integration.However, since these copies appear only at large field strengths we can neglectthem in a perturbative treatment around either a vanishing or a classical fieldconfiguration.

The integral (13.46) is gauge invariant. This can be seen by writing the

Jacobian as an integral and starting from a gauge-field AU

M−1[AU

µ ] =

DU δ

F

AU

µ

U . (13.53)

We now change the variables from U to U = UU and get, because theintegration measure is gauge-invariant,

M−1[AU

µ ] =

DU δF

AU

µ

= M−1[Aµ] , (13.54)

since U is just an integration variable that could be renamed into U .

M−1

is thus indeed gauge-invariant.We can now transform the original path integral (13.39) by inserting the

1 from (13.46). This gives

Z [0] =

DA eiS [A] =

DA M[Aµ]

DU δF

AU µ

=1

eiS [A] (13.55)

with the actionS =

d4x L . (13.56)




We now perform a gauge transformation AU µ→

Aµ. Since S ,

Mand

DA

are invariant under this transformation the integrand no longer depends onU and we can take the integral over U out of the remaining integral

Z [0] =

DU

DA M[Aµ] δ [F (Aµ)] eiS . (13.57)

This functional determines our theory in the F (Aµ) = 0 gauge. The integral DU (13.58)

is just the infinite factor we wanted to pull out from the expression.

Since this factor is cancelled when normalizing the functional, we canfrom now on work with the generating functional

Z [J ] =

DA M[Aµ] δF l

Akµ

ei

d4x (L+J lµAl µ) . (13.59)

This functional can be used for a perturbation theoretical treatment becauseit is finite. This was not possible for the original functional (13.39) whichdiverges because of the integration over physically equivalent gauge fields.

13.3.1 Gauge Fixing

For a derivation of the perturbation theory rules, i.e. the Feynman rules, thefunctional (13.59) is still not easy to handle because of the presence of thegauge-fixing δ-functional. As a guideline for a simplification we can use theeasier example of an Abelian Theory, treated in Sect. 13.1. There we hadsucceeded in (13.21) to write down a meaningful generating functional thatdid not contain such a δ-functional, but instead a gauge-fixing term in theLagrangian.

Taking this result as guideline we, therefore, in the following paragraphsreformulate Z [J ] into a more tractable form. To do this we consider some-

what more specialized gauge conditions

F m(Aµ(x)) − C m(x) = 0 , (13.60)

where C m(x) is an arbitrary function of space-time. Note that for given A,C m and F m this condition determines a gauge-transformation. The definitionof M[A] changes correspondingly

M−1[Aµ] =

DU δF m(AU µ (x)) − C m(x)

. (13.61)




Suppose, we looked at a field for which the argument of the δ-functional

with given C m(x) vanishes and then looked at this argument again, but witha different function C m(x), so that the argument no longer vanishes. Wecould then find a gauge transformation that makes the argument of the δ-functional zero again. Since the rest of the integral is invariant under gaugetransformations (and we integrate anyway over all gauge transformations)the value of the integral does not change. Thus, the FP determinant M doesnot change under this replacement and is even independent of the specificfunction C m(x)). Then, obviously, also

Z [J ] = DA

M[Aµ] δ [F m(Aµ)

−C m(x)] ei

LJ d

4x (13.62)

(with LJ = L + J · A) is independent of C (x). We are thus free to changethe integrand by a weighting factor

e−i2λ

[C m(x)]2d4x (13.63)

and integrate functionally over the function C ; this will only change thenormalization of the integral. We then get

Z [J ] =

DC

DA M[Aµ] δ [F m(Aµ) − C m(x)] ei

(LJ − 12λ (C (x))2) d4x .

(13.64)Now the functional integration over C can be performed since C appearsin the δ-functional. This gives

Z [J ] =

DA M[Aµ] ei

(LJ − 12λ

[F m(Aµ)]2) d4x . (13.65)

In this form L has again picked up a ‘penalty potential’ (see discussion after(13.14) on p. 159)). This generating functional is finite due to the presenceof the gauge fixing term and can, therefore, be used for a derivation of theFeynman rules for perturbation theory. As we have seen in the last sectionfor the case of QED the divergence of the original path integral was linkedto the zero eigenvalues of the matrix in the quadratic part of the gauge-fieldaction so that no one-particle propagator could be defined. The vanishingeigenvalues were connected to the infinite integration over the gauge trans-formation degree of freedom. Here now this infinity has been removed byfixing the gauge and the propagators are finite.

Notice that the appearance of the gauge-fixing term in the action is notthe only change. In a non-Abelian gauge-field theory an additional factor,the FP determinant M, appears in the integrand of the path integral. Innon-Abelian theories the FP determinant in general depends on the fields and




thus cannot be pulled out of the path integral. The deeper reason for this

difference of Abelian and non-Abelian theories lies in the different structureof the field tensors in both theories. Due to the presence of the selfinterac-tion term in the non-Abelian field tensor (13.35) the kinetic part of the fieldenergy becomes field-dependent and fields and conjugate momenta no longerdecouple in the Hamiltonian. This is then exactly the situation that we haveencountered already in Sect. 1.3.2 in the framework of nonrelativistic quan-tum mechanics where a coordinate dependent term also led to an additionalfactor in the path integral (cf. (1.54)).

While everything else in the integrand of the path integral (13.65) is gaugeinvariant, the source term and the gauge-fixing terms break this invariance.

Thus, the specific rules of the perturbation theory, e.g. the Feynman rules,will depend on the gauge. In particular also the Green’s functions obtainedas functional derivatives of Z [J ] are not gauge invariant. This, however, isno problem since these Green’s functions only appear at intermediate stagesof the calculation. The S matrix elements, which are physical quantities, arethen gauge invariant again. We have seen an example for that in the lastsection where we discussed the QED propagators; in calculating the S ma-trix the gauge-dependent propagators are always contracted with conservedcurrents and the gauge-dependent terms then drop out.

13.4 Feynman Rules

13.4.1 Faddeev-Popov Determinant

It now remains to evaluate M−1[Aµ]. In order to understand the meaning of the integral in (13.46) better and to be able to actually evaluate it we writeit for n discretized space coordinates3

M−1n =

d1d2 . . . dn δn[F (Aµ)] (13.66)

with n = (xn). In order to evaluate this integral we now change the inte-

gration variables from dn to dF . This gives

d1d2 . . . dn det M ij = dF (x1) dF (x2) . . . dF (xn) , (13.67)

where det M is the Jacobian for this transformation of the integral measure.We thus get

M−1n =

dF (x1) dF (x2) . . . dF (xn) δn[F ]

=1

(det M ij)−1 . (13.68)

3To facilitate the notation we drop here the group indices.




The integral is identically equal to 1 so that

Mn is just the Jacobian

Mn = det M ij = det

δF (xi)

δ(x j)

F =0

. (13.69)

We now generalize these equations to the continuum case. Since M[Aµ]is gauge invariant we can always choose a field Aµ that fulfills the conditionF (Aµ) = 0. This allows us to replace the condition F = 0 for the derivativein (13.69) by the simpler condition U = 1. We thus have to consider onlyinfinitesimal gauge transformations

U ≈ 1 − ilT l (13.70)

for the evaluation of the derivative. The gauge condition F (A) can then beTaylor-expanded

F m(AU µ (x)) = 0 +

d4y M ml(x, y)εl(y) + O(ε2) (13.71)

with

M ml(x, y) =δF m

AU µ (x)

δl(y)

|=0 . (13.72)

We thus get for M−1 at a field close to one fulfilling the gauge condition

M−1[Aµ] = DU δ

F m

AU µ

= DU δ

d4y M ml(x, y)l(y)

. (13.73)

For these integrations close to U = 1 we can use the integration measure(13.48) and obtain (see (13.69))

M−1[Aµ] =det M ml(x, y)

−1=

det

δF m(Aλ(x))

δl(y)

=0

−1

. (13.74)

The determinant here, the so-called Faddeev-Popov (FP) determinant , hasto be calculated with respect to both the space-time indices (x, y) and the

N

2

− 1 SU (N ) group indices (m, l). Note that the FP determinant dependsin general on the gauge fields Akµ.We can now replace M[Aµ] in (13.59) by the Faddeev-Popov determinant

(13.74) and obtain as the generating functional

Z [J ] =

DA det

M ml(x, y)

δ[F l(Aµ)] ei LJ d

4x (13.75)

with LJ = L + J · A. This generating functional is the central result of thischapter. The presence of the FP-determinant in (13.75) is a typical featureof non-Abelian gauge field theories.




13.4.1.1 Explicit forms of the FP determinant

Abelian fields. We have seen earlier that the non-Abelian gauge field the-ory contains the Abelian U (1) symmetry group as a special case. We thereforestart by analyzing M for an Abelian gauge field theory such as QED. Weobtain from the definition (13.72)

M (x, y) =δF

AU µ (x)

δ(y)

F =0

=δF

Aµ(x) + 1

g∂ µ(x)

δ(y)

=0

. (13.76)

Choosing, for example, the covariant gauge ∂ µAµ = 0 gives

M (x, y) =1

g

2δ(x)

δ(y)=

1

g2δ4(x − y) . (13.77)

Thus, M is independent of Aµ, and therefore its determinant can be pulledout of the path integral in (13.75), changing only the normalization. Thegenerating functional thus reduces to

Z [J ] =

DA δ [F (Aµ)] ei d4xLJ . (13.78)

In this form the functional contains the free Lagrangian without any gauge-

fixing terms. It is nevertheless finite because of the presence of the δ-functional. Note the difference of this functional to that written down in(13.21); we will discuss this difference in the next Section.

Non-Abelian fields. For a non-Abelian field the first term in the gaugetransformation (13.31) has a more complicated form so that it will in generalcontribute an Aµ-dependent factor. We start by again expanding F (AU µ )around a field that fulfills the gauge condition, i.e. F m(Aµ) = 0 and U ≈ 1(cf. (13.71)). Under Aµ → AU µ = Aµ + δAµ the condition F changes into

F m(AU µ ) = 0 + d4z δF m

(Aλ(x))δAkµ(z)

δAkµ(z)

=1

g

d4z

δF m (Aλ(x))

δAkµ(z)(Dµ(z))k (13.79)

Here we have used (13.33) to replace δA with the covariant derivative

Dµ = ∂ µ + igT nAnµ . (13.80)




Taking now the derivative according to (13.72) we get

M ml(x, y) =δF m (Aλ(x))

δl(y)

=0

=1

g

d4z

δF m (Aλ(x))

δAkµ(z)

δ

δl(y)(Dµ(z))k

=1

g

d4z

δF m (Aλ(x))

δAkµ(z)Dklµ (z)δ4(z − y)

F =0

. (13.81)

This is a more general form for the FP-matrix. Its actual value depends onthe gauge used.

Lorentz gauge. The Lorentz gauge is given by

∂ µAnµ = 0 . (13.82)

The FP-matrix is then obtained from (13.81) as

M ml(x, y) =1

g

d4z

δ

∂ λAmλ (x)


F =0

(13.83)

= 1g d4z δmk ∂ µxδ4(x − z)Dklµ (z)δ4(z − y)

F =0.

By using ∂ µxδ4(x − z) = −∂ µz δ4(x − z) and partial integration we obtain

M ml(x, y) =1

g

d4z δmkδ4(x − z)∂ µzDkl

µ (z)δ4(z − y)

F =0

=1

g

∂ µxDml

µ (x)δ4(x − y)F =0

. (13.84)

Since we have to evaluate this expression for F m = ∂ µAmµ = 0 we can inter-

change the order of the two derivatives and obtain

M ml(x, y) =1

g

Dmlµ ∂ µxδ4(x − y)

. (13.85)

For the case of QED this expression indeed reduces to the form already foundearlier (see (13.77)). This is so because in this case δAµ = 1

g∂ µ in (13.79)

and thus in this case instead of Dµ simply ∂ µ appears in (13.85).




Axial gauge. It is interesting to note that the FP-determinant is inde-

pendent of the fields Aµ not only for Abelian gauge theories, but also in anon-Abelian theory, if the so-called axial gauge is chosen. This axial gaugeis defined by the constraint

n · A = 0 , (13.86)

where n is a fixed four-vector; this form comprises the last two cases in(13.43). In this case we obtain from (13.81)

M ml(x, y) =1

g

d4z

δ

nλAmλ (x)


F =0

= 1g

δmknµ d4z δ4(x − z) δkl∂ zµ + ig (T n)kl Anµ(z) δ4(z − y)F =0

=1

gδmlnµ∂ xµδ4(x − y) , (13.87)

because F = nµAnµ = 0. Thus in this case the FP-determinant is againindependent of Aµ so that it can, as in the Abelian case, be pulled out of thepath integral and put into the normalization.

13.4.2 Ghost fields

We now continue with the discussion of the general non-Abelian case (13.65).The trick now is to use (11.56) to express the Faddeev-Popov determinant byanother path integral over (hypothetical) Grassmann fields η(x). We have

det(iM ) =

Dη Dη e−i ηm(x)M ml(x,y)ηl(y) d4xd4y . (13.88)

Here the Grassmann fields η and η carry the indices of the internal symmetrygroup SU (N ). The generating functional of the theory then becomes (cf.(13.65))

Z [J ] = DA det(M ) e

i (LJ − 12λ

[F m(Aµ)]2)d4x

=

DA Dη Dη ei

(LJ − 12λ

[F m(Aµ)]2− ηMη d4y)d4x . (13.89)

If we remember that the FP determinant owes its existence to the gaugefixing condition, then we see that the introduction of the fields η(x) amountsto a dynamical treatment of the gauge invariance. We therefore expect thatthis treatment generates also new Feynman graphs involving the fields η(x).

In order to be able to calculate their Green’s functions we now supplementthis generating functional with sources also for the so-called ”ghost field” η






developed in Chap. 11 together with the proper source terms. The fermion-

gauge field coupling is mandated by local gauge invariance (cf. (13.30)) andappears through the covariant derivative.

Since the generating functional (13.90) is finite and free of the overcount-ing of gauge fields discussed at the start of this chapter it can be taken asa starting point to derive the Feynman rules for non-Abelian gauge fieldtheories.

Gauge field couplings. For the gauge field and its couplings we obtainthe Feynman-rules by isolating the selfinteraction terms in L. We get

L = −14

F lµν F lµν

= −1

4

∂ µAlν − ∂ ν A

lµ − gf lmnAmµ Anν

∂ µAlν − ∂ ν Alµ − gf lopAoµA pν

= − 1

4

∂ µAlν − ∂ ν A

lµ

∂ µAlν − ∂ ν Alµ

+

1

2gf lmnAmµ Anν


− 1

4g2f lmnAmµ Anν f lopAoµA pν . (13.92)

The first term alone just looks like the Lagrangian of an Abelian field. Itis the only term quadratic in the fields, so that the gauge boson propagatoris just that of the photon found in (13.19), except for the additional SU(N)labels. We thus have for the gauge boson two-point function (cf. (13.24))

1)

l

µ ν

m

k= iDlm

µν =−iδlm

k2 + i

gµν + (λ − 1)

kµkν k2

.

(13.93)Note that D has to be diagonal in the intrinsic quantum numbers be-cause L involves a trace over the group indices.

The second and third line in (13.92) give the cubic self-coupling terms of the gauge field. The cubic term

1

2gf lmnAmµ Anν


= gf lmnAmµ Anν ∂ µAlν (13.94)

can be represented by the graph shown in Fig. 13.2. Its Feynman rules canbe obtained by considering the generating functional of lowest order in the




coupling constant g. Quite analogous to the situation in the φ4 model (cf.

section 9.1, (9.8)) the only term of Z that contributes to the three-pointGreen’s function is that of order O(J 3); it is obtained by just replacing thefields by those generated by the external source. This gives

Z [J ] = −igf lmn

Dmkµλ (x − y)J kλ(y)d4y

Dnkνλ(x − y)J kλ(y)d4y

× ∂ µ

Dlkνλ(x − y)J kλ(y)d4y

+ terms of lower order in J

d4x

× exp− i

2

J jκ(x)D jkκλ(x − y)J kλ(y)d4x d4y

. (13.95)

Here the Latin superscripts refer to the intrinsic SU (N ) degrees of freedom,the Greek subscripts and superscripts are those of the Lorentz group. Thethree-point Green’s function can now be obtained by differentiating Z [J ]

Ghijπρσ(x1, x2, x3) =

1

i

3 δ3Z

δJ hπ(x1)δJ iρ(x2)δJ jσ(x3)

J =0

. (13.96)

This gives

Ghijπρσ(x1, x2, x3)

= igf lmn

d4x

iDmjµσ (x − x3)iDni

νρ(x − x2)∂ µiDlh ν π (x − x1)

+ permutations of external legs

. (13.97)

Equation (13.97) becomes in momentum space (with G = iD)

Ghijπρσ( p,q,k) = igf lmn

Gmjµσ (k)Gniνρ(q)(−i pµ)Glh ν π ( p)

+ permutations of external legs (13.98)= gf hji

pµgνλG jjµσ(k)Giiνρ(q)Ghhλπ( p)

+ permutations of external legs

.

In the Feynman gauge we obtain the gauge field three-point function

Ghijπρσ ( p,q,k) = gf hij−i

k2

−i

p2

−i

q2(13.99)

×

(k − q)π gρσ + (q − p)σ gρπ + ( p − k)ρ gπσ

.




p

q

k

π1h

ρ2i

σ3 j

Figure 13.2: Cubic gauge coupling term. The first symbols at each gluon linegive the Lorentz indices, the second number the lines and the third denotethe SU (N ) labels.

From this expression the vertex factor can be read off. We then have therules

2) The vertex factor for the three-point vertex is

gf hij (k

−q)π gρσ + (q

− p)σ gρπ + ( p

−k)ρ gπσ . (13.100)

3) The four-point vertex in Fig. 13.3 gives similarly the factor

−ig2 [ f his f jks (gπσgρτ − gπτ gρσ) (13.101)

+ f hjs f kis (gπτ gρσ − gπρgστ )

+ f hks f ijs (gπρgστ − gπσgρτ ) ] .

π1h

τ 4k

ρ2i

σ3 j

Figure 13.3: Quartic gauge coupling graph.




λ l

Figure 13.4: Fermion-gauge boson vertex. The fermion line is represented bya solid line.

Fermion-gauge coupling. If there are also particles present, then thegauge bosons couple to them through the minimal substitution (13.30)

Dµ = ∂ µ + igT lAlµ (13.102)

and the Feynman rules for these couplings can directly be obtained from ourconsiderations in Chaps. 8.3 and 11. In particular, any fundamental fermionfield couples to the gauge fields according to the rule

4) The fermion-gauge vertex (Fig. 13.4) for a gauge boson with Diracindex λ and group index l is given by:

−ig (γ λ)βα

T lfi

(13.103)

Ghost couplings. We now discuss the rules for handling those terms inthe Lagrangian that involve the ghost field.

The ghost-part of the Lagrangian reads (cf. (13.91) and (13.85))

Z ghost[ξ, ξ] = Dη Dη e−i (ηMη)d4xd4y+i (ξη+ηξ) d4x

=

Dη Dη

× exp

− i

g

ηm(x)

Dmlµ (x)∂ µxδ4(x − y)

ηl(y)

d4x d4y

+ source terms

=

Dη Dη




× exp−i

g η

m

(x)D

ml

µ ∂

µ

η

l

(x) d

4

x + source terms=

Dη Dη

× exp

− i

g

ηm(x)

δml∂ µ + ig (T n)ml Anµ(x)

∂ µηl(x)

d4x

+ source terms

. (13.104)

Using the regular representation (13.29) for the generators we get for theexponent (without the source terms)

. . . = − i

g

ηm(x)2ηm(x) d4x +

ηm(x)

−if nml

Anµ(x) ∂ µηl(x) d4x

= − i

g

ηm(x)2ηm(x) d4x + if lmn

ηm(x)Anµ(x) ∂ µηl(x) d4x .

After rescaling the ghost field (η → √gη) we are thus left with the following

ghost contribution to the generating functional

Z ghost[ξ, ξ] (13.105)

= Dη Dη exp−i d4

x ηl2

ηl

− gf lmn

An

µ ηm

∂ µ

ηl

+¯ξl

ηl

+ ηl

ξl .

Note that the ghost field couples only to the gauge field and not to any of the other fields, like e.g. fermions, in the theory. Also, we remember thatthe ghost fields η and η cannot appear on external lines because the ghostsfollow the wrong spin-statistics relation. This ensures that the ghosts areintegrated out so that the FP determinant is generated.

The Feynman rules for the ghost fields can now be read off from (13.105).

6) The ghost propagator is given by

l mk = δlm i

k2 + i. (13.106)

Note that the ghost line carries an arrow; this reflects the presence of ghostsand antighosts, because the FP matrix is in general not hermitean. Thenotation is then such that particles always run with the time axis from bottomto top and antiparticles run against it. More loosely speaking, the ghost lineruns from η to η.




k

p

q

σ3 j

1h

2i

Figure 13.5: Ghost-gauge coupling. The wavy line denotes a gauge bosonwhereas the dashed line denotes the ghost. The first symbol on the gaugefield line (σ) denotes the Lorentz index, the second just numbers the particleand the third ( j) gives the SU(N) label.

Finally, according to (13.105) the ghost can couple to the gauge fieldas depicted in Fig. 13.5. The vertex factor is obtained by considering theghost-ghost-gauge boson three-point function. As for the gauge three-pointfunction just discussed it can be obtained by considering the lowest order (in

the gauge-ghost coupling) expression

Z [ξ, ξ, J ] = −i

d4x−gf lmn

1

i

δ

δJ nµ(x)

1

i

δ

δξm(x)∂ µ

1

i

δ

δξl(x)eiS 0[ξ,ξ,J ]

= − igf lmn

Dnkµν (x − y)J kν (y) d4y

ξkDkm(x − y) d4y

× ∂ µ

Dlk(x − y)ξk(y) d4y

+ terms of lower order in J

d4x eiS 0[ξ,ξ,J ] (13.107)

This generating functional determines the following rule for the gauge boson-ghost coupling vertex:

7) The gauge boson-ghost vertex (Fig. 13.5) has to be associated with afactor

gf hijqσ (13.108)

where q is the outgoing momentum of the ghost. The four-momentumconservation ( p + k = q) is understood here; it is not written explicitly.




In addition, we have, as for real, physical fermions, the rule

8) All ghost loops carry a sign (−1).

We also have to remember that ghosts cannot appear on external lines.



Chapter 14

EXAMPLES FOR GAUGE

FIELD THEORIES

The Feynman rules developed in the preceding chapter are of a ‘generic’nature: they apply to all non-Abelian gauge field theories. The particu-lar physics content of a gauge field theory is determined by specifying thesymmetry group SU (N ).

In the preceding chapter we have referred to Quantum Electrodynamics(QED), Quantum Chromodynamics (QCD) and the theory of electroweakInteractions. In this chapter we, very briefly, summarize the physical contentsand the Lagrangian of these theories.

14.1 Quantum Electrodynamics

Quantum Electrodynamics is the most simple gauge field theory. It de-scribes the interactions between electrons and the electromagnetic field, i.e.the Coulomb and Lorentz forces. Its ingredients are discussed at the end of appendix 4 and in Sects. 13.1 and 13.2. It is based on the U (1) gauge groupand is thus an Abelian theory. Its Lagrangian is given in (4.58) in appendix

4.

14.2 Quantum Chromodynamics

Quantum Chromodynamics is the theory of the strong interactions. Its foun-dations, first discussed in the 60’s, lie in the observation that the observedhadron spectrum can be understood in terms of quarks, mesons being com-posed of quark-antiquark pairs and baryons of three quarks; the quarks carryspin 1/2 and a so-called flavor quantum number. For certain baryons, such

184



CHAPTER 14. EXAMPLES FOR GAUGE FIELD THEORIES 185

as the ∆++, the total wavefunction, i.e. the product of space-part, spin-part

and flavor-part, one obtains in such a scheme ( flavor SU(3)) a completelysymmetric wavefunction for the three quarks, in glaring contradiction to thePauli principle. This problem was solved by the introduction of a new, ad-ditional intrinsic degree of freedom for the quarks, called color , in which thequarks can differ (for a detailed discussion see [MOSEL]). The Pauli principlethen requires a completely antisymmetric wavefunction in this new intrin-sic space. In experiments that produce mesons, i.e. quark-antiquark pairs,through lepton-annihilation the number of colors has been found to be three.The completely antisymmetric color state has therefore been identified withthe completely antisymmetric singlet state of a color SU(3) group. Gauging

this group has led to the non-Abelian theory of Quantum Chromodynamics(QCD).

The Lagrangian of QCD is then very simply given by

LQCD = −1

4F cµν F cµν +

f

[qf (iγ µDµ − mf )qf ] . (14.1)

Here the fundamental triplets of SU (3)C are given by the quark spinors

qf =

qf rqf g

qf b

, (14.2)

which transform according to the fundamental (= lowest-dimensional) rep-resentation of SU (3)C . The indices r, g and b stand for the three colors (red,green and blue), the index f for the flavor. The covariant derivative in (14.1)is given by (cf. (13.30))

Dµ = ∂ µ + igλc

2Gcµ . (14.3)

Here the eight matrices λc are the so-called Gell-Mann matrices, the gener-ators of SU (3), and the fields Gc represent the eight gauge-fields, here calledthe gluon fields, which form an SU (3) octett.

The field tensors F cµν have the general form of a non-Abelian theory(13.35). The gluons thus also interact among themselves. One of the pre-dictions of this theory is, therefore, the existence of so-called glueballs, thatconsist only of gauge-field quanta.

The theory, furthermore, has the property of asymptotic freedom , i.e.its running coupling constant (cf. the discussion at the end of Sect. 10.2.2)becomes larger with decreasing gluon momentum. This leads to a quark-quark potential that at small distances is Coulomb-like, but at large distances




increases linearly, and thus provides an explanation for the fact that free

quarks and gluons do not seem to exist, i.e. that they are confined inside thehadrons.

14.3 Electroweak Interactions

While the Lagrangian of QCD has the rather simple, generic form (14.1), thatof the electroweak interactions is considerably more complicated. Its founda-tions go back to the early seventies when a gauge-field theory was found thatcomprised both ‘classical’ Quantum Electrodynamics and the phenomena of weak interactions, responsible, for example, for the nuclear β -decay.

This theory is based on a product of two gauge-groups, one an AbelianU (1) and the other one a non-Abelian SU (2). For the latter the left-handedparts of the observed fermions, quarks and leptons,

ψL =1

2(1 − γ 5) ψ (14.4)

are assumed to form doublets

Ψk =

ν kL

ekL

or

ukL

dkL

(14.5)

for the left-handed fields of the kth family of leptons and quarks. The right-handed parts

Ψk = ekR or qkR (14.6)

are assumed to form SU (2) singlets. In (14.5) we have used generic abbrevi-ations

ν kek

for

ν ee−

,

ν µµ−

or

ν τ τ −

(14.7)

and

uk

dk for u

d ,c

s or t

b . (14.8)

The U (1) gauge group is very similar to QED, but it is not connectedwith the electric charge, but instead a so-called weak hypercharge y = 2q−2t3

where q is the electrical charge of the fermion and t3 its weak isospin ± 12

,depending on the position of the particle in the two-component spinors (14.7),(14.8), just as in the usual Pauli-spinors.




The complete Lagrangian of this theory is then given by

L = −1

4Gµν · Gµν − 1

4F µν F µν

+ M 2W W †µW µ

1 +ϕ

v

2

+1

2M 2Z Z µZ µ

1 +

ϕ

v

2

+k

Ψkiγ µDµΨk + LY (14.9)

+1

2(∂ µϕ) (∂ µϕ) − 1

2m2H ϕ

2

1 +

ϕ

v+

1

4

ϕ

v

2

.

The boldfaced field tensors are vectors in the internal space and their productcontains a summation over the group indices.

Since there are now two gauge groups present, we also have two gaugefields appearing in the Lagrangian: Gµν for the non-Abelian SU (2)L and F µν for the Abelian U (1)Y. Thus, also the covariant derivative contains both of these fields with two different coupling constants

DµΨk =

∂ µ + igtlW lµ + ig1

2ykBµ

Ψk , (14.10)

where yk is the hypercharge quantum number of Ψk and the tl are the gen-erators of SU (2). The three fields W l and the single field B are the gauge

fields for SU (2)L and U (1)Y, resp. By linearly combining them one obtainsthe 2 physical fields W µ and W †µ, the field Z µ as well as the photon field Aµ.With the covariant derivative (14.10) the first term in the third line describesmassless fermions interacting with the gauge fields in the standard way.

The gauge fields W and B normally have to be massless, since an explicitmass term breaks gauge invariance. Since these fields transmit the weakinteraction the latter would then be long ranged. This, however, presentsan immediate problem because the weak interaction, since Fermi’s theory, isknown to be very short-ranged. A way out of this difficulty is provided bythe so-called Higgs mechanism. The main ingredient of this mass generation

mechanism is the existence of a scalar Higgs field ϕ, yet to be discovered,with non-linear self-couplings (last line in (14.9); these self-couplings lead toa non-vanishing vacuum field. The second ingredient is a Yukawa couplingof the gauge-fields to this Higgs field that generates their masses M W andM Z in the same way as described in Sect. 12.3. Similarly, the term LY in(14.9), generates masses for all the fermions through a Yukawa coupling tothe Higgs field. For the fermions it has the simple structure

L = −gf v√

2ΨΨ

1 +

ϕ

v

, (14.11)




from which the fermion mass can be read off as

mf = gf v√

2. (14.12)



Appendix A

Units and Metric

A.1 Units

While we work in this book in the first 3 chapters, the ’classical’ quantummechanics part, with the usual system of units, it is customary in elementaryparticle physics and field theories to choose the system of units such thatthe resulting expressions assume a simpler form. In particular, instead of working with the usual three mechanical units for length, mass and time oneintroduces instead three basic units for velocity, action and length.

The choice of the velocity of light, c, as the unit for the velocity impliesthatc = 1 . (A.1)

This in turn means that in such a system length and time have the samedimensions and are equivalent units. With c = 1 the relativistic energy–momentum relation assumes the simple form

E 2 = p2 + m2 . (A.2)

Choosing next the unit of action, h, such that

h = 1 (A.3)connects the dimensions of mass, time and length.

Since time and length are equivalent, mass assumes the dimension of aninverse length; the same holds consequently for the momentum and the en-ergy (A.2). The choice of 1 fm = 10−13 cm as a length unit, for example,leads to masses, momenta and energies all given in fm−1. Another often-usedunit for energy is that of 1 MeV, i.e. the energy that an electron acquireswhen it is accelerated by the voltage of 1 MV. The transformation of the ex-pressions back to the standard MKSA system can be achieved by multiplying

189



APPENDIX A. UNITS AND METRIC 190

all quantities with the appropriate combinations of h and c. A particularly

useful relation for this conversion is

hc ≈ 197.3 MeV fm . (A.4)

A.2 Metric and Notation

All vectors, both ordinary three-component vectors and those in some inter-nal space, are distinguished by boldface italic print in this book.

In relativistic expressions four-vectors are always written as

Aµ = A0, A = A0, A1, A2, A3 (A.5)

Four-vectors with a superscript are called contravariant vectors. Covariant vectors, denoted by subscripts, are then defined by

Aµ = gµν Aν (A.6)

with the metric tensor gµν that reads in Cartesian coordinates in Minkowskispace

1 0 0 00

−1 0 0

0 0 −1 00 0 0 −1

. (A.7)

The metric tensor with superscripts is used to raise the indices of four-vectors;it is thus the inverse of g and is given by

gµν = gµν . (A.8)

The productgµν gνλ = gµλ = 1 (A.9)

is often also abbreviated by the delta-function

δµλ = gµλ . (A.10)

A scalar product of 2 four-vectors is then defined by

A · B = AµBµ = AµBµ = A0B0 − A · B (A.11)

if Aµ and Bµ are defined by

Aµ = (A0, A) , Bµ = (B0, B) . (A.12)



APPENDIX A. UNITS AND METRIC 191

The invariant square of the four-vector is thus given by

A2 = AµAµ = A20 − A2 . (A.13)

Here, as always in the text, the Einstein convention of summing over doubleindices, one lower and one upper, is used. The components of four-vectorsare generally labeled by Greek indices, whereas roman indices are used torefer specifically to the last three (vector) components.

The space-time four-vector is

xµ = (x0, x) = (t, x) , (A.14)

and the four-momentum is given by

pµ = (E, p) . (A.15)

Finally, the components of the contravariant four-gradient are abbreviatedby

∂ µ ≡ ∂

∂xµ=

∂

∂t, −

, (A.16)

so that the four-momentum operator is given by

ˆ pµ

= i∂ µ

, (A.17)

and the Lorentz-invariant d’Alembert operator by

∂ µ∂ µ =∂ 2

∂t2− 2 = 2 . (A.18)

Other important four-vectors are the current

jµ = (ρ, ) (A.19)

and the electromagnetic potential

Aµ =

φ, A

. (A.20)



Appendix B

Functionals

B.1 Definition

Very generally speaking a functional is a mapping from a space of functionsinto the real or complex numbers. For example, the integral +∞

−∞f (x)dx (B.1)

is a real or complex number that depends on the special function f (x) in the

integrand; other functions in general give other values for the integral. Thisdependence of the integral on the function f is called a functional dependenceof the integral which itself is called a functional of f :

F [f ] = +∞

−∞f (x)dx . (B.2)

In order to stress that not a special value of f , but instead the whole functionf is the argument of the functional square parentheses are used here to showthis functional dependence.

In physics one often deals with such functionals. For example, in classical

mechanics the action S =

L(q(t), q(t))dt (B.3)

integrated along a trajectory q(t) plays a special role. The action dependsobviously on the trajectory and is thus a functional integral of it: S = S [q(t)].

The Lagrange equations of motion are derived by investigating the changesof S with the trajectory q(t): the physical trajectory leads to a stationaryvalue of S . To find this stationary value, and thus the physical trajectory,requires taking the derivative of S [q] with respect to q. We thus need todefine also the so-called functional derivative.

192



APPENDIX B. FUNCTIONALS 193

B.2 Functional Integration

A functional integration can be introduced as an integration over a space of functions

df F [f (x)] , (B.4)

which is defined as a limit n → ∞ of an integration over finite n-dimensionalsubspaces. A typical example is given by the path integral for quadraticHamiltonians

K (x, t; xi, ti) = N Dx e

ih

t ti

L(x,x)dt

= limn→∞

m

i2πhη

n+12

(B.5)

× nk=1

dxk exp

i

hηn j=0

m

2

x j+1 − x j

η

2

− V (x j)

.

B.2.1 Gaussian integrals

In this book we are often confronted with the evaluation of path integralsover exponential functions whose exponents are quadratic in the coordinates.

It is thus useful to generalize the Gaussian integral formula+∞ −∞

e−12ax

2

dx =

2π

a(a > 0) (B.6)

to this case.From the definition of the path integral it is obvious that we have to

consider products of such integrals

exp−

1

2

n

k=1

akx2k dx1 dx2 . . . dxn =

(2π)n

nk=1 √ak

. (B.7)

We next assume that the n numbers ak are all positive and form the elementsof a diagonal matrix A. We thus have

det(A) =nk=1

ak (B.8)

andnk=1

akx2k =

nk=1

xkAkkxk = xT Ax , (B.9)




where x is a column vector

x =

x1

x2...

xn

.

Thus (B.7) becomes

e−

12xT Ax dnx =

(2π)n2

det(A)

. (B.10)

So far, we have derived this equation only for a diagonal matrix A. Itis, however, valid for a more general class of matrices. This can be seen bynoting that for each real, symmetric matrix B there exists a real, orthogonalmatrix O that diagonalizes B to A

OT BO = A (O real, orthogonal) (B.11)

or, equivalently

B = OAOT (B real, symmetric) . (B.12)

We thus get e−

12y

T By dny =

e−12y

T OAOT y dny =

e−12x

T Ax dnx

=(2π)

n2

det(A)=

(2π)n2

det(B), (B.13)

where we have substituted y = Ox and have used the fact that the Jacobianof an orthogonal transformation is 1 (because det(O) = 1). The last step ispossible because the determinant of a matrix is invariant under an orthogonal

transformation. Equation (B.10) is thus valid also for general symmetricmatrices with positive eigenvalues; it can also be shown to hold for complexmatrices with positive real parts.

This result can also be extended to more general quadratic forms in theexponent. For a one-dimensional integral of such type we have

+∞ −∞

e−ap2+bp+cd p =

π

aeb2

4a+c , (B.14)




We now assume an n-dimensional integral over such a form where the expo-

nential is given bye−F (x) = e−( 1

2xT Ax+BT x+C ) (B.15)

where A is a real symmetric matrix, B is a vector and C a constant. Webring F (x) into a quadratic form by writing

F (x) =1

2(x − x0)T A(x − x0) + F (x0) (B.16)

where x0 is given byx0 = −A−1B (B.17)

and F 0 = F (x0) = C − 12 BT A−1B. Setting now y = x − x0 gives e−( 1

2xT Ax+BT x+C ) dnx =

e−

12y

T Ay−F 0 dny

=(2π)

n2

det(A)e12B

T A−1B−C . (B.18)

Similar to the Gaussian integral (B.7) the following integral relation,which can be proven by induction from n to n + 1, holds also

+∞ −∞

dx1 . . . dxn expiλ (x1 − a)2 + (x2 − x1)2 + . . . + (b − xn)2

=

inπn

(n + 1)λn

12

exp

iλ

n + 1(b − a)2

. (B.19)

Complex integrals. We can generalize these formulas now to complexintegration by noting that (B.6) can be squared and then be written as

π

a=

e−ax2

dx

e−ay2

dy =

e−a(x2+y2)dx dy . (B.20)

Introducing now the complex variable z = x + iy gives

π

a=

1

2i

e−az

∗zdz∗dz . (B.21)

This can be generalized as before to many coordinates (by replacing orthog-onal matrices by unitary ones). We obtain

e−z

†Az dnz∗ dnz =

e−z∗i Aijzj dnz∗ dnz =

(2πi)n

det(A), (B.22)




where A is a Hermitian matrix with positive eigenvalues.

Another convenient, often used relation in this context is

ln det A = trln A , (B.23)

most easily proven for diagonal matrices. In this relation ln A is defined byits power series expansion

ln A = ln(1 + A − 1) = A − 1 − (A − 1)2

2+

(A − 1)3

3!− . . . . (B.24)

With the help of this relation we have

e−z

†Az dnz∗ dnz = (2πi)ne−tr lnA . (B.25)

B.3 Functional Derivatives

Suppose that F [f ] is a functional of the function f (x). The functional deriva-tive of F is then defined by

δF [f (x)]

δf (y)= limε→0

F [f (x) + εδ(x − y)] − F [f (x)]

ε. (B.26)

For example, let us takeF [f ] = 2f (x) , (B.27)

where 2 is the d’Alembert operator. We get from the definition (B.26)

δF

δf (y)= limε→0

1

ε[2 (f (x) + εδ(x − y)) −2f (x)] = 2δ(x − y) (B.28)

Another example is

F [f ] =

+∞ −∞

f (x)dx . (B.29)

According to the definition just given we have

δF

δf (y)= lim

ε→0

1

ε

[f (x) + εδ(x − y)] dx −

f (x) dx

(B.30)

=

δ(x − y) dx = 1

A second example is given by

F [f ] = ei f (x)xdx (B.31)




with

δF

δf (y)= lim

ε→0

1

ε

ei

[f (x)+εδ(x−y)]xdx − ei f (x)xdx

= limε→0

1

εei f (x)xdx

eiεy − 1

= iy ei

f (x)xdx . (B.32)

A general formula that we will need quite often involves functionals F [J ]of the form

F [J ] =

dx1 . . .

dxn f (x1, . . . , xn) J (x1)J (x2) . . . J (xn) (B.33)

with f symmetric in all variables. Then the functional derivative with respectto J has the form

δF [J ]

δJ (x)= n

dx1 dx2 . . . dxn−1 f (x1, x2, . . . , xn−1, x)

× J (x1)J (x2) . . . J (xn−1) . (B.34)

In our later considerations the functional may also be defined by a powerseries expansion

φ[J ] =

∞n=1

1

n! dx1 . . . dxn φn(x1, . . . , xn)J (x1)J (x2) . . . J (xn) . (B.35)

Then the functional derivative is given by

δkφ[J ]

δJ (y1)δJ (y2) . . . δ J (yk)

J =0

=1

k!

p

φk (y p1, y p2, . . . , y pk) , (B.36)

where the sum runs over all permutations p1, . . . , pk of the indices 1, . . . , k.If we assume that φk is a symmetric function under exchange of any of thecoordinates x1, . . . , xk, then the functional derivative is given by

δkφ[J ]

δJ (y1)δJ (y2) . . . δ J (yk)

J =0

= φk(y1, . . . , yk) , (B.37)

just as in a normal Taylor series.



Appendix C

RENORMALIZATION

INTEGRALS

In Chapt. 10 we encountered divergent integrals in the calculation of higher-order two- and fourpoint functions. To evaluate these integrals analyticallyin n dimensional Minkowski space is the purpose of this appendix. We startwith a discussion of the Gamma function whose properties play a role in theevaluation of the integral and its expansion into n ≈ 4 dimensions.

Properties of the Gamma function. The Gamma function is definedby an integral representation

Γ(x) = ∞

0e−ttx−1 dt ; (C.1)

it is single-valued and analytic everywhere except at the points z = n =0, −1, −2, . . . where it has a simple pole with residue (−1)n/n!. It can thusbe expanded around z = 0

Γ(z) =1

z− γ + O(z) , (C.2)

where γ is known as the Euler-Mascheroni constant (γ ≈ 0.577 . . .). Sincethe Gamma function also obeys the relation

Γ(z + 1) = zΓ(z) = z! (C.3)

we obtain

Γ(−1 + z) =Γ(z)

z − 1≈ −(1 + z)

1

z− γ

= −1

z− 1 + γ . (C.4)

198



APPENDIX C. RENORMALIZATION INTEGRALS 199

It obeys the relation

Γ(z) = (z − 1)Γ(z − 1) ≈ (z − 1)−1

z + 1, (C.5)

where the second part of this equation is correct close to the pole at z = −1.

Evaluation of integrals with powers of propagators in n dimensions.

The typical integral to be evaluated reads

I l =

dnq1

(q2 − m2 + iε)l=

dq

dq01

(q20 − q 2 − m2 + iε)

l . (C.6)

with one timelike (q0) and n−1 spacelike (qk) coordinates; the vector symboldenotes a (n − 1)-dimensional vector q = (q1, q2, . . . , qn−1). l is an integerparameter and m2 a real, positive number (mass squared).

We first consider the integration over q0. The situation here is exactlyas in Sect. 6.1.2. The integrand has its only poles in the second and fourthquadrant of the complex q0 plane at q0 = ± [( q 2 + m2) − iδ] (cf. Fig. 5.1).Since the integral behaves as 1/(q0)2l−1 for large q0 the integration alongthe q0-axis can be closed in the lower half of the complex q0 plane withoutchanging the integral’s value.

According to Cauchy’s theorem this integration contour can now be chan-

ged into one that runs along the imaginary q0 axis and closes the contour inthe right half of the complex q0 plane. Because the contour still encloses thesame pole (the one in the fourth quadrant of the complex plane) the valueof the integral does not change.

This then gives the equality +∞

−∞dq0

1

(q20 − q2 − m2 + iε)

l = +i∞

−i∞dq0

1

(q20 − q2 − m2 + iε)

l (C.7)

since in both cases the contribution of the half-circle that closes the contourvanishes. In the integral on the rhs the integration runs along the imaginary

axis in the complex q0 plane. On that axis q0 is purely imaginary; the inte-grand has thus been analytically continued from the originally purely real q0

to a complex one.Using the transformations (6.22), (6.23)

qn = −iq0 dnqE ≡ dqn d q = −idnq , (C.8)

with real qn we obtain the integral

I l = (−)li

d q +∞

−∞dqn

1

(q2E + m2)l

= (−)li

dnqE1

(q2E + m2)l

(C.9)




with

q2E =

ni=1

qi2 = q 2 + q2

n . (C.10)

(C.8) is just the Wick rotation discussed in Sect. 5.1.1.We now introduce “polar coordinates” by defining an n − 1 dimensional

solid angle element dΩn by the relation

dnqE = qn−1E dqE dΩn . (C.11)

and get for the integral1

I l = (−)li dΩn

∞ 0

qn−1dq 1(q2 + m2)l

. (C.12)

The integral over the solid angle can be analytically performed and yields

dΩn =

2πn/2

Γn2

. (C.13)

This can be seen by considering the n-th power of the Gaussian integral

√πn

= ∞0 dx e

−x2n = dx1dx2 . . . dxne−n

k=1x2k

=

dΩn

∞0

xn−1e−x2

dx =

dΩn1

2

∞0

d(x2)

x2n−2

2 e−x2

=

dΩn1

2Γ

n

2

. (C.14)

Here the integral representation (C.1) of the Gamma function has been usedin the last step.

This gives for the integral

I l = (−)li2π

n2

Γn2

∞ 0

qn−1dq1

(q2 + m2)l . (C.15)

The remaining integral can be evaluated with the help of Euler’s β func-tion

B(x, y) =Γ(x)Γ(y)

Γ(x + y)= 2

∞ 0

dt t2x−1(1 + t2)−x−y , (C.16)

1In order to simplify the notation we are no longer denoting the Euclidean vectors bythe subscript E.




(see Abramowitz (6.2.1)); we obtain

qn−1

(q2 + m2)ldq =

1

2mn−2lB(

n

2, l − n

2) =

1

2mn−2l

Γn2

Γ

l − n2

Γ(l)

. (C.17)

Thus the complete integral is now given by

I l = (−)li2π

n2

Γn2

1

2mn−2l

Γn2

Γ

l − n2

Γ(l)

= (−)liπn2 mn−2l

Γ

l − n2

Γ(l)

. (C.18)

In this expression the dependence of the integral on the dimension n is ex-

plicit; it can analytically be continued to the physical case n = 4 where ithas a pole.





APPENDIX D. GRASSMANN INTEGRATION FORMULA 203

We can now continue with the evaluation of I (n) in (D.4) and get

I (n) =

d1 . . . dn (−)n2

1n2

!

T M

n2 , (D.5)

because only that term in the expansion of the exponental can contribute ina n-dimensional integral that has exactly n factors of i. In detail, we have

I (n) =(−)

n2

n2

!

d1 . . . dn

αM αβ β

n2 . (D.6)

Since M

has the special, block-diagonal form given above (D.2), we haveM αβ = 0, except for α = 2α, β = α − 1 or α = 2α − 1, β = α + 1. Again,there can be no two equal s under the integral, if this integral is to benonzero. We thus have

I (n) =(−)

n2

n2

!

d1 . . . dn

2αM 2α,2α−12α−1

− 2α−1M 2α−1,2α2α

n2

=2n2 (−)n

n2! d1 . . . dn 2α−1M 2α−1,2α2αn2 , (D.7)

since M is antisymmetric. Here the index α still runs from 1 to n/2; n iseven so that the phase disappears in the following.

We now perform the exponentiation

I (n) =2n2n2

!

d1 . . . dn

α

β

. . .ν

n sums

2α−12α · · · 2ν −12ν

× M 2α−1,2αM 2β −1,2β · · · M 2ν −1,2ν . (D.8)

The product of the factors M 2α−1,2α · · · M 2ν −1,2ν just gives

det(M ). Sincethe i appear always pairwise, they can be commuted to normal ordering(n · · · 1) without sign-change; the sums then give (n/2)! times the sameresult. Thus

I (n) = 2n2

d1 . . . dn n · · · 1

det(M )

= 2n2

det(M ) . (D.9)



APPENDIX D. GRASSMANN INTEGRATION FORMULA 204

The last step is possible because we have for each blockmatrix M βα separately

det(M βα) = − (M 2α−1,2α) (+M 2α,2α−1)

= + (M 2α−1,2α)2 , (D.10)

and for det(M )det(M ) =

α

det(M βα) . (D.11)

Using (D.3) we thus get the desired result

d1 . . . dn e−

T M = 2n2

det(M ) . (D.12)

Equation (D.12) corresponds to (B.10) for the boson case. Note that here –in contrast to the bosonic case – the determinant appears in the numerator!





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