overall constants have no physical signi¬cance.

5 A convergence factor in the exponent, ’ φ2 , for security, can be assumed absorbed in the inverse

free propagator.

318 10. E¬ective action

no bearing on the physics they describe, resulting in6

D• eiS0 [•] + i• J .

Z0 [J] = = (10.22)

Since our interest is the real-time treatment of non-equilibrium situations, the closed

time path guarantees the even stronger normalization condition of the generator,

Z0 [J] = 1, provided that the sources on the two parts of the closed time path are

taken to be identical, such as for example is the case for coupling to an external

classical ¬eld.

To treat functional integration over a complex function, we ¬rst consider integra-

tion over the real and imaginary parts of an N -tuple with complex entries and have,

for the multiple Gaussian integral,

’1

A

’ 1 z † Az

†

dz dz e = det , (10.23)

2

2π

where † denotes in addition to transposition complex conjugates, i.e. hermitian con-

jugation. We note the additional square root power of the determinant in comparison

with the Gaussian integral over real variables, Eq. (10.15).

For the case of a complex function ψ(x, t), the functional integral becomes

∞ N,N

Dψ — (x, t)Dψ(x, t) F [ψ — (x, t), ψ(x, t)] = — —

lim dψM dψM F [ψ1 , . . . , ψN ]

N,N ’∞

’∞ M,M =1

(10.24)

and for the Gaussian integral (shu¬„ing again irrelevant constants)

1

—

A’1 ψ

Dψ — (x, t) Dψ(x, t) e’ 2 ψ

i

= (10.25)

DetA

where A is a hermitian and positive de¬nite matrix.

Just as for zero-dimensional quantum ¬eld theory, i.e. quantum mechanics, where

path integrals allow us to write down the solution of the Schr¨dinger equation in

o

explicit form, so functional integration allows us to write down explicitly the solution

of the functional di¬erential equation specifying a quantum ¬eld theory as considered

in Section 9.2.2. In Section 10.2, we show how this is done by introducing the

concept of action and show how it can be used to get a useful functional integral

representation of the full theory. But ¬rst functional integration over Grassmann

variables is introduced in order to cope with fermions.

6 Thenormalization of the free generator is in the canonical or operator formalism of equilibrium

zero temperature quantum ¬eld theory the statement that for a quadratic action the addition of the

coupling to the source does not produce a transition from the vacuum state.

10.1. Functional integration 319

10.1.3 Fermionic path integrals

To treat a fermionic ¬eld theory in terms of path integrals, we shall need to introduce

integration over anti-commuting objects. The most general function of a Grassmann

variable, ·, is (recall Section 9.4) the monomial

f (·) = c0 + c1 · (10.26)

and integration with respect to a Grassmann variable is de¬ned as the linear operation

d· f (·) = c1 (10.27)

or

d· 1 = 0 (10.28)

and

d· · = 1 . (10.29)

Integration with respect to a Grassmann variable, Berezin integration, is thus iden-

tical to di¬erentiation.

We note that the basic formula of integration, that the integral of a total di¬er-

ential vanishes,

df (·)

d· = 0, (10.30)

d·

also holds for Berezin integration as d·/d· = 1. The equivalent is true for the

conjugate Grassmann variable · — (recall Section 9.4).

For a general function of two conjugate Grassmann variables, Eq. (9.45), we then

have according to the de¬nitions of integration over Grassmann variables

’ d· d· — f (·, · — ) = d· — d· f (·, · — ) = c3 . (10.31)

For the basic Gaussian integral for Grassmann ¬elds we have

—

d· — d· ei· A·

= (DetiA) (10.32)

as after transforming to diagonal form

—

i ·M AM M ·M

— — —

d·M d·M e = d·M d·M 1+i ·M AMM ·M

M

M M M

= iAMM = Det(iA) , (10.33)

M

320 10. E¬ective action

where the ¬rst equality sign follows from the property (· — ·)2 = 0 for anti-commuting

numbers (recall Section 9.4), and the second equality sign follows from the de¬nition

of integration with respect to Grassmann variables. Thus the Gaussian integral over

Grassmann variables gives the inverse determinant in comparison with the case of

complex functions.7

10.2 Generators as functional integrals

In the previous chapter we showed how all the diagrammatics of a theory, non-

equilibrium situations included, could be captured in a generating functional, ex-

pressing the whole theory in terms of a single di¬erential equation. The Green™s

functions were obtained by di¬erentiating the generating function, thereby obtain-

ing the equations of motion for all the Green™s functions. We now introduce the

functional integral expression for the generating functional, thereby obtaining ex-

plicit integral representations for the Green™s functions, i.e. explicit solutions of the

functional di¬erential equations. Needless to say, only the Gaussian integral can be

evaluated, and in practice we are back to perturbation theory and diagrams. But

the path integral has its particular bene¬ts as we shall explore in this chapter, and

is very useful when it comes to exploit the symmetry of a theory.

We now turn to obtain the functional integral expression for the generating func-

tional for the case where interactions are present. Operating with the inverse bare

propagator on the fundamental equation for the dynamics, Eq. (9.32), we get accord-

ing to Eq. (9.5) the functional di¬erential equation

N ’1

δ N ’1

1 δZ[J] 1 1

(G’1 )12 = g12...N + J1 Z[J]

(N ’ 1)! δJN · · · δJ3 δJ2

0

i δJ2 i

N

(10.34)

where we consider a theory with an arbitrary number of vertices.8

We introduce the Fourier functional integral representation of the generating func-

tional

Dφ Z[φ] eiφ J ,

Z[J] = (10.35)

where for the dummy functional integration variable we use the notation φ to distin-

guish it from the average ¬eld considered in the previous chapter for which we used

the notation •.

7 This is the trick behind the use of supersymmetry methods to avoid the denominator problem

in the study of quenched disorder [51]. However, the supersymmetry trick has the disadvantage of

not being able to cope with the case of interactions. Anyway, we have confessed our preference to

avoid the denominator problem by using the real-time technique.

8 In Eq. (10.34) we performed the shift δ/δJ ’ δ/iδJ for proper quantum ¬eld theory notation

as dictated by the functional Fourier transform. Details of the transition between Euclidean and

Minkowski (contour-time) ¬eld theories are stated in the next section.

10.2. Generators as functional integrals 321

The functional Fourier transformation turns the fundamental dynamic equation

into the form9

δZ[φ] 1

’(G’1 )12 φ2 +

’i g12...N φ2 · · · φN

= Z[φ] . (10.36)

(N ’ 1)!

0

δφ1

N

The term on the left originates from the term J1 Z[J], and results from a functional

partial integration.

We refer to φ also as the ¬eld, and it starts out as just a dummy functional

integration variable as introduced in Eq. (10.35), but immediately got a life to itself,

Eq. (10.36), through the dynamics of the theory.

We then introduce the action (this at a proper place, but recall also Section 9.7)

1 1

S[φ] ≡ ’ φ1 (G’1 )12 φ2 + g12...N φ1 φ2 · · · φN (10.37)

0

2 N!

N

for a theory with vertices of arbitrary high connectivity. The compact matrix notation

covers the action being an integral with respect to space-time (or for non-equilibrium

situations contour time) and a summation with respect to internal degrees of freedom

(and with respect to the real-time dynamical or Schwinger“Keldysh indices if traded

for the contour time). We can therefore introduce the Lagrange density

d1 L(φ, φ ) .

S[φ] = (10.38)

We note that the e¬ective action, Eq. (9.80), has the same functional form as the

action except that one-particle irreducible vertex functions appear instead of the bare

vertices and in the e¬ective action appears the average ¬eld.

Since the bare propagator is chosen symmetric in all its variables, i.e. in particular

with respect to the dynamical indices as we are treating non-equilibrium states, so

is its inverse, and Eq. (10.36) can be written on the form

δZ[φ] δS[φ]

=i Z[φ] (10.39)

δφ1 δφ1

and immediately solved (up to an overall constant which can be ¬xed by comparing

with the free theory) as

Z[φ] = eiS[φ] , (10.40)

and we have the path integral representation of the generating functional (up to a

source independent normalization factor)10

Dφ eiS[φ]+iφ J .

Z[J] = (10.41)

9 We e¬ortlessly interchange functional integration and di¬erentiation, amounting here to func-

tional integration being a linear operation.

10 A virtue of the path integral formulation is the ease with which symmetries of the action leads

to important relations between Green™s functions as discussed in Appendix B.

322 10. E¬ective action

We note that in the path integral formulation of a quantum ¬eld theory, the

fundamental dynamic equation, Eq. (10.34), can be stated in terms of the basic

theorem of integration, the integral of a derivative vanishes

δ iS[φ]+iφ J

Dφ

0= e . (10.42)

δφ

In the treatment of non-equilibrium states in the real-time technique, a symmetric

representation of the bare propagator should thus be used, say

GA

0 0

G0 = , (10.43)

GR GK

0 0

in order for the path integral to be well-de¬ned. Since our interest is the real-

time treatment of non-equilibrium situations, the closed time path guarantees the

normalization condition of the generator, Z[J] = 1, provided that the sources on the

two parts of the closed time path are taken as identical.

The action is speci¬ed solely in terms of the (inverse) bare propagators and the

bare vertices and captures, according to Eq. (10.37), all the information of the theory,

just like the diagrammatics and the generating functional technique, but now in a

di¬erent way through Eq. (10.41). For a scalar boson ¬eld theory we thus have a

new formulation not in terms of the quantum ¬eld, an operator, but in terms of a

scalar ¬eld φ, a real function of space-time. The price paid for having this simpler

object appear as the basic quantity is that to calculate the amplitudes of the theory

we must perform a functional integral. In this formalism, the superposition principle

manifests itself most explicitly as a summation over all intermediate alternative ¬eld

con¬gurations. For the case of fermions, the role of the real ¬eld is taken over by

conjugate pairs of Grassmann ¬elds in order to respect the anti-symmetric property

of amplitudes for fermions.

The amplitudes of the theory are obtained by di¬erentiating the generating func-

tional with respect to the source, and they now appear in terms of functional integral

expressions11

Dφ φ1 φ2 · · · φN eiS[φ] .

A12...N = (10.44)

In the functional integral representation of a quantum ¬eld theory, the amplitudes

are thus moments of the ¬eld weighted with respect to the action. We note that in

the functional integral representation, the amplitudes are automatically the contour

time-ordered amplitudes (or in zero temperature quantum ¬eld theory, the time-

ordered amplitudes), because of the time slicing involved in the de¬nition of the

functional integral, as we also recall from Eq. (A.16) of Appendix A.12

For the generator of connected Green™s functions

i W [J] = ln Z[J] (10.45)

11 Theappearance of the imaginary unit for one™s favorite choice of de¬ning Green™s functions are

suppressed. As usual they are part of one™s private set of Feynman rules.

12 Normal ordering of interactions on the other hand, has to be enforced by hand.

10.2. Generators as functional integrals 323

we then have

eiW [J] = N ’1 Dφ eiS[φ] eiφ J , (10.46)

where N denotes the normalization factor guaranteeing that W [J] vanishes for van-

ishing source, W [J = 0] = 0. Or in the real-time non-equilibrium technique, the

generator of connected Green™s functions vanishes, W [J] = 0 if the source is taken

to be equal on the two parts of the contour, J’ = J+ .

From the Legendre transform relating the generator of connected Green™s func-

tions to the e¬ective action, Eq. (9.85), and the functional integral representation

of the generating functional, Eq. (10.41), a functional integral representation of the

e¬ective action, the generator of one-particle irreducible vertices, is obtained (rein-

stating for once )

i i

Dφ e

“[•] (S[φ]+(φ’•) J)

e = , (10.47)

where the normalization factor has been absorbed in the de¬nition of the functional

integral.

By inspecting the path integral expression of the generating functional for the

theory in question

’1

Dφ e’ 2 • G0

i

•

eiSi [φ] eiφ J

Z[J] = (10.48)

one can envisage the perturbation theory diagrams: expand all exponentials except

the one containing the inverse free propagator, and perform the Gaussian integrals.

We shall do this in Section 10.2.2, but before that we discuss the relationship between

the Euclidean and Minkowski versions of ¬eld theories.

10.2.1 Euclid versus Minkowski

The exposition in the previous chapter was mostly explicitly for the Euclidean ¬eld

theory or thermodynamics. We left out the annoying imaginary unit irrelevant to

the functioning of the generating functional technique. In that case, the Green™s

functions are given by

Dφ φ1 φ2 · · · φN eS[φ] ,

A12...N = (10.49)

where the action is a real functional specifying in equilibrium statistical mechanics

the probability for a given con¬guration of the ¬eld, the Boltzmann factor.

In a quantum ¬eld theory, the transformation between source and ¬eld is Fourier

transformation involving the imaginary unit.13 Anyone is entitled to deal with this

through one™s favorite choice of Feynman rules. We followed the standard choice in

Section 4.3.2 where we included the imaginary unit in the de¬nition of the Green™s

13 For a quantum ¬eld theory expressed in the operator formalism, the imaginary unit will also

appear through the time evolution operator, recall Section 4.3.2.

324 10. E¬ective action

functions, recall for example Eq. (4.39) or Eq. (3.61), and we have for the transition

between Euclidean, i.e. imaginary-time ¬eld theory and real-time quantum ¬eld

theory the connections

G0 ” ’iG0 g ” ig J ” iJ

, ,

i

S[φ] ” S[φ] . (10.50)

Equation (9.32) thus transforms into Eq. (10.34).14

For a quantum ¬eld theory we have for the generating functional ( is later often

discarded)

i i

S[φ] + i φ J

Dφ Z[φ] e Dφ e

φJ

Z[J] = = (10.51)

and Green™s functions are generated according to our choice

N

δZ[J]

N ’1

(’i) = A12...N . (10.52)

δJ1 δJ2 · · · δJN

i

J=0

We can swing freely between using real-time and imaginary-time formulation, all

formal manipulations being analogous.

In the real-time or closed time path technique there is no denominator problem,

but otherwise in order to have proper normalization we should write

i i

Dφ e S[φ] φJ

e

Z[J] = (10.53)

i

Dφ e S[φ]

but often such an overall constant are incorporated in the de¬nition of the functional

integral.

10.2.2 Wick™s theorem and functionals

We now show how perturbation theory falls out very easily from the functional for-

mulation, viz. Wick™s theorem becomes a simple matter of di¬erentiation.

We note the relationship

δ

eiS[φ] eiφ J = eiS[’i δ J ] eiφ J , (10.54)

which is immediately obtained by expanding the exponential of the action on the

right-hand side and noting that di¬erentiating with respect to the source substitutes

the φ-variable, and re-exponentiating gives the exponential of the action as on the

left-hand side.

14 The form of the propagator also changes when Wick rotating from real to imaginary time,

changing the analytical properties of the propagator.

10.2. Generators as functional integrals 325

Let us in the action split o¬ the trivial quadratic term

1

S0 [φ] = ’ φ(G(0) )’1 φ (10.55)

2