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encounter in¬nite products, the functional determinant, a highly divergent object, but happily such
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 z † Az

dz dz e = det , (10.23)

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


A’1 ψ
Dψ — (x, t) Dψ(x, t) e’ 2 ψ
= (10.25)

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
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)


d· 1 = 0 (10.28)


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)

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


= iAMM = Det(iA) , (10.33)
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
i δJ2 i


where we consider a theory with an arbitrary number of vertices.8
We introduce the Fourier functional integral representation of the generating func-
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)!

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)
2 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

0= e . (10.42)

In the treatment of non-equilibrium states in the real-time technique, a symmetric
representation of the bare propagator should thus be used, say

0 0
G0 = , (10.43)
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
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
By inspecting the path integral expression of the generating functional for the
theory in question
Dφ e’ 2 • G0

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
, ,

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
i i
S[φ] + i φ J
Dφ Z[φ] e Dφ e
Z[J] = = (10.51)

and Green™s functions are generated according to our choice
N ’1
(’i) = A12...N . (10.52)
δJ1 δJ2 · · · δJN

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
Z[J] = (10.53)
Dφ e S[φ]

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

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
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
S0 [φ] = ’ φ(G(0) )’1 φ (10.55)

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