1

g12...N φ1 φ2 · · · φN .

Si [φ] = (10.56)

N!

N

The functional integral expression for the generating functional

(0) ’1

Dφ eiSi [φ] ’ 2 φ(G

i

) φ + iφ J

Z[J] = (10.57)

then, in accordance with Eq. (10.54), becomes

δ

Z[J] = eiSi [’i δ J ] Z0 [J] , (10.58)

where Z0 [J] is the generating function for the free theory

(0)

i

Z0 [J] = e 2 JG J

. (10.59)

We have thus achieved expressing the generating functional in terms of the generator

of the free theory. Formula Eq. (10.58) expresses the perturbation theory of the

theory in a compact form, and in a very di¬erent form compared to how in the

operator formulation the full theory was expressed in terms of the free theory as we

recall from Section 4.3.2. We now unfold this formula and show that it leads to the

diagrammatic perturbation theory from which we started out in this chapter, and

of course expressions equivalent to the non-equilibrium diagrammatic perturbation

theory we derived in the canonical operator formalism in Chapters 4 and 5 by use of

Wick™s theorem on operator form.

The exponential containing the interaction is then expanded, for example consider

a 3-vertex theory for which we get

(’i)3 δ 3 (’i)3 δ 3 1 (’i)3 δ 3

1 11

+ ···

Z[J] = 1 + g123 + g123 g1 2 3

3! δJ1 δJ2 δJ3 2! 3! δJ1 δJ2 δJ3 3! δJ1 δJ2 δJ3

(0)

i

— e 2 JG J

. (10.60)

A derivative brings down from the exponential a source contracted with a free prop-

agator and another derivative must eliminate this source if the terms are to survive

when at the end the source is set to zero. An odd number of di¬erentiations will thus

lead to a vanishing expression, and the derivatives must thus group in pairs, and this

can be done in all possible ways.

Before arriving at Wick™s theorem, we note that the generator can be related to

vacuum diagrams. We expand both exponentials multiplied in Eq. (10.58), again

326 10. E¬ective action

considering a 3-vertex theory,

(’i)3 δ 3 (’i)3 δ 3 1 (’i)3 δ 3

1 11

+ ···

Z[J] = 1 + g123 + g123 g1 2 3

3! δJ1 δJ2 δJ3 2! 3! δJ1 δJ2 δJ3 3! δJ1 δJ2 δJ3

2

i 1 i

— 1 + JG(0) J + JG(0) J + .... . (10.61)

2 2! 2

Now operating with the terms, we get strings of di¬erentiations which will attach

free propagators to vertices. Setting the source to zero in the end, J = 0, we obtain

that Z[J = 0] is the sum of all vacuum diagrams constructable from the vertices and

propagators of the theory.

Exercise 10.1. Obtain the perturbative expansion of the generating functional at

zero source value, Z[J = 0], to fourth order in the coupling constant for a 3-vertex

theory and draw the corresponding vacuum diagrams.

The amplitudes of the theory are generated by taking derivatives of the generating

functional, for example for the 2-state amplitude we encounter the further derivatives

δ 2 Z[J]

A12 [J] = i

δJ1 δJ2

δ2 g123 (’i)3 δ 3 1 g123 (’i)3 δ 3 g1 2 3 (’i)3 δ 3

+ ···

= i 1+ +

δJ1 δJ2 3! δJ1 δJ2 δJ3 2! 3! δJ1 δJ2 δJ3 3! δJ1 δJ2 δJ3

2

i 1 i

— + ···

1 + JG(0) J + JG(0) J . (10.62)

2 2! 2

The resulting perturbative expressions for the amplitude upon setting the source to

zero are precisely the ones which corresponds to the original diagrammatic de¬nition

of the 2-state Green™s function, correct factorials and all. This is Wick™s theorem

expressed in terms of functional di¬erentiation and obtained by using the functional

integral representation of the generating functional. The above scheme gives us

back by brute force the diagrammatics in terms of free propagators and vertices we

started out with. However, as a calculational tool, the procedure becomes quickly

quite laborious. Using the generating functional equations of the previous chapter is

more e¬cient, as we demonstrated in Section 9.7.

Exercise 10.2. Obtain the perturbative expansion of the 2-state Green™s function,

A12 , to lowest order in the coupling constants for a 3- plus 4-vertex theory and draw

the corresponding diagrams.

Exercise 10.3. Obtain the perturbative expansion of the 2-state Green™s function,

A12 , to fourth order in the coupling constant for a 3-vertex theory and draw the

corresponding diagrams.

10.2. Generators as functional integrals 327

We now turn to show how Wick™s theorem can be formulated in the functional

integral approach. The amplitudes or Green™s functions of a quantum ¬eld theory

are in the functional integral representation of the Green™s functions speci¬ed by av-

erages over the ¬eld, such as in Eq. (10.44). Also, in an expansion of the exponential

containing the interaction term in Eq. (10.57), such averages will appear, and we en-

counter arbitrary correlations with a Gaussian weight. Let us therefore ¬rst consider

the N -dimensional integral

∞ ∞

dx1 . . . dxN xp1 · · · xp2N e’ 2 x

T

1

Ax

I(p1 , . . . , p2N ) = (10.63)

’∞ ’∞

where A denotes the symmetric matrix of Section 10.1.2, and xpM denotes any vari-

able picked from the N -tuple (x1 , . . . , xN ) and allowed to appear any number of the

possible 2N times. We have chosen a string of even factors, since the integral vanishes

if an odd number of xs occurred, as seen immediately by diagonalizing the quadratic

form. The correlation function to be evaluated can be rewritten

∞N

‚ ‚

dxM e’ 2 x e’ib

T T

1

··· i Ax x

I(p1 , . . . , p2N ) = i

‚bp1 ‚bp2N

’∞ M=1

b=0

‚ ‚

··· i

=i I(A; b) (10.64)

‚bp1 ‚bp2N

b=0

and according to Eq. (10.15)

’1/2

A ‚ ‚ 1 T ’1

e’ 2 b A b

··· i

I(p1 , . . . , p2N ) = det i . (10.65)

2π ‚bp1 ‚bp2N

b=0

The expression on the right can be evaluated by use of the formula, valid for arbitrary

functions f and g,

‚ ‚

f (c) e’ib

T

c

f i g(b) = g i , (10.66)

‚b ‚c

c=0

which is immediately proved by the help of Fourier transformation, i.e. by showing

the formula for plane wave functions. Employing Eq. (10.66) we obtain

’1/2

A A’1 ‚c

cp1 · · · cp2N e’ib

T T

1

e 2 ‚c c

I(p1 , . . . , p2N ) = det

2π

c=0 b=0

’1/2

A A’1 ‚c

T

1

cp1 · · · cp2N

e 2 ‚c

= det , (10.67)

2π

c=0

328 10. E¬ective action

where the last equality is obtained as the terms originating from di¬erentiating the

exponential eventually vanish when b is set equal to zero. The only surviving term

on the right comes from the term in the expansion of the exponential containing 2N

di¬erentiations giving

’1/2

A 1

(‚c A’1 ‚c )N cp1 · · · cp2N

T

I(p1 , . . . , p2N ) = det . (10.68)

N

2π N !2

c=0

In each of the N double di¬erentiation operators, we must choose pairs in the pick

of the factors on the right thereby uniquely exhausting the pick in order to get

a non-vanishing result upon setting c = 0. Then upon di¬erentiating and setting

c = 0, a product of N terms of the form (A’1 )pi ,pj occurs with the chosen pairings

as indices. Permuting which pair is related to which double di¬erentiation operator

gives N ! identical products. Furthermore, since A is a symmetric matrix so is A’1

(transposition and inverting of a matrix are commuting operations) and we obtain

’1/2

A

(A’1 )pi ,pj ,

I(p1 , ..., p2N ) = det (10.69)

2π a.p.p.

where the sum is over all possible pairings of the indices in the pick p1 , . . . , p2N ,

without distinction of the ordering within a pair, explaining in addition the canceling

of the factor 1/2N . The above observation is the equivalent of Wick™s theorem.

With the usual convention of absorbing the functional determinant in the de¬ni-

tion of the functional integral we get, in accordance with Eq. (10.63) and Eq. (10.69),

that the amplitudes of the free theory are obtained according to

’1

Dφ φ1 φ2 · · · φ2N e’ 2 • G0

i

•

A12...2N =

(iG’1 )p1 p2 (iG’1 )p3 p4 · · · (iG’1 )p2N ’1 p2N .

= (10.70)

0 0 0

a.p.p.

By inspecting the path integral expression for the generating functional

’1

Dφ e’ 2 • G0

i

•

eiSi [φ] eiφ J

Z[J] = (10.71)

one can envisage its perturbation expansion and corresponding Feynman diagrams by

this recipe: expand all exponentials except the one containing the inverse free propa-

gator, the Gaussian term, and evaluate the averages according to the above formula,

Eq. (10.70). This recipe for functional integration of products of ¬elds weighted by

their Gaussian form provides Wick™s theorem, but now in the functional or path

integral formulation of the ¬eld theory. From this observation we can immediately

recover the non-equilibrium Feynman diagrammatics of a quantum ¬eld theory by

expanding the exponential containing the interaction in Eq. (10.41).

10.2. Generators as functional integrals 329

The limiting procedure used in Section 10.1 to de¬ne functional integration can

be made rigorous only for the Euclidean case. For the quantum ¬eld theory case, an

alternative now o¬ers itself, viz. to de¬ne the functional integrals in terms of their,

as above, perturbative expansions in the non-Gaussian interaction part.

Exercise 10.4. If the Gaussian part of the integrand in Eq. (10.63) is interpreted as

a probability distribution for the random or stochastic variable x, then Eq. (10.69)

is the statement that any correlation function of a Gaussian random variable, with

zero mean, is expressed in terms of all possible products of the two-point correlation

function.

Show that the generating function, i.e. the Fourier transform of the normalized

probability distribution

’1/2

A

e’ 2 x

T

1

Ax

P (x) = det , (10.72)

2π

is

A’1 k

P (k) = e’ 2 k

T

1

. (10.73)

Exercise 10.5. Consider a set of independent stochastic variables {xn }n=1,...,N ,

each with arbitrary probability distributions except for zero mean and same ¬nite

√

variance, say σ. Show that the stochastic variable X = (x1 + · · · + xN )/ N will then

obey the central limit theorem, i.e. in the limit N ’ ∞, the stochastic variable X

will be Gaussian distributed with variance σ.

Another application of the formula Eq. (10.66), allows us to rewrite Eq. (10.58)

(0) 1δ δ

δ i G0

Z[J] = eiSi [’i δ J ] e 2 JG J

eiSi [φ] + i• J

= e 2 δφ (10.74)

δφ

thereby giving the following functional integral expression

’1 1δ δ

1

Dφ e 2 δ φ G0

Z[J] = e 2 Tr ln G0 eiSi [φ] + i• J . (10.75)

δφ

φ=0

From here we see directly that Z[J] is the sum of all the vacuum diagrams for the

theory in question in the presence of the source J. This observation is again the

equivalent of Wick™s theorem, but here at its most expedient form involving both

functional integration and di¬erentiation.

Introducing the generator of connected Green™s functions

Z[J] = eiW [J] (10.76)

and recalling the combinatorial argument of Section 9.5, the above important ob-

servation gives that iW [J] is the sum of all the connected vacuum diagrams in the

presence of the source J.

For the connected Green™s functions we then obtain the functional integral ex-

pression

Dφ φ1 φ2 · · · φN eiS[φ]

≡ φ1 φ2 · · · φN .

G12...N = (10.77)

Dφ eiS[φ]

330 10. E¬ective action

Often the denominator, which cancels all the disconnected contributions in the nu-

merator, is left implicit as a normalization factor in the de¬nition of the functional

integral.

For the average or classical ¬eld, •1 , considered in Section 9.6, we thus have the

functional integral expression for the Euclidean case

Dφ φ1 eS[φ]

≡ φ1 ≡ φ1 ,

•1 = (10.78)

Dφ eS[φ]

the reason for calling •1 ≡ G1 the average ¬eld now being obvious.

The diagrammatics obtained by the above procedures are of course naive per-

turbation theory, expressed in terms of the bare propagators and vertices. A rep-

resentation which contains the full propagators and the e¬ective vertices is a better

representation since it expresses the physics of a particular situation, viz. the state

under consideration. This representation can be obtained at the diagrammatic level

by topological arguments, leading to the so-called skeleton diagrams as discussed in

Section 4.5.2. In Section 9.8, we followed another way and employed the e¬ective

action to show how easily the skeleton diagrammatics is obtained from the analytical

functional di¬erentiation formalism. In the next section, we show how the partially

re-summed perturbation expansion of Green™s functions, the skeleton diagrammatic

representation, is expressed in the functional integral formalism.

10.3 Generators and 1PI vacuum diagrams

In the previous section we showed that the generating functionals had perturbative

expansions corresponding diagrammatically to the sum of all vacuum diagrams ex-

pressed in naive perturbation theory. In this section we shall exploit the functional

integral representation of a quantum ¬eld theory to relate the various generators to

classes of one-particle irreducible vacuum diagrams.

We therefore turn to show that the generator of connected Green™s functions

can be expressed in terms of the e¬ective action and a restricted functional integral.

A restricted functional integral is a functional integral interpreted in terms of its

perturbative expansion or equivalently the corresponding Feynman diagrams, and

where only certain topological classes of diagrams are retained. First, we recall

the result derived diagrammatically, the relationship displayed in Figure 9.48: that

the tadpole, the ¬rst derivative of the generator of connected amplitudes, has a

diagrammatic expansion in terms of only tree diagrams, tadpoles attached to one-

particle irreducible vertices. This means that the generator of connected amplitudes,

W [J], itself is given by the irreducible vertices attached to tadpoles. This suggests

that the generator of connected amplitudes, W [J], can be speci¬ed in terms of the

e¬ective action, “[φ]. We now turn to show that it is indeed the case and this in

terms of a functional integral where the e¬ective action appears instead of the action

and the functional integral is restricted:

Dφ ei“[φ] +iφ J ,

i W [J] = (10.79)

CTD

10.3. Generators and 1PI vacuum diagrams 331

where CTD indicates that only the Connected Tree Diagrams should be kept of all the

vacuum diagrams generated by the perturbative expansion of the functional integral.

Tree diagrams contain no loops, they are contained within the 1PI vertices, and tree

diagrams can be cut in two by cutting a single line of a tadpole.

To keep track of the number of loops in the diagrams generated by the unrestricted

functional integral in Eq. (10.79), we introduce the parameter a

’1

˜

Dφ eia

eiWa [J] = (“[φ] +φ J)

. (10.80)

The vacuum diagrams generated by this functional integral can be characterized as

follows. Separate out in the e¬ective action the quadratic term, which according

to Eq. (9.83) is the inverse of the full Green™s function of the theory multiplied by

a’1 . Then expand the rest of the exponential and use Wick™s theorem according to

the previous section, or rather the just derived procedure for Gaussian averaging of

products of ¬elds to obtain the perturbative expansion of the functional integral in

Eq. (10.80), and its corresponding Feynman diagrams. A Green™s function has thus