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the free part, and the interaction part of the action, S = S0 + Si , is in general
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

c=0 b=0



’1/2
A A’1 ‚c
T
1
cp1 · · · cp2N
e 2 ‚c
= det , (10.67)

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)

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

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