of the 2-state one-particle irreducible vertex according to the diagrammatic expansion

as depicted in Figure 9.53.

= +

+ +...

Figure 9.53 Self-energy representation of 2-state propagator.

The reason for this is, that any 2-state diagram is uniquely classi¬ed topologically

according to whether it can not be cut in two or can be cut in two by cutting an

internal particle line at only one place, or at two, three, etc., places. By construction

we thus uniquely exhaust all the possible diagrams for the 2-state propagator. The

2-state one-particle irreducible vertex is also called the self-energy.

The diagrammatic equation in Figure 9.53 can be expressed in the form depicted

in Figure 9.54, which is seen by iterating the equation in Figure 9.54.

= +

Figure 9.54 Dyson equation for the 2-state Green™s function.

The diagrammatic equation depicted in Figure 9.54 corresponds analytically to

the equation for the 2-state Green™s function expressed in term of the 2-state irre-

9.6. One-particle irreducible vertices 303

ducible vertex, the self-energy (recall we absorbed the inverse free propagator in “12 ,

(’G’1 + “)12 ’ “12 , i.e. Σ12 denotes the one-particle irreducible 2-state vertex)

0

(0) (0)

G12 = G12 + G13 Σ34 G42 (9.93)

or equivalently the equation

(0) (0)

G12 = G12 + G13 Σ34 G42 (9.94)

by iterating from the other side. We have obtained the non-equilibrium Dyson equa-

tions.42

We now show how the Dyson equation can be obtained by using the e¬ective

action. More importantly, we show that we can use Eq. (9.84) to obtain the equation

for the connected 2-state amplitude in the presence of the source where it is expressed

in terms of the derivative of the e¬ective action. This will lead to a simple method

whereby all amplitudes can be expressed in terms of the 2-state connected amplitude,

the full propagator, and one-particle irreducible vertices.

The Legendre transformation, according to Eq. (9.78), gives rise to the relation

δ 2 W [J] δ

δ δ•2 δ δ

= = = G12 . (9.95)

δJ1 δJ1 δ•2 δJ1 δJ2 δ•2 δ•2

Taking the derivative of Eq. (9.84) with respect to the source then gives

δ 2 W [J] δ 2 “[•]

’ = δ12 . (9.96)

δJ2 δJ3 δ•3 δ•1

Adding and subtracting in the e¬ective action the so-called free term, “[•] ≡

’ 2 •1 (G(0) )’1 •2 +“i [•],

1

i.e. splitting o¬ again the inverse propagator term we previ-

12

ously included in “12 , sothat “i now denotes the original e¬ective action introduced

in Eq. (9.80), provides the self-energy

δ 2 “i [•]

Σ12 = (9.97)

δ•1 δ•2

in the presence of the source, as expressed through the ¬eld. Inserting into Eq. (9.96)

gives

δ 2 “[•] δ 2 “i [•]

(0) ’1

= ’(G )12 + (9.98)

δ•1 δ•2 δ•1 δ•2

and inserting into Eq. (9.96) gives

δ 2 W [J] δ 2 “i [•]

’(G(0) )’1

’ + = δ12 . (9.99)

21

δJ2 δJ1 δ•2 δ•1

42 Recovering the non-equilibrium Dyson equations thus makes contact with quantum ¬eld theory

studied by canonical means in the previous chapters. For the non-equilibrium states we studied in

the previous chapters, we had valid approximate expressions for the self-energy, and did not need

to go further into the diagrammatic structure of higher-order vertices.

304 9. Diagrammatics and generating functionals

Matrix multiplying by the bare propagator from the right gives43

δ 2 W [J] δ 2 W [J] δ 2 “i [•] (0)

(0)

= G12 + G, (9.100)

δJ1 δJ3 δ•3 δ•4 42

δJ1 δJ2

which in terms of diagrams has the form depicted in Figure 9.55.

•

= +

Figure 9.55 Dyson equation for the 2-state Green™s function in the presence of the

source.

Iterating the equation gives the full propagator

δ 2 W [J] 2 2 2

(0) δ “i [•] (0) δ “i [•] (0) δ “i [•]

(0) (0) (0)

= G11 + G12 G31 + G12 G34 G51 + ...

δJ1 δJ1 δ•2 δ•3 δ•2 δ•3 δ•4 δ•5

(9.101)

the analog of the Dyson equation depicted in Figure 9.53, but now for the case where

the source is present. The second derivative relationship between the generator of

connected Green™s functions and the e¬ective action can compactly be rewritten

suppressing the matrix indices, i.e. the two state labels occurring upon di¬erentiation

are now only indicated by the primes, in the form

1

W [J] = (9.102)

G’1 ’ “i [•]

0

as we recall the formula for a matrix X

1

= 1 + X + X 2 + X 3 + ... . (9.103)

1’X

This is the relationship between the full propagator and the self-energy we arrived

at earlier by topological classi¬cation of diagrams, expressing the connected 2-point

Green™s function in terms of the self-energy. Here we have constructed the functional

analog in terms of a functional di¬erential equation.

By taking further source derivatives of Eq. (9.96), we express the higher-order

connected Green™s functions in terms of the full propagator and the higher-order

one-particle irreducible vertices.

Taking the derivative of Eq. (9.96) with respect to the source and using Eq. (9.84)

gives

δ 3 W [J] δ 2 W [J] δ 2 W [J] δ 2 W [J] δ 3 “[•]

= . (9.104)

δJ1 δJ2 δJ3 δJ1 δJ1 δJ2 δJ2 δJ3 δJ3 δ•1 δ•2 δ•3

In terms of diagrams we have for Eq. (9.104) the relation depicted in Figure 9.56.

43 This equation is of course immediately recognized as the Dyson equation, Eq. (4.141), G12 =

(0) (0)

G12 + G13 Σ34 G42 .

9.6. One-particle irreducible vertices 305

•

=

Figure 9.56 Connected 3-state diagram expressed by the 1P-irreducible 3-vertex.

Exercise 9.10. Show by taking further source derivatives of Eq. (9.96) that the

equation obtained for the 4-state connected Green™s function has the diagrammatic

form (for a theory with 3- and 4-connector vertices) depicted in Figure 9.57.

•

• +

=

•

• • +

+

• •

Figure 9.57 Connected 4-state diagram expressed by 1P-irreducible vertices.

If in the above equation we set the source to zero, and thereby the ¬eld to zero,

instead of encountering quantities depending on the source and ¬eld, we will obtain

expressions for the connected Green™s functions in term of the full 2-point Green™s

function and the irreducible vertex functions. Since the full 2-point Green™s function

is the one into which we can feed our phenomenological knowledge of the mass of

a particle, these equations are basic for the renormalization procedure. The bare

306 9. Diagrammatics and generating functionals

Green™s function with its bare mass, and the bare vertices have thus left the theory

explicitly, leaving room for the trick of renormalization.

In Section 9.8, we shall use the equations, Eq. (9.78) and Eq. (9.84), the Legendre

transformation between source and ¬eld variables, to replace source-derivatives by

¬eld-derivatives and thereby obtain the equations satis¬ed by the e¬ective action

and the diagrammatics of the one-particle irreducible vertices. But ¬rst we turn to

show how equations very e¬ciently relating the connected Green™s function can be

generated.

9.7 Diagrammatics and action

In this section we show how the fundamental di¬erential equation for the dynamics,

Eq. (9.32), can be turned into an equation from which the relationships between the

connected Green™s functions can easily be obtained. This is done by introducing the

action, which is de¬ned in terms of the inverse propagator and the bare vertices of

the theory according to44

1 1

S[φ] ≡ ’ φ1 (G’1 )12 φ2 + g12...N φ1 φ2 · · · φN , (9.105)

0

2 N!

N

here for a theory with vertices of arbitrary high connectivity. The fundamental

equation, Eq. (9.32), expressing the dynamics of a theory can then be written in the

form (for an arbitrary theory speci¬ed by the above action)

δ

δS[ i δJ ]

0= + J1 Z[J] , (9.106)

δφ1

where by de¬nition

δ

δS[ i δJ ] δS[φ]

= (9.107)

δφ1 δφ1

δ

φ’ i δJ

i.e. the action is di¬erentiated and then the source derivative is substituted for the

¬eld. We have written the equation in a form having a quantum ¬eld theory in

mind but shall immediately shift to the Euclidean version, or simply suppressing the

appearance of /i by absorbing the factor in the source derivative.

44 At this junction in the generating functional formulation of a quantum ¬eld theory the solemnity

of the action is scarcely noticed, but as just another formal construction. In the next chapter we

show how the action in the functional integral formulation of a quantum ¬eld theory naturally

appears as the fundamental quantity describing the dynamics. The action can also be given a

fundamental status in the operator formulation of the generating functional technique (recall Section

3.3), if the dynamics is based on Schwinger™s quantum action principle [50]. However, the point of

the presentation in this chapter is to base the dynamics directly on diagrams and then by simple

topological arguments construct the generating functional technique.

9.8. E¬ective action and skeleton diagrams 307

Since the generator of connected diagrams, W , is the logarithm of Z, we have the

relation valid for an arbitrary functional F

1δ δW [J] δ

(Z[J] F [J]) = + F [J] (9.108)

Z[J] δJ1 δJ1 δJ1

and by repetition

δN δW [J] δ δW [J] δ

···

Z[J] = Z[J] + + , (9.109)

δJ1 · · · δJN δJ1 δJ1 δJN δJN

where operator notation has been used, i.e. the operations are supposed to operate

on a functional F .

Since the action is a sum of polynomials we have according to Eq. (9.109)

δ

δS[ δW + δ

δS[ δJ ] δJ ]

δJ

Z[J] = Z[J] . (9.110)

δφ1 δφ1

The fundamental equation, Eq. (9.106), can thus be written in the form

δS[ δW + δ

δJ ]

δJ

0= + J1 . (9.111)

δφ1

Using the explicit form of the action for an arbitrary theory we have

δS[ δW + δ

δJ ] δW δ

’(G(0) )’1

δJ

= +

12

δφ1 δJ2 δJ2

1 δW δ δW δ

···

+ g12...N + +

(N ’ 1)! δJ2 δJ2 δJN δJN

N

(9.112)

and using Eq. (9.111) and performing the di¬erentiations and lastly multiply by the

bare propagator we immediately recover Eq. (9.77) (for the 3- plus 4-vertex theory).

Having the fundamental equation on the form speci¬ed in Eq. (9.111) turns out

in practice to be very useful for generating the relations between the connected full

Green™s functions, and exempli¬es the expediency and powerfulness of the generating

functional formalism.

9.8 E¬ective action and skeleton diagrams

In this section, we shall use the equations, Eq. (9.78) and Eq. (9.84), the Legen-

dre transformation between source and ¬eld variables, to replace source-derivatives

by ¬eld-derivatives and thereby obtain the equations obeyed by the e¬ective action.

Upon setting the ¬eld to zero, • = 0, we then obtain the skeleton diagrammatic

308 9. Diagrammatics and generating functionals

equations satis¬ed by the one-particle irreducible vertices. Instead of using topolog-

ical diagrammatic arguments to obtain the skeleton diagrammatics, we turn to use

the generating functional method to achieve the same goal.

On the right side in Eq. (9.112) we can introduce the average ¬eld and obtain

δS[ δW + δ

δ2W

δJ ] δ

= ’(G(0) )’1 •2 +

δJ

12

δφ1 δJ2 δJ2 δ•2

δ2W δ2W

1 δ δ

· · · •N +

+ g12...N •2 + ,

(N ’ 1)! δJ2 δJ2 δ•2 δJN δJN δ•N

N

(9.113)

where we in addition have used Eq. (9.95) to substitute the ¬eld derivative for the

source derivative.

Inserting Eq. (9.111) into Eq. (9.84) and using Eq. (9.78) thus gives the relation

between the action and the e¬ective action

δ

δS • + W [J]

δ“[•] δ•

=’ , (9.114)

δ•1 δ•1

where the right-hand side is short for the right-hand side in Eq. (9.113).

For a 3- plus 4-vertex theory we obtain

δ 2 W [J]

δ“[•] 1 1

’(G’1 )12 •2 +

= g123 •2 •3 + g123

0

δ•1 2 2 δJ2 δJ3

δ 2 W [J]

1 3

+ g1234 •2 •3 •4 + g1234 •4

3! 3! δJ2 δJ3

δ 2 W [J] δ 2 W [J] δ 2 W [J] δ 3 “[•]

1

+ g1234 (9.115)

3! δJ2 δJ5 δJ3 δJ6 δJ4 δJ7 δ•5 δ•6 δ•7

as the last term emerges upon noting

δ δ 2 W [J] δ 3 W [J] δ 2 “[•] δ 3 W [J]

δJ2

= =

δ•2 δJ3 δJ4 δ•2 δJ2 δJ3 δJ4 •2 •2 δJ2 δJ3 δJ4

δ 2 W [J] δ 2 W [J] δ 3 “[•]

= , (9.116)

δJ3 δJ5 δJ4 δJ6 δ•2 δ•5 δ•6

where in obtaining the last equality we have used Eq. (9.96) and Eq. (9.104). The

relationship expressed in Eq. (9.116) has the diagrammatic representation depicted

in Figure 9.58.

9.8. E¬ective action and skeleton diagrams 309

•

δ =

δφ1

1

Figure 9.58 Average ¬eld dependence of the propagator.

The implicit dependence of the propagator on the average ¬eld, through the source,

is thus such that taking the derivative inserts a one-particle irreducible vertex in

accordance with the relation depicted in Figure 9.58.

The equation for the ¬rst derivative of the e¬ective action, Eq. (9.115), has for a

3- plus 4-vertex theory the diagrammatic representation depicted in Figure 9.59.

• 1

=’ + 2

1 1

+ +

3! 2

•

1

+

1

+ 3!

2

Figure 9.59 Diagrammatic relation for the ¬rst derivative of the e¬ective action for

a 3- plus 4-vertex theory.

310 9. Diagrammatics and generating functionals

The stubs on the bare vertices in Figure 9.59 indicates the uncontracted state label

identical to the state label on the left.

Here we ¬nd the origin for calling “[•] the e¬ective action: if thermal or quantum

¬‚uctuations are neglected, leaving only the ¬rst three terms on the right in Figure

9.59, the (derivative of the) e¬ective action reduces to the (derivative of the) action.

This corresponds to dropping the W -terms in Eq. (9.114). In other words, the

exact equation of motion for the ¬eld, Eq. (9.84), can be obtained from the equation

determining the classical ¬eld (where S is the action given in Eq. (10.37))

δS[φ]

0 = J1 + (9.117)

δφ1

by substituting the one-particle irreducible vertices for the bare vertices in the action

S. We recall that in the absence of the source, J = 0, the ¬eld makes the e¬ective

action stationary, Eq. (9.86). The classical theory is given by the ¬eld speci¬ed by

making the action stationary

δS[•]

=0 (9.118)

δ•1

the classical equation of motion.

We note that the terms containing loops in Figure 9.59 are the quantum correc-

tions to the classical action.

Exercise 9.11. In this exercise we elaborate the statement that the classical ap-

proximation corresponds to neglecting all loop diagrams. Consider a theory with 3-

and 4-connector vertices. Obtain the classical equation of motion for the ¬eld. In-

terpret the equation diagrammatically, and note that no loop diagrams appear, only

so-called tree diagrams.

At this point we appreciate the e¬ciency of the generating functional method:

it provides us immediately with equations containing only full propagators and ver-

tices, i.e. the derived equations correspond to skeleton diagram equations, and in¬-

nite partial summations of the diagrams of naive perturbation theory are obtained

automatically.

To obtain the equation satis¬ed by the second derivative of the e¬ective action, we

can now take one more derivative of Eq. (9.116) with respect to the ¬eld. However,

this is done automatically at the diagrammatic level of Figure 9.59. A tadpole is

the ¬eld and the •-derivative removes it; the derivative thus reduces the number

of tadpoles present by one. For the ¬eld dependence of the propagator we use the

relation depicted in Figure 9.58. For the second derivative of the e¬ective action, we

thus ¬nd that it satis¬es the equation depicted in Figure 9.60.

9.8. E¬ective action and skeleton diagrams 311

• 1

=’ + + 2

•

1 1

+ +

2 2

•

• 1

1 +

+ 2

2

•

•

1

+ 3!

Figure 9.60 Diagrammatic relation for the second derivative of the e¬ective action.

Taking further derivatives, we obtain the equations satis¬ed by the higher deriva-

tives of the e¬ective action, and upon setting the ¬eld to zero, • = 0, we obtain the

skeleton diagrammatics for the one-particle irreducible vertices.

In the next chapter we shall study the e¬ective action formalism in detail, and

give a functional integral evaluation which gives an interpretation of the e¬ective

action in terms of vacuum diagrams.

312 9. Diagrammatics and generating functionals

9.9 Summary

In this chapter we have taken the diagrammatic description of quantum dynamics as

a basis, representing the amplitudes of quantum ¬eld theory by diagrams, and stating

the laws of nature in terms of the propagators of species and their vertices of interac-

tion. The quantum dynamics then follows in this description from the superposition

principle and the two exclusive options: to interact or not. The fundamental dia-

grammatic dynamic equation of motion, relating the amplitudes of a theory, is then

trivial to state. The diagrammatic structure of a theory was organized by intro-

ducing generators, encrypting the total information of the theory which is assessed

by functional di¬erentiation of the generator. Simple and easy visually understood

topological arguments for diagrammatics were used to turn the fundamental dynamic

equation of motion into nontrivial functional di¬erential equation for the generator.

Generators of connected Green™s functions and one-particle irreducible vertices were

introduced by diagrammatic arguments, and shown to be exceedingly e¬cient tools

to generate the equations on the form corresponding to the skeleton diagrammatic

representation. We shall now take the use of the e¬ective action a level further, and

although the content of the next chapter can be obtained staying within the for-

malism of functional di¬erential equations, the introduction of functional integrals

will ease derivations. The intuition of path integrals as usual strengthens the use of

diagrammatics.

10

E¬ective action

In the previous chapter we introduced the one-particle irreducible e¬ective action

by collecting the one-particle irreducible vertex functions into a generator whose

argument is the ¬eld, the one-state amplitude in the presence of the source. The

e¬ective action thus generates the one-particle irreducible amputated Green™s func-

tions. We shall now enhance the usability of the non-equilibrium e¬ective action by

establishing its relationship to the sum of all one-particle irreducible vacuum dia-

grams. To facilitate this it is convenient to add the ¬nal mathematical tool to the

arsenal of functional methods, viz. functional integration or path integrals over ¬eld

con¬gurations. We are then following Feynman and instead of describing the ¬eld

theory in terms of di¬erential equations, we get its corresponding representation in

terms of functional or path integrals. This analytical condensed technique shall prove

powerful when unraveling the content of a ¬eld theory. The loop expansion of the

non-equilibrium e¬ective action is developed, and taken one step further as we intro-

duce the two-particle irreducible e¬ective action valid for non-equilibrium states. As

an application of the e¬ective action approach, we consider a dilute Bose gas and a

trapped Bose“Einstein condensate.

10.1 Functional integration

Functional di¬erentiation has its integral counterpart in functional integration. We

shall construct an integration over functions and not just numbers as in elementary

integration of a function. We approach this in¬nite-dimensional kind of integration

with care (or, from a mathematical point of view, carelessly), i.e. we base it on

our usual integration with respect to a single variable and take it to a limit. To

deal with any function, •(x, t), of continuous variables such as space-time, (x, t), the

continuous variables must be discretized, i.e. space-time is divided into a set of small

volumes of size ” covering all or the relevant part of space-time, and the value of the

function • is speci¬ed in each such small volume or equivalently on the corresponding

mesh of N lattice sites, •M , M = 1, 2, . . . , N . This is immediately incorporated into

313

314 10. E¬ective action

our condensed state label notation

1 ≡ (s1 , x1 , t1 , σ1 , . . .) (10.1)

if the space and time variables are now interpreted as discrete. To treat arbitrary

non-equilibrium states, a real-time dynamical or Schwinger“Keldysh index is included

or the time variable is replaced by the contour time variable for treating general non-

equilibrium situations. We shall ¬rst consider a real scalar ¬eld, and in each cell the

¬eld can then take on any real value.

The functional integral of a functional, F [•], of a real function •, is then de¬ned

as the limit1

∞ N

D• F [•] ≡ lim d•M F (•1 , . . . , •N ) . (10.2)

N ’∞

’∞ M=1

The functional integral is a sum over all ¬eld con¬gurations.

(0)

Shifting each of the integration variables a constant amount, •M ’ •M + •M ,

leaves the integrations invariant, and we have the property of a functional integral

D• F [•] ≡ D• F [• + •0 ] . (10.3)

Quantum ¬eld theory describes a system with in¬nitely many degrees of freedom

and the functional integral is the in¬nite dimensional version of the path integral

formulation of quantum mechanics, the zero-dimensional quantum ¬eld theory, which

is discussed in Appendix A.

10.1.1 Functional Fourier transformation

The main functional integral tool will be that of functional Fourier transformation,

and to obtain that we recall that usual Fourier transformation of functions is equiva-

lent to the integral representation of Dirac™s delta function in terms of the exponential

function.2

The delta functional, i.e. the functional δ satisfying for any functional F

DJ (2) F [J (2) ] δ[J (1) ’ J (2) ] ,

F [J (1) ] = (10.4)

is construed as a product of delta functions over all the cells, and is constructed as

the limit of a product of delta functions, each of which can be represented in terms

of its usual integral expression

∞

N N

N

2π (1) (2)

’JM )

(1) (2)

’ d•M ei” •M (JM

δ(JM JM ) = . (10.5)

”

M=1 ’∞ M=1

1 The functional integral over a complex function, a complex ¬eld, is de¬ned analogously, involving

integration over the real and imaginary parts of the ¬eld.

2 For a discussion of Dirac™s delta function and Fourier transformation we refer to Appendix A

of reference [1].

10.1. Functional integration 315

For the integration over space and contour time we introduce the notation

N

•J ≡ dxdt •(x, t) J(x, t) = lim ” •M JM . (10.6)

N ’∞

M=1

We thus obtain the following functional integral representation of the delta functional

’J (2) )

(1)

δ[J (1) ’ J (2) ] = D• ei•(J , (10.7)

where the normalization factor limN ’∞ (2π/”)N has been incorporated into the def-

inition of the functional integral. The delta functional expresses according Eq. (10.4)

the identity of two functions, i.e. the equality of the two for any value of their

argument.

Having the integral representation of the delta functional at hand, Eq. (10.7), we

immediately have for the functional Fourier transformation

DJ e’i• J F [J]

F [•] = (10.8)

the inverse relation

D• eiJ • F [•] .

F [J] = (10.9)

Functional Fourier transformation is thus the product of ordinary Fourier transforms

over each cell.

The mathematical job performed by functional Fourier transformation is, just as

in usual Fourier transformation, to change, now functional, di¬erential equations

into algebraic equations. As far as physics is concerned, the functional integral

provides an explicit interpretation, in terms of the superposition principle, of the

dynamics of quantum ¬elds, the propagation of quantum ¬elds, viz. as a sum over all

intermediate ¬eld con¬gurations leading from an initial to a ¬nal state of the ¬eld,

quite analogous to the path integral in quantum mechanics, the zero-dimensional

quantum ¬eld theory, as discussed in Appendix A.

10.1.2 Gaussian integrals

The mathematics of quantum mechanics of a single particle resides in the one-

dimensional Gaussian integral

∞

2π

dx e’ 2 ax =

2

1

I(a) = (10.10)

a

’∞

or by completing the square

∞

2π b 2

dx e’ 2 ax ±bx

2

1

= e 2a , (10.11)

a

’∞

316 10. E¬ective action

where the integral is convergent whenever a is not a negative real number, i.e. I(a)

is analytic in the complex a-plane except at the branch cut speci¬ed by that of the

square root. This message holds true for the functional integrals of quantum ¬eld

theory.

The functional integral is treated as the limit of a multi-dimensional integral and

we consider the N -dimensional Gaussian integral

∞

dx1 . . . dxN e’C(x1 ,...,xN )

I(A; b) = (10.12)

’∞

speci¬ed by the quadratic form

N N

1 1T

xM AM,M xM ± x A x ± bT x .

C(x) = bM xM = (10.13)

2 2

M,M =1 M

Here xT denotes the row tuple xT = (x1 , . . . , xN ), and x the corresponding column

tuple, and similar notation for the N -tuple b. We assume that the matrix A is real,

symmetric, AT = A, and positive, so that it can be diagonalized by an orthogonal

matrix S, S ’1 = S T , and D = S T A S has then only positive diagonal entries dM .

The Jacobian, | det S|, for the transformation x = Sy is thus one, and the integral

becomes the elementary integral, Eq. (10.11), occurring N -fold times,

∞

N N

2π 2d1 (S T b)2

’ 1 dM yM ± yM (S T b)M

2

I(A; b) = dyM e = eM M

2

dM

M=1’∞ M=1

N

(S T b)2

1

M

det(2πD’1 )

2d M

= e . (10.14)

M =1

Using (S T AS)’1 = D’1 to express A’1 = SD’1 S T , or in terms of matrix elements

(A’1 )MM = M1 SMM1 dM (S T )M1 M , and using that det A = det D, we arrive at

1

1

the expression for the multi-dimensional Gaussian integral

’1/2

A A’1 b

1T

e2b

I(A; b) = det . (10.15)

2π

Again the result can be generalized by analytical continuation to the case of a com-

plex symmetric matrix, A, with a positive real part, the branch cut in the complex

parameter space being speci¬ed by the square root of the determinant.

The Gaussian functional integral is then perceived in the limiting sense of Eq. (10.2)

and we have3

1

i

D• e 2 • A • = √ , (10.16)

DetA

√

3 Here we have included extra factors in the de¬nition of the path integral, viz. a factor 1/ 2πi

for each integration d•M , explaining the absence of ’i and 2π in front of A on the right-hand side.

The imaginary unit and 2π can thus be shu¬„ed around.

10.1. Functional integration 317

where the limiting procedure introduces the meaning of the functional determinant

distinguished by a capital D in Det. Using the identity ln det A = Tr ln A we have4

D• e 2 • A • = e’ 2 Tr ln A .

1

i

(10.17)

Similarly, we obtain from the above analysis

’1

D• e 2 • A • + i• J = e’ 2 Tr ln A e’ 2 J A

1

i i

J

(10.18)

or by in the Gaussian integral, Eq. (10.16), shifting the variable, • ’ • + A’1 J.

The generating functional for the free theory, Eq. (9.35), can thus be expressed

in terms of a functional integral

’1 ’1

D• e’ 2 • G0

1 i

Z0 [J] = = e 2 Tr ln G0 • + i• J

. (10.19)

We have thus made the ¬rst connection between functional integrals and the gener-

ating functional and thereby to diagrammatics. In the treatment of non-equilibrium

states in the real-time technique, the real-time representation in the form Eq. (5.1)

or the more economical symmetric representation of the bare propagator should thus

be used

GA

0 0

G0 = . (10.20)

GR GK

0 0

in order to have a symmetric inverse propagator as demanded for the functional

integral to be well-de¬ned.

The functional

1

S0 [•] = ’ • G’1 • (10.21)

0

2

is called the free action, or action for the free theory.5

The normalization constant, guaranteeing the normalization of the generator

Z0 [J = 0] = 1, is often left implicit as overall constants of functional integrals have

4 The identity is obvious for a diagonal matrix, and therefore for a diagonalizable matrix

which is the case of interest here. The identity follows generally from the product expansion

of the exponential function, det eA = det limn’∞ (I + A/n)n = limn’∞ (det(I + A/n))n =

limn’∞ (1 + TrA/n) + O(1/n2 ))n = eTrA . Or, by changing the parameters in a matrix gives for the

variation ln det(A + ”A) ’ ln det A = ln det(I + A’1 ”A) = ln(1 + Tr(A’1 ”A) + O((A’1 ”A)2 )) =

ln(Tr(A’1 ”A)) + O((A’1 ”A)2 ), and thereby the sought relation as the overall constant not deter-

mined by the variation of the function is ¬xed by considering the identity matrix as det I = 1 and

ln I = (ln(I ’ (I ’ I)) = ’ ∞ (I ’ I)n /n = 0I. In connection with functional integrals we thus

n=1

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

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

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

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