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