+

= 4

1 1

+ +

8 4

1

1 +

+ 4

4

1

+ 8

Figure 9.18 The 2-state amplitude equation for a 3-line-vertex theory exposed to

second order in the coupling.

268 9. Diagrammatics and generating functionals

In this way an amplitude is expressed in terms of higher-order amplitudes ap-

pearing as the vertices launch propagator lines into states represented by amplitudes

of ever higher state numbers. We can in this fashion systematically develop the dia-

grammatic perturbation expansion order by order in the coupling constants. Let us

do it for the 2-state amplitude for a 3-line vertex theory up to second order in the

interaction. Using the diagrammatic expansion of the 2-state amplitude obtained

in Figure 9.18 for a 3-line vertex theory, the diagrammatic expansion of the 2-state

amplitude to second order in the 3-vertex can now explicitly be identi¬ed by neglect-

ing e¬ects higher than second order. The 2-state amplitude to second order in the

coupling thus has the diagrammatic expansion depicted in Figure 9.19.

1

= + 2

1

1 +

+ 4

2

Figure 9.19 The 2-state amplitude to second order for a 3-vertex theory.

We have noted the feature that the sum of vacuum diagrams will overall multiply

the zeroth and all second-order diagrams and can be separated o¬. Proceeding in

this fashion, the perturbative expansion of the 2-state amplitude (or in general any

N -state amplitude) to arbitrary order in the interaction can be generated.

Exercise 9.2. Consider a theory with both 3- and 4-vertex interaction and obtain the

diagrammatic expansion of the 2-state amplitude to second order in the interactions.

Another systematic characterization of the plethora of diagrams in perturbation

theory is exposing them according to the number of loops that appear in a diagram.

From the diagrammatic expansion of the 2-state amplitude in Figure 9.18 we obtain

that, to two-loop order, the 2-point amplitude in a 3-vertex theory is given by the

diagrams depicted in Figure 9.20.15

15 This

type of expansion, the loop expansion, will give rise to a powerful systematic approximation

scheme as discussed in Section 10.4. In quantum ¬eld theory it corresponds to a power series

expansion in , the number of loops in a diagram corresponds to the power in , and is thus a way

systematically to include quantum ¬‚uctuations.

9.1. Diagrammatics 269

+1 1

+1

= + 16

2 2

3

1 1 +

+4 + 16

16

1

3

1 +

+

+ 8

16

8

1 1

+ ···

+ +

16 16

Figure 9.20 The 2-state amplitude to two-loop order for a 3-line-vertex theory.

In low order perturbation theory, we have noticed the feature that the sum of all

vacuum diagrams separates o¬, and we show in the Section 9.5 that all amplitudes

can be expressed in terms of their corresponding connected amplitude times, the

amplitude representing the sum of all the vacuum diagrams.16

Exercise 9.3. Consider a theory with both 3- and 4-vertex interaction and obtain

the diagrammatic expansion of the 2-state amplitude to one-loop order.

Exercise 9.4. Consider a theory with both 3- and 4-vertex interaction and obtain

the diagrammatic expansion of the 2-state amplitude to two-loop order.

In this section we have proceeded from simplicity, the simple rules of diagram-

matics, to complexity, the multitude of systematically generated diagrams by the

simple law of dynamics, to interact or not to interact. However, this scheme soon

16 From the canonical version of non-equilibrium perturbation theory considered in Chapter 4, we

know that the sum of all the vacuum diagrams is an irrelevant number to the theory, in fact just one.

But in standard zero-temperature formulation and ¬nite temperature imaginary-time formulation

of perturbation theory they appear, and to include these cases we include them in the diagrammatic

discussion. Vacuum diagrams can be of use in their own right as discussed and taken advantage of

in Chapters 10 and 12.

270 9. Diagrammatics and generating functionals

gets messy; just try your luck in the previous exercise to muscle out all the diagrams

for a 3- plus 4-vertex theory. In order not to be blinded by all the trees in the forest

we shall now proceed to get a total view of the jungle, and in this way we return to

simplicity. We shall introduce an object that contains all the amplitudes of a theory

and the vehicle for extracting any desired amplitude of the theory. This object is

called the generating functional and the vehicle for revealing amplitudes will be dif-

ferentiation, and we shall obtain a formulation of the diagrammatic theory in terms

of di¬erential equations.

9.2 Generating functional

We now embark on constructing the analytical theory describing e¬ciently the to-

tality of all the diagrams describing the amplitudes, the quantities containing the

information of the theory. The complete set of all amplitudes possible in a given

theory can conveniently be collected into a generating functional

∞

1

A12...N J1 J2 · · · JN ,

Z[J] = (9.8)

N!

N =0

where summation over repeated indices is implied, or as we shall say state labels

appearing twice are contracted.17 The function of the possible particle states, J, is

called the source (or current).18 We have used a square bracket to remind us that we

are dealing not with a function but a functional.19 The expansion coe¬cients are the

amplitudes of the theory. Here the generating functional or generator is considered

to generate all the probability amplitudes of the quantum ¬eld theory in question.20

In the diagrammatic approach, the (N = 0) -term, the value of Z[J = 0], shall

by de¬nition be taken to be the amplitude representing the sum of all the vacuum

diagrams of the theory in question.21

17 For the continuous parts of the compound state label index the summation is actually integra-

tion, summation over small volumes. We shortly elaborate on this, but for simplicity we let this

feature be implicit using matrix contraction for convolution.

18 The source functions not only as a source for particles, but also as a sink, i.e. particle lines not

only emanate from the source but can also terminate there, a feature we bury in the indices and

need not display explicitly in the diagrammatics.

19 A functional maps a function, here J, into a number.

20 Actually, quantum ¬eld theory requires the substitution J ’ iJ, but for convenience we leave

out at this stage the imaginary unit since it is irrelevant for the ensuing discussion. The imaginary

unit is fully installed in Chapter 10.

21 In a T = 0 quantum ¬eld theory, the sum of all vacuum diagrams equals according to the Gell-

Mann“Low theorem, Eq. (4.20), a phase factor of modulus one. In the closed time path formulation,

which we shall always have in mind, the sum of all vacuum diagrams are by construction equal to

one. The (N = 0)-term can therefore be set equal to one, i.e. giving the normalization condition

Z[J = 0] = 1. 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, viz. Z[J] = 1,

provided that the sources on the two parts of the closed time path are taken as identical. When

calculating physical quantities, the sum of all vacuum diagrams in fact drops out as an overall

factor, a feature we have already encountered in low order perturbation theory in the previous

section. However, vacuum diagrams can in themselves be a useful calculational device, a feature

we shall employ when employing the e¬ective action approach in Chapter 10 and Chapter 12. In

9.2. Generating functional 271

This way of collecting all the data of a theory into a single object, the genera-

tor of the theory, is indeed quite general. In equilibrium statistical mechanics the

generating functional will be the partition function in the presence of an external

¬eld, the source (recall the general relation between quantum theory and thermody-

namics as discussed in Section 1.1 (there displayed explicitly only for the simplest

case of a single particle, the general case being obtained straightforwardly). The

construction of the generating functional is also analogous to how the probabilities

in a classical stochastic theory are collected into a generating function that generates

the probabilities of interest of the stochastic variable (in that case the (N = 0) -term

is one by normalization). In that context the generating function is usually reserved

to denote the generator of the moments of the probability distribution involving a

Fourier transformation of the probability distribution. This avenue we shall also take

advantage of in the context of quantum ¬eld theory of non-equilibrium states when

we introduce functional integration in Chapter 10.

Since the values of the source function J in di¬erent states are independent,

varying the magnitude of the source for a given state in¬‚uences only the source for

the state in question and we have for such a variation (a formal discussion of the

involved functional di¬erentiation is given in the next section)

δJM

= δ1M , (9.9)

δJ1

i.e. the Kronecker function which vanishes unless 1 = M . Di¬erentiating the gener-

ating functional with respect to the source function J and subsequently setting J = 0

therefore generates the amplitudes of the theory of interest, for example

δ N Z[J]

= A12...N , (9.10)

δJ1 δJ2 . . . δJN

J=0

where the factorial in Eq. (9.8) is canceled by the same number of equal terms

appearing due to the symmetry, Eq. (9.7), of the probability amplitude.22 In the

particle picture of quantum ¬eld theory the function J acts as a source for creating

or absorbing a particle in the state speci¬ed by its argument.

For continuous variables, such as space and (contour or for real forward and

return) time, the summation in Eq. (9.8) is actually short for integration, and we

encounter instead of the Kronecker function, Eq. (9.10), Dirac™s delta function,23 say

in the spatial variable

δJx

= δ(x ’ x ). (9.11)

δJx

However, this feature will in our notation be kept implicit for continuous variables.

We have used the symbol δ to designate that the type of di¬erentiation we have in

mind is functional di¬erentiation, the strength of the source is varied for given state

label.

the present chapter, the starting point is diagramatics and for that reason the (N = 0)-term is by

de¬nition taken to be the sum of all the vacuum diagrams.

22 We ¬rst discuss the Bose case, the Fermi case needs the introduction of Grassmann numbers,

as discussed in Section 9.4.

23 For a discussion of Dirac™s delta function we refer to appendix A of reference [1].

272 9. Diagrammatics and generating functionals

Thus functional generation of the amplitudes is achieved by functional di¬eren-

tiation. We therefore dwell for a moment on the mathematical rules of functional

di¬erentiation. However, in the intuitive approach of this chapter, we could in view

of Eq. (9.9) simply de¬ne functional di¬erentiation as the sorcery: cutting open the

contraction of the source and amplitude, thereby exposing the state.

9.2.1 Functional di¬erentiation

Functional di¬erentiation maps a functional, F [J], into a function according to the

limiting procedure

F [J(x ) + µδ(x ’ x )] ’ F [J]

δF [J]

= lim . (9.12)

δJ(x) µ’0 µ

More precisely, into a function of x and in general still a functional of J. Since we shall

be dealing with functionals which have Taylor expansions, i.e. have a perturbation

expansion in terms of the source, an equivalent de¬nition is

δF [J]

F [J + δJ] ’ F [J] = dx δJ(x) + O(δJ 2 ) . (9.13)

δJ(x)

The functional derivative measures the change in the functional due to an in¬nitesi-

mal change in the magnitude of the function at the argument in question.

The operational de¬nition of Dirac™s delta function

dx δ(x ’ x ) J(x )

J(x) = (9.14)

is thus seen to be identical to the functional derivative speci¬ed in Eq. (9.11) if in

Eq. (9.12) or Eq. (9.13) we choose F to be the functional

F [J] = J(x) (9.15)

for ¬xed x, or returning to our index notation F [J] = Jx .

For the functional de¬ned by the integral

F [J] ≡ dx f (x) J(x) (9.16)

we get for the functional derivative

δF [J]

= f (x) (9.17)

δJ(x)

exposing the kernel.

As regards the discrete degrees of freedom we have instead of Eq. (9.13)

j

”F [J]

F [J + ”J] ’ F [J] = ”Jσ1 (9.18)

”Jσ1

σ1 =’j

9.2. Generating functional 273

and if we choose F to be the functional

F [J] = Jσ1 (9.19)

the functional derivative becomes

”Jσ1

= δσ1 ,σ1 (9.20)

”Jσ1

i.e. the Kronecker part in Eq. (9.9). The δ on the right-hand side in Eq. (9.9) is thus

a product of delta and Kronecker functions in the continuous respectively discrete

variables.

As usual in theoretical physics, to be in command of formal manipulations one

needs only to be in command of the exponential function. In the context of functional

di¬erentiation, we note that the functional di¬erential equation

δF [J] δG[J]

= F [J] (9.21)

δJ(x) δJ(x)

has the solution

F [J] = eG[J] , (9.22)

which is proved directly using the expansion of the exponential function or follows

from the chain rule for functional di¬erentiation

δ δG[g] ‚f (G)

f (G[g]) = (9.23)

δg(x) δg(x) ‚G

for arbitrary functional G and function f .24

Of particular importance is the case

F [J] ≡ e dx f (x) J(x)

, (9.24)

where f is an arbitrary function, and in this case we get for the functional derivative

δF [J]

= f (x) F [J] . (9.25)

δJ(x)

Exercise 9.5. Standard rules for di¬erentiation applies to functional di¬erentiation.

Verify for example the rule

δ δ δ

(F [f ]G[f ]) = F [f ] G[f ] + F [g] G[f ] (9.26)

δf (x) δf (x) δf (x)

and the functional Taylor series expansion

δ

dx f2 (x)

F [f1 + f2 ] = e F [f1 ] . (9.27)

δ f 1 (x )

24 Inequations Eq. (9.12) and Eq. (9.13) we deviate from our general notation that capital letters

represent functionals whereas lower capital letters denote functions.

274 9. Diagrammatics and generating functionals

9.2.2 From diagrammatics to di¬erential equations

We shall now show how to capture the whole diagrammatics in a single functional

di¬erential equation. We introduce the diagrammatic notation for the generating

functional, Z, displayed in Figure 9.21.

Z[J] =

Figure 9.21 Diagrammatic notation for the generating functional.

According to the de¬nition of the generating function in terms of amplitudes and

sources, Eq. (9.8), we have the relation as shown in Figure 9.22.

+ ···

1 1

+ + 2! + 3!

=

Figure 9.22 Diagrammatic representation of the generating functional.

The ¬rst term on the right of Figure 9.22 is the sum of all vacuum diagrams and

independent of the source, and we have introduced the diagrammatic notation that

a cross designates the source, the label of the source being that of the state indicated

by the corresponding dot as shown in Figure 9.23.

=J