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1
+
= 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

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