the equation in Figure 9.45 diagrammatically, the options for entering into connected

diagrams for say propagators emerging from the 4-vertex is either into a 3-state

diagram, or 2- and 1-state diagrams. The prefactor of the next to last diagram on

the right in Figure 9.45 is caused by the three identical diagrams with the appearance

of the 2-state diagram.

We have thus expressed the 1-state connected Green™s function, in the presence

of the source, in terms of higher-order connected Green™s function and the free prop-

agators and vertices of the theory. By taking further derivatives in Eq. (9.77) we

can obtain the di¬erential equation satis¬ed by any connected Green™s function and

immediately write down its diagrammatic analog. We have thus made the full circle

back to the canonically derived non-equilibrium Feynman diagrammatics of Chap-

ters 4 and and 5, where the diagrams represented averages of the quantum ¬elds, but

now we are armed in addition with the powerful tool of a functional formulation of

non-equilibrium quantum ¬eld theory.

The amplitudes which in this chapter were de¬ned in terms of the diagrams

are thus for the case of quantum ¬eld theory the expectation values of products of

the quantum ¬elds of the theory. For example, the 2-state connected amplitude is

296 9. Diagrammatics and generating functionals

the 2-point Green™s function (normal or anomalous for the superconducting state as

dictated by the Nambu index), etc. The 1-state amplitude is the average value of

the quantum ¬eld. The 1-state amplitude, the tadpole, thus vanishes for a state with

a de¬nite number of particles, but can be non-vanishing for, for example, photons

in a coherent state. However, even when treating a system with a de¬nite number

of bosons, it can be convenient to introduce states where the average value of the

bose ¬eld is non-vanishing. This will be the case when we discuss the Bose“Einstein

condensate in Section 10.6.

Exercise 9.9. Show by taking one more source derivative of Eq. (9.77) that the

equation for the 2-state connected Green™s function in the presence of the source, for

a 3-vertex theory, has the diagrammatic form depicted in Figure 9.46.

= +

1

+ 2

Figure 9.46 Equation for the 2-state connected Green™s function for a 3-vertex

theory.

Then argue instead diagrammatically from the equation in Figure 9.45 to obtain the

above equation. One then learns to appreciate the skill of di¬erentiation.

9.6 One-particle irreducible vertices

In order to get a handle of the totality of connected diagrams we shall further exploit

their topology for classi¬cation. We introduce the concept of one-particle irreducible

diagrams (1PI-diagrams). All diagrams can then be classi¬ed uniquely by the topo-

logical property: they can be cut in two by cutting zero (1PI), one, two, etc., internal

bare lines. This will lead to the appearance of the one-particle irreducible vertices,

and to the important formulation of the theory in terms of the e¬ective action.

Consider the 1-state connected Green™s function in the presence of the source, i.e.

the derivative of the generator of connected Green™s functions

δW [J]

•1 = . (9.78)

δJ1

9.6. One-particle irreducible vertices 297

We shall refer to this function as the ¬eld.36 Besides being a function of the state

exposed by di¬erentiation, the ¬eld is also a functional of the source, •1 = •1 [J]. We

shall leave this feature implicit. However, in the diagrammatic notation we shall keep

the source dependence explicit, through the cross, as we introduce the diagrammatic

notation depicted in Figure 9.47 for the ¬eld.

•1 = 1

Figure 9.47 Diagrammatic representation of the 1-state connected Green™s function

or average ¬eld, the tadpole.

The state label of the ¬eld, exposed by di¬erentiating the generator of connected

Green™s functions, launches a free propagator which in its further propagation has two

options. The trivial one is where it propagates directly to the source in accordance

with the ¬rst diagram on the right in Figure 9.45, this option being represented

by the ¬rst diagram on the right in Figure 9.48. The other option corresponds to

interaction and the corresponding diagrams can be uniquely classi¬ed topologically

into distinct classes as follows: the exposed state where a propagator is launched can

enter into a connected diagrammatic structure which has the property that it can

not be cut in two by cutting only one internal bare propagator line, i.e. excepting

the launched propagator. By de¬nition such diagrams must not end on the source,

and these diagrams are thus a subset of the set of diagrams described by the ¬rst

diagram on the right in Figure 9.44, and are referred to as one-particle irreducible

diagrams, 1PI-diagrams. Diagrammatically this set of diagrams is represented by the

second diagram on the right in Figure 9.48. The next option is that the launched

propagator enters into a one-particle irreducible diagrammatic part and emerges into

a diagrammatic part such that the total diagram can be cut into two parts by cutting

one internal line at exactly one or two or three, etc., places, all of these lines therefore

emerging into the 1-state connected Green™s function in the presence of the source,

the ¬eld.37 Diagrammatically these sets of diagrams are therefore represented by the

third, etc., diagrams on the right in Figure 9.48 (combinatorial factors are inherited

from our convention, here expressed in the starting equation depicted in Figure 9.44).

The 1-state connected Green™s function, the tadpole, is thus represented in terms of

the one-particle irreducible vertices with attached tadpoles as depicted in Figure 9.48.

36 Or average ¬eld or classical ¬eld. The reason for this terminology will become clear in the

next chapter (or by comparison with the diagrammatic representation of the canonical operator

formalism). The diagrammatic structure of the theories considered are identical to those of the

quantum ¬eld theories we studied in Chapter 4. Therefore, interpreting the diagrammatic theory

as a quantum ¬eld theory, the 1-state amplitude is the average value of the quantum ¬eld (in

the presence of the source). For photons in a coherent state it describes the classical state of the

electromagnetic ¬eld.

37 Two (or more) lines can not enter into the same connected diagram, since then it is part of the

one-particle irreducible part.

298 9. Diagrammatics and generating functionals

= + +

1 1 + ···

+ +

2! 3!

Figure 9.48 The 1P-irreducible vertex representation of the 1-state connected

Green™s function in the presence of the source.

We could continue and construct the one-particle irreducible vertex representation

for any N -state amplitude, but we do not pause for that and relegate it to Section

9.6.2.

The 1-state connected amplitude in the presence of the source which by de¬nition

had the diagrammatic expansion depicted in Figure 9.44 and for a 3- plus 4-vertex

theory was shown to satisfy the diagrammatic equation depicted in Figure 9.45 has

now been organized into a di¬erent diagrammatic classi¬cation by introducing the

one-particle irreducible vertex functions, “1,2,...,N , which diagrammatically are rep-

resented by cross-hatched circles with amputated lines protruding and the dots as

usual represent the states where lines can end up or emerge from, as shown in Figure

9.49.

2

1

“12...N =

N

Figure 9.49 One-particle irreducible N -vertex function.

The diagrams on the right in Figure 9.48 correspond to a di¬erent re-grouping

of the diagrams compared to those on the right-hand side in Figure 9.45, the re-

grouping being based on a topological feature easily visually recognizable for any

diagram: its 0, 1, 2, etc., irreducibility with respect to internal cutting. According

to the topological construction, the one-particle irreducible vertices do not depend

on the source J. They are uniquely speci¬ed in diagrammatic perturbation theory

in terms of the bare vertices and bare propagators (and their topological property

of one-particle irreducibility). As we shall see in the next section, they provide yet

9.6. One-particle irreducible vertices 299

another way of capturing the content of the diagrammatic perturbation theory. The

virtue of the diagrammatic relationship expressed in Figure 9.48 is that no loops

appear explicitly, they are all buried in the one-particle irreducible vertices.

The diagrammatic structure of the equation expressed in Figure 9.48 should be

stressed: the tadpole in the presence of the source is expressed in terms of tad-

poles in the presence of the source attached to 1PI-irreducible vertices, i.e. in terms

of so-called tree diagrams, diagrams that become disconnected by cutting just one

propagator line. This observation shall be further developed in Section 10.3.

Analytically, the diagrammatic equation in Figure 9.48 reads

1 1

(0)

•1 = G12 J2 + “2 + “23 •3 + “234 •3 •4 + “2345 •3 •4 •5 + ... . (9.79)

2 3!

To write Eq. (9.79) in a compact form, we collect the one-particle irreducible vertices

into a generator, the generator of the one-particle irreducible vertex functions, the

e¬ective action38

∞

1

“[•] ≡ “12...N •1 •2 · · · •N (9.80)

N!

N

so that the one-particle irreducible vertices, or one-particle irreducible amputated

Green™s functions, are obtained by functional di¬erentiation

δ N “[•]

“12...N = . (9.81)

δ•1 δ•2 · · · δ•N

•=0

Recall that the one-particle irreducible vertices, “12...N , by construction do not de-

pend on the source, and the ¬eld is a function we can vary as it is a functional of the

source which is at our disposal to vary.

We can then rewrite Eq. (9.79) as

δ“[•]

(0)

•1 = G12 J2 + . (9.82)

δ•2

We introduce the diagrammatic notation depicted in Figure 9.50 for the e¬ective

action, the generator of 1PI-vertices.

•

“[•] =

Figure 9.50 Diagrammatic notation for the e¬ective action.

We introduce the diagrammatic notation for the functional derivative of the ef-

fective action depicted in Figure 9.51.

38 The e¬ective action is also referred to as the e¬ective potential for the theory. We shall return

to the reason for the terminology in Section 9.8. In the next chapter we develop the e¬ective action

approach, developing functional integral expressions for the e¬ective action.

300 9. Diagrammatics and generating functionals

•

δ“[φ]

=

δφ1 1

Figure 9.51 Diagrammatic notation for the ¬rst derivative of the e¬ective action.

The dot in Figure 9.51 signi¬es as usual a state label and the functional depen-

dence on the ¬eld is made explicit. Similarly, diagrams containing additional dots

represent additional functional derivatives with respect to the ¬eld, and give, upon

setting the ¬eld equal to zero, • = 0, the one-particle irreducible vertices depicted

diagrammatically in Figure 9.49.

Operating on both sides of Eq. (9.79) with the inverse free propagator according

to Eq. (9.5) thus gives

1 1

0 = J1 + “1 + (’G’1 + “)12 •2 + “123 •2 •3 + “1234 •2 •3 •4 + ... (9.83)

0

2 3!

and we can rewrite Eq. (9.83) in the form (upon absorbing the inverse free propagator

in the de¬nition of the 2-state irreducible vertex (’G’1 + “)12 ’ “12 ):

0

δ“[•]

0 = J1 + . (9.84)

δ•1

Diagrammatically, Eq. (9.84) is represented as depicted in Figure 9.52.

•

0= +

Figure 9.52 Source and e¬ective action relationship.

The content of Eq. (9.78) and Eq. (9.84) is that up to an overall constant, the

e¬ective action is the functional Legendre transform of the generator of connected

Green™s functions39

“[•] = W [J] ’ J • (9.85)

and the Legendre transformation thus determines the overall value, “[• = 0].

We note that in the absence of the source, J = 0, Eq. (9.84) becomes40

δ“[•]

= 0, (9.86)

δ•1

the e¬ective action is stationary with respect to the ¬eld. This is an equation stating

that the possible values of the ¬eld can be sought among the ones which make the

e¬ective action stationary.

39 In equilibrium statistical mechanics, the e¬ective action “ is thus Gibbs potential or free energy,

i.e. the (Helmholtz) potential or free energy in the presence of coupling to an external source, a

J-reservoir.

40 In the applications to non-equilibrium situations we consider in Chapter 12, this option is not

available as part of the source is an external classical force, the classical force that drives the system

out of equilibrium, and we shall employ Eq. (9.84).

9.6. One-particle irreducible vertices 301

9.6.1 Symmetry broken states

Having introduced the e¬ective action according to Eq. (9.80) we are considering the

normal state, i.e. we assume that the ¬eld vanishes in the absence of the source

δW [J]

•1 = = 0. (9.87)

δJ1

J=0

These, however, are not the only type of states existing in nature, there exist states

with spontaneously broken symmetry, i.e. states for which41

δW [J]

≡ •cl = 0 .

•1 = (9.88)

1

δJ1

J=0

We shall consider precisely such a situation and use the formalism presented in this

chapter when we discuss Bose“Einstein condensation in Section 10.6. In Chapter

8 we already encountered the generic symmetry broken state, the superconducting

state. It can be discussed as well in the present formalism by just allowing the ¬eld

or order parameter to be a composite object. We discuss this case in Section 10.5

where we in addition to a one-particle source include a two-particle source.

For a symmetry broken state we shall de¬ne the e¬ective action according to

∞

1

“[•] ≡ “12...N [•cl ] (•1 ’ •cl )(•2 ’ •cl ) · · · (•N ’ •cl ) . (9.89)

1 1 2 N

N!

N

This means that according to Eq. (9.84) we again have

δ“[•]

= ’J1 . (9.90)

δ•1

• =•cl

The e¬ective action vanishes for vanishing source.

By a shift of variables, • ’ •cl ’ •, we can of course rewrite Eq. (9.89) as

∞

1

“12...N •1 •2 · · · •N ,

“[•] = (9.91)

N!

N

where now the vertices “12...N = “12...N [•cl = 0] are evaluated in the normal or

1

so-called disordered or symmetric state where the classical ¬eld vanishes, •cl = 0.

1

We thus realize the fundamental importance of the e¬ective action: it allows us to

41 Such states are well-known in equilibrium statistical mechanics, for example from the existence

of ferro-magnetism, the appearance below a de¬nite critical temperature of an ordered state with

a magnetization in a de¬nite direction despite the rotational invariance of the Hamiltonian. These

spontaneously broken symmetry states were ¬rst studied in the mean ¬eld approximation, the

Landau theory, and the full theory of phase transitions, critical phenomena, were obtained by

Wilson using ¬eld theoretic methods. Super¬‚uid phases are broken symmetry states, and even more

fundamentally, the masses of quarks are the result of the Higgs ¬eld having a nonzero value.

302 9. Diagrammatics and generating functionals

explore the existence of symmetry broken states by searching for extrema of the

e¬ective action, i.e. solutions of

δ“[•]

=0 (9.92)

δ•1

for which the ¬eld is di¬erent from zero, • = 0.

We shall also encounter symmetry broken states created by a simpler mechanism,

viz. owing to the presence of an external classical ¬eld, but again the e¬ective

action approach shall prove useful for such non-equilibrium states, as we elaborate

in Chapter 12.

9.6.2 Green™s functions and one-particle irreducible vertices

In this section we shall show that since the generator of connected Green™s functions

and the e¬ective action are related by a Legendre transform we can, by using the

functional methods, easily obtain the systematic functional di¬erential equations

expressing connected Green™s functions in terms of the one-particle irreducible vertex

functions. But ¬rst let us argue for such equations at the purely diagrammatic level.

The connected 2-state amplitude or Green™s function is expressed solely in terms

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