for spectral densities, non-equilibrium distribution functions and the order parame-

ter, which of course in general are inaccessible to analytic treatment, but which can

be handled by numerics. The versatility of the quasi-classical Green™s technique to

understand non-equilibrium phenomena in super¬‚uids is testi¬ed by the wealth of

results obtained using it. For the reader interested in non-equilibrium superconduc-

tivity, we give the general references where further applications can be trailed [48]

[49].

9

Diagrammatics and

generating functionals

At present, the only general method available for gaining knowledge from the funda-

mental principles about the dynamics of a system is the perturbative study. Accord-

ing to Feynman, as described in Chapter 4, instead of formulating quantum theory in

terms of operators,1 the canonical formulation, for calculational purposes quantum

dynamics can conveniently be formulated in terms of a few simple stenographic rules,

the Feynman rules for propagators and interaction vertices.

In Chapters 4 and 5, we showed how to arrive at the Feynman rules of diagram-

matic perturbation theory for non-equilibrium states starting from the Hamiltonian

de¬ning the theory. The feature of non-equilibrium states, originally carried by the

dynamical indices, could be expressed in terms of two simple universal vertex rules for

the RAK-components of the matrix Green™s functions. We are thus well acquainted

with diagrammatics even for the description of non-equilibrium situations. However,

for the situations studied using the quantum kinetic equations in Chapters 7 and

8, only the Dyson equation was needed, i.e. the self-energy, the 2-state one-particle

irreducible amputated Green™s function. No need for higher-order vertex functions

was required, and the full ¬‚ourishing diagrammatics was not put into action. In

this chapter we shall proceed the other way around. We shall show that the dia-

grammatics of a physical theory, including the description of non-equilibrium states,

can be obtained by simply stating quantum dynamics, the superposition principle,

as the two exclusive options for a particle: to interact or not to interact! From this

simple Shakespearean approach we shall construct the Feynman diagrammatics of

non-equilibrium dynamics. Thus starting with bare propagators and vertices de¬n-

ing a physical theory, and constructing its dynamics in diagrammatic perturbation

theory, we then show how to capture all of the diagrammatics in terms of a single

functional di¬erential equation. In this way we shall by simple topological arguments

for diagrams construct the generating functional approach to quantum ¬eld theory of

non-equilibrium states. The corresponding analytic generating functional technique

1 Orequivalently for that matter in terms of path or functional integrals as we discuss in the next

chapter.

253

254 9. Diagrammatics and generating functionals

is originally due to Schwinger [50]. In the next chapter, we shall then follow Feynman

and instead of describing the dynamics of a theory in terms of di¬erential equations,

describe its corresponding representation in terms of path integrals. These analytical

condensed techniques shall prove very powerful when unraveling the content of a ¬eld

theory. The methods were originally developed to study equilibrium state proper-

ties, in fact strings of ¬eld operators evaluated in the vacuum state as relevant to

the Green™s function™s of QED, and later taken over to study equilibrium properties

of many-body systems. In the following we shall develop these methods for general

non-equilibrium states.

A point we wish to stipulate is that diagrammatics and the equivalent functional

methods are a universal language of physics with applications ranging from high to

low energies: from particle physics over solid state physics even to classical stochastic

physics and soft condensed matter physics, as we shall exemplify in the following

chapters. In the next chapter, we shall eventually use the e¬ective action approach

to study Bose“Einstein condensation, viz. the properties of a trapped Bose gas.

In Chapter 12, we shall consider classical statistical dynamics, classical Langevin

dynamics, where the ¬‚uctuations are caused by the stochastic nature of the Langevin

force, a problem which, interestingly enough, mathematically is formally equivalent

to a quantum ¬eld theory.

9.1 Diagrammatics

According to the Feynman rules, the quantum theory of particle dynamics is de¬ned

by its bare propagators and vertices, specifying the possible transmutations of parti-

cles and thereby describing how any given particle con¬guration can be propagated

into another one. In the standard model, the elementary particles consist of mat-

ter constituents: leptons (electron, electron neutrino and their heavier cousins) and

quarks (up and down and their heavier cousins, three families in all), all spin-1/2

particles and therefore fermions, and the force carriers which are bosons and medi-

ate interaction through their exchange between the matter constituents, or realize

transmutation of particles through decay. The electro-weak force is mediated by the

photon and the heavy vector bosons, and the strong force is mediated by gluons.

In condensed matter physics, electromagnetism or simply the Coulomb interaction is

the relevant interaction; typically the interactions of electrons with photons, phonons,

magnons and other electrons are of chief interest. In statistical physics, thermal as

well as quantum ¬‚uctuations are of interest but the diagrammatics are the same,

even for non-equilibrium states, the emphasis of this book. In equilibrium statisti-

cal mechanics, thermodynamics, thermal and not quantum ¬‚uctuations are often of

chief importance, and the use of diagrams are also of great e¬ciency, for example

in understanding phase transitions. In Chapter 12 we demonstrate the usefulness

of Feynman diagrams even in the context of classical physics, viz. in the context

of classical stochastic dynamics. At the level of diagrammatics there is no essen-

tial di¬erence in the treatment of di¬erent physical systems and di¬erent types of

¬‚uctuations, and all cases will here be dealt with in a uni¬ed description.

A generic particle physics experiment consists of colliding particles in certain

9.1. Diagrammatics 255

states and at a later time detecting the resulting debris of particle content in their

respective states, or rather reconstructing these since typically the particle content

of interest has long ceased to exist once the detector signals are recorded.2 To

any possible outcome only a probability P can according to quantum mechanics be

attributed. To each possible process (a ¬nal con¬guration of particles given an initial

one) is thus associated a (conditional) probability P . The probability for a certain

process occurring, is according to the fundamental principle of quantum mechanics,

speci¬ed by a probability amplitude A, a complex number, giving the probability for

the process as the absolute square of the probability amplitude3

P = |A|2 . (9.1)

In order not to clutter diagrams and equations with indices, a compound label is

introduced

1 ≡ (s1 , x1 , t1 , σ1 , . . .) (9.2)

for a complete speci¬cation of a particle state and it thus includes: species type s,

space and time coordinates (x, t), internal (spin, ¬‚avor, color) degrees of freedom

σ, . . ., or say in discussing superconductivity a Nambu index. Most importantly,

since we also allow for non-equilibrium situations the index 1 includes a dynamical or

Schwinger“Keldysh index in addition, or equivalently we let the temporal coordinate

t become a contour time „ on the contour depicted in Figure 4.4 or Figure 4.5.

However, we shall for short refer to the labeling 1 as the state label. Instead of the

position, the complementary momentum representation is of course more often used

owing to calculational advantages or experimental relevance, say in connection with

particle scattering, but for the present exposition one might advantageously have the

more intuitive position representation in mind.

We now embark on constructing the dynamics of a non-equilibrium quantum ¬eld

theory in terms of diagrams, i.e. stating the laws of nature in terms of the propagators

of species and their vertices of interaction.

9.1.1 Propagators and vertices

Feynman has given us a lucid way of representing and calculating probability am-

plitudes in terms of diagrams. In this framework a theory is de¬ned in terms of

the particles it describes, their propagators and their possible interactions. Each

(0)

particle is attributed a free or bare propagator, G12 , the probability amplitude to

freely propagate between the states in question, say spin states and space-time points

(x1 , σ1 , t1 ) and (x2 , σ2 , t2 ). The corresponding free propagator or Green™s function,

2 Indeed any dynamics of particles can be viewed as caused by collisions, i.e. interactions, and the

following diagrammatic discussion is valid for any in-put/out-put kind of machinery. The dynamics

need not be dictated by the laws of physics for diagrammatics to work, it can be the result of any

mechanism of choice, say a random walk. The diagrammatic approach can therefore also be used

to study statistical mechanics models, and for the brave perhaps models of evolution or climate, or

the stock market for the greedy.

3 For almost 100 years, no mechanics beyond these probabilities has been found despite many

brave attempts. Furthermore, we stress the weird quantum feature that the probabilities have to be

calculated through the more fundamental amplitudes, which are the true carriers of the dynamics

of the theory.

256 9. Diagrammatics and generating functionals

the amplitude for no interaction, is represented diagrammatically by a line as shown

in Figure 9.1, where a dot signi¬es a state label.

1 2 (0)

= G12

Figure 9.1 Diagrammatic notation for bare propagators.

In the context of quantum theory, the propagator or Green™s function is the

conditional probability amplitude for the event 1 to take place given that event 2 has

taken place. All states have equal status and the bare propagators are symmetric

(0) (0)

functions of the state labels, G12 = G21 . The free propagator is species speci¬c

(0)

G12 ∝ δs1 s2 , (9.3)

a free particle can not change its identity.4

In the treatment of non-equilibrium states in the real-time technique, the real-time

forward and return contour matrix representation, Eq. (5.1), or better the economical

and more physical symmetric representation of the bare propagator, should thus be

used, the latter having the following additional matrix structure in the dynamical or

Schwinger“Keldysh indices:

GA

0 0

G0 = . (9.4)

GR GK

0 0

The bare vertices describe the possible interactions allowed to take place, and

generic examples, the three- and four-line attachment or connector vertices, are dis-

played in Figure 9.2.

1

1 2

2

= g1234

= g123

3

4

3

Figure 9.2 Diagrammatic notation for bare vertices.

(0)

4 If,

say, there is no spin dynamics then G12 ∝ δσ 1 σ 2 . Sometimes it is convenient to include in

the free propagator the change in the internal degrees of freedom of the particle; for example, if

the spin of the particle is coupled to an external magnetic ¬eld. The chosen notation is seen to be

capable of dealing with any kind of dynamics.

9.1. Diagrammatics 257

Without risking confusion, we have in accordance with standard notation also

used a single dot in connection with vertices (and, say, not a triangle with three

attached dots or a box with four attached dots), and here the dot does not specify a

single state label but several, as speci¬ed by the protruding stubs to which propaga-

tors can be attached. The rationale for this is that quantum ¬eld theories are local

in time, so that at least all time labels of the propagators meeting at a vertex are

identical. In the 3-connector vertex, the single dot with its three protruding stubs

thus represents three state labels where propagators can be attached and they can

all be di¬erent. The form of the vertices, as speci¬ed by the indices, describes how

particle species are transmuted into other particle species or how a particle changes

its quantum numbers owing to interaction. The numerical value of the vertex, the

amplitude for the process speci¬ed by g, the coupling constant or charge, gives the

strength of the process.

The two ingredients, propagators and vertices, are the only building blocks for

constructing the Feynman diagrams. In condensed matter physics, the corresponding

amplitudes represented by the propagators and vertices are the only ones needed to

specify the theory. These numbers are taken from experiment, for example from

the measured values of the mass and charge of the electron. However, in relativistic

particle physics they are only bare parameters, i.e. rendered unobservable quantities

owing to the presence of interactions. For example, the value of the mass entering a

bare propagator is a quantity unreachable by experiment (i.e. has no manifestation in

the world of facts) since it corresponds to the non-existent situation where the particle

is not allowed to interact. The interaction causes the mass to change, and in order to

make contact with experiment the knowledge of the measured masses (and charges)

must be introduced into the theory through the scheme of renormalization.5 The

expressions for the bare propagators are known a priori, since they are speci¬ed by the

space-time symmetry, and the forms of the vertices are given by the symmetry of the

theory, but their numerical values must be taken from comparison with experiment.6

In elementary particle physics, only the two types of vertices displayed in Figure

9.2 occur, the 4-connector vertex being relevant only for the gluon“gluon coupling.

The 3-connector vertex is ubiquitous, for example describing electron“photon inter-

action or pair creation such as in QED. In fact, in QED, the theory restricted to

the multiplet of electron and its anti-particle, the positron, and the photon, the ver-

tex is nonzero for various species combinations, describing both electron or positron

emission or absorption of a photon, or pair creation or destruction. In condensed mat-

ter physics, the 3-connector could for example describe electron“phonon, electron“

electron or electron“magnon interactions, as discussed in Section 2.4. In statistical

physics, where the propagators describe both thermal and quantum ¬‚uctuations and

5 Of course, the interactions encountered in condensed matter physics in the same manner lead

to renormalization of, say, the electron mass, as we have calculated in Section 7.5.2. However, this

is a ¬nite amount on top of the in¬nitely renormalized bare mass. Usually this is an e¬ect of only

a few percent of the electron mass, except in for example the case of heavy fermion systems.

6 In relativistic quantum theory the forms of the propagators are speci¬ed by Lorentz invariance.

For a massive particle the propagator or Green™s function is speci¬ed by its bare mass and the type

of particle in question. Also the form of the interaction can be obtained from the symmetry and

Lorentz invariance of the theory, whereas the strength of the coupling constants are phenomenolog-

ical parameters, i.e. they are obtained by comparison of theory and experiment.

258 9. Diagrammatics and generating functionals

for example e¬ects of quenched disorder, vertices of arbitrary complexity can occur.7

In the theory of phase transitions, which is an equilibrium theory, the diagrams de-

scribe transitions, i.e. thermal ¬‚uctuations, between the possible states of the order

parameter relevant to the transition and critical phenomenon in question. However,

we shall frame the arguments in the appealing particle representation, but since ar-

guments are about the topological character of diagrams the formalism applies to

any representation and any type of ¬‚uctuations and thus to any kind of ¬eld theory.

9.1.2 Amplitudes and superposition

Consider an amplitude A1234...N speci¬ed by N external states, an N -state amplitude.

It could, for example, describe the transition probability amplitude for collision of two

particles in states 1 and 2, respectively, to end up in a particle con¬guration described

by the states 3, 4, . . . , N , or the decay of a particles in state 1 into particles in states

2, 3, . . . , N , etc. This general conditional probability amplitude is represented by the

N -state diagram shown in Figure 9.3.8

3

2

1 N

Figure 9.3 Diagrammatic notation for the N -external-state amplitude A1234...N .

Specifying any amplitude is done by following the laws of Nature, quantum dy-

namics, which at the diagrammatic level of bare propagators and vertices is the basic

rule that a particle has two options: to interact or not to interact!9 The probability

amplitude for a given process, characterized by the ¬xed initial and ¬nal state labels,

is then construed as represented by the multitude of topologically di¬erent diagrams

that can be constructed using the building blocks of the theory, viz. all the topolog-

7A case in question within the context of classical stochastic phenomena will be discussed in

Chapter 12. The simplest vertex, a two-line vertex, is of course also relevant, viz. describing a

particle interacting with an external classical ¬eld, but it is trivial to include, as will become clear

shortly and we leave it implicit in the discussion for the moment.

8 In statistical mechanics the diagrams can represent probabilities directly, say transitions between

con¬gurations of the order parameter.

9 The former option is evident since otherwise the particle would live undetected, devoid of

in¬‚uence. The latter option is required by the fact that not all particles can interact directly.

9.1. Diagrammatics 259

ically di¬erent diagrams that the vertices and bare propagators allow. Examples of

diagrams for the 4-state amplitude are shown in Figure 9.4 for the theory de¬ned by

having only a 4-connector vertex.

3

2

3

2

···

+

=

1 4

1 4

3

2

···

+ +

1 4

3

2

···

+

+

1 4

Figure 9.4 Generic types of diagrams.

The numerical value represented by a diagram is obtained by multiplying together

the amplitudes for each component, propagators and vertices, constituting the dia-

gram,10 and in accordance with the superposition principle summation occurs over

all internal labels, adding up all the alternative ways the process can be e¬ected,

for example summation over all the alternative space-time points where interaction

could take place is performed.11 The ¬rst diagram on the right in Figure 9.4 thus

10 This rule is often left implicit, but represents the multiplication rule of quantum mechanics:

that amplitudes for events e¬ected in a sequence should be multiplied in order to get the amplitude

for the sequence of events. The expression of causality in quantum mechanics.

11 Only topologically di¬erent diagrams appear, interchanging the labeling of interaction points,

i.e. permutation of vertices, are not additionally counted. This is precisely how the diagrammatics

of (non-equilibrium) quantum ¬eld theory turned out as discussed in Chapter 4; the important

260 9. Diagrammatics and generating functionals

represents the analytical expression as displayed in Figure 9.5, and we have intro-

duced the convention that repeated indices are summed over, or as we shall say state

labels appearing twice are contracted.

3

2

(0) (0) (0) (0)

= G11 G22 g1 G3 G4

234 3 4

1 4

Figure 9.5 Numerical and diagrammatic correspondence.

The basic principle of quantum mechanics, the superposition principle, entails

further the diagrammatic rule: the probability amplitude for a real process is rep-

resented by the sum of all diagrams allowed, i.e. constructable by the vertices and

propagators de¬ning the theory. In accordance with the superposition principle,

the amplitudes obtained from each single diagram are then added, adding up the

contributions from all the di¬erent internal or virtual ways the initial state can be

connected to the ¬nal state in question. The sum gives the amplitude for the process

in question.

The diagrammatic representation of any amplitude consists of three topologi-

cally di¬erent classes of diagrams: connected diagrams, disconnected or unlinked

diagrams, and diagrams accompanied by vacuum ¬‚uctuations, the virtual processes

where particles pops out and back into the vacuum. For the amplitude with four

external states, the three classes for the theory de¬ned by having only a 4-connector

vertex are exempli¬ed in Figure 9.4.

The last diagram in Figure 9.4 represents the type of diagrams where a diagram