8 |T ’ Tc |

In the superconducting state the relaxation of the order parameter is, according

to Eq. (8.211), determined by the non-equilibrium distribution of the quasi-particles,

8.5. Charge imbalance 249

which in turn is in¬‚uenced by the time dependence of the order parameter. In the

spatially homogeneous situation where h2 vanishes, the kinetic equation, Eq. (8.191),

reduces to

N1

™ ™

N1 h1 + R2 ”‚E h0 = ’ h1 . (8.219)

„in

Calculating the control function gives

π ™

χ= „in ” (8.220)

4Tc

and according to the time-dependent Ginzburg“Landau equation the relaxation time

for the order parameter, ” „in 1, is

π 3 Tc

„R = „in . (8.221)

7ζ(3) ”

Experimental observation of the relaxation of the magnitude of the order parameter

can been achieved by driving the superconductor out of thermal equilibrium by a

laser pulse [44].

8.5 Charge imbalance

Under non-equilibrium conditions in a superconductor a di¬erence in the electro-

chemical potential between the condensate and the quasi-particles can exist, re¬‚ecting

the ¬nite rate of conversion between supercurrent and normal current. For example,

charge imbalance occurs when charge from a normal metal is injected into a super-

conductor in a tunnel junction. As an application of the theory of non-equilibrium

superconductivity, we shall consider the phenomenon of charge imbalance generated

by the combined presence of a supercurrent and a temperature gradient. We shall

limit ourselves to the case of temperatures close to the critical temperature where

analytical results can be obtained.23

The charge density is in the real ”-gauge, recall Section 8.1.3,

⎛ ⎞

∞

ρ = 2eN0 ⎝¦ + dE N1 (E) f2 (E)⎠ , (8.222)

’∞

where the condensate electro-chemical potential in general is ¦ = χ/2 ’ e•, χ being

™

the phase of the order parameter. We have introduced distribution functions related

to the original ones according to h1 = 1 ’ 2f1 and h2 = ’2f2 .24 We could insert

instead the full distribution function, f = f1 + f2 , in Eq. (8.222) as f1 is an odd

function, and thereby observe that the charge of the quasi-particles described by the

distribution function is the elementary charge, the full electronic charge.

23 We essentially follow the presentation of reference [45]. For general references to charge imbal-

ance in superconductors, as well as other non-equilibrium phenomena, we refer to the articles in

reference [42].

24 In reference [46], they are referred to as the longitudinal and transverse distribution functions.

250 8. Non-equilibrium superconductivity

The strong Coulomb force suppresses charge ¬‚uctuations, but it is possible to

have a charge imbalance between the charge carried by the condensate of correlated

electrons and the charge carried by quasi-particles

∞

Q— = 2eN0 dE N1 (E) f2 (E) . (8.223)

’∞

The presence of a temperature variation, T (r) = T + δT (r) creates a non-

equilibrium distribution in the thermal mode

E ‚f0

δf = f1 ’ f0 = ’ δT (8.224)

T ‚E

where f0 is the Fermi function. The presence of a supercurrent, ps = mvs = ’(∇χ +

2eA)/2, couples via the kinetic equation for the charge mode the thermal and the

charge mode. For a stationary situation with f2 homogeneous in space we have

according to Eq. (8.192)

2N2 ”f2 ’ 4D0 N2 R2 ps · ∇f1 = K2 [f2 ] . (8.225)

The ¬rst term on the left gives rise to conversion between the supercurrent and the

current carried by quasi-particles, while the second term is a driving term propor-

tional to vs · ∇T . Close to the transition temperature, Tc , the collision integral is

dominated by energies in the region E T . In this energy regime we have ± 1

˜˜

and β, γ · where we have introduced the notation · = ”(T )/T for the small

parameter of the problem. The collision integral then becomes proportional to the

inelastic collision rate

⎛ ⎞

∞

N1 ⎝ ‚f0

dE N1 (E) f2 (E)⎠ .

K2 [f2 ] = ’ f2 + (8.226)

„ (E) ‚E

’∞

The last term is proportional to the charge imbalance, and we get the following

kinetic equation

N1 ‚f0 E ‚f0

(2”„ (E)N2 + N1 ) f2 + Q— = ’4mD0 „ (E)N2 R2 vs · ∇T. (8.227)

2N0 ‚E T ‚E

Integrating with respect to the energy variable gives

Q— 4mD0 A

=’ vs · ∇T , (8.228)

T 1’B

2N0

where

∞

‚f0 N1 N2 R2 E„ (E)

A=’ dE (8.229)

‚E N1 + 2”„ (E)N2

’∞

8.6. Summary 251

and

∞

2

‚f0 N1

B=’ dE . (8.230)

‚E N1 + 2”„ (E)N2

’∞

Assuming weak pair-breaking the quantities can be evaluated. To zeroth order in ·

we have B 1 as N1 1 and N2 0. From the structure of the densities of states

it is apparent that the main correction contribution comes from the energy range ”

up to a few ”. We can therefore use the high-energy expansion

”“

N2 = , (8.231)

E 2 + “2

where

1 1 1

+ D0 (p2 ’ ”’1 ∇2 ”)

“= + (8.232)

s

2„ (E) „s 2

is the pair-breaking parameter. In the limit of weak pair-breaking, “ ”, and ”

is small as we assume the temperature is close to the critical temperature, so that

”(„ (E)“)1/2 T , and we get

π”

B=1’ (2„ (E)“)1/2 . (8.233)

4T

In the BCS-limit, A is logarithmically divergent due to the singular behavior of

the density of states, but the pair-breaking smears out the singularity and gives a

logarithmic cut-o¬ at ln(4”/“), and we have

” 4”

+ 2(2„ (E)“ ’ 1)1/2 arctan((2„ (E)“ ’ 1)1/2 )

A= ln . (8.234)

8T “

In the limit where electron“phonon interaction provides the main pair breaking mech-

anism, “ ∼ 1/2„E , the charge imbalance thus becomes

2 pF l

Q— = 2N0 (vs · ∇T ) ln(8”„ (E)) . (8.235)

3π T

For a discussion of the experimental observation of charge imbalance we refer the

reader to reference [47].

8.6 Summary

In this chapter we have considered non-equilibrium superconductivity. By using the

quasi-classical Green™s technique, a theory with an accuracy in the 1% range was

constructed that were able to describe the non-equilibrium states of a conventional

superconductor. This is a rather impressive achievement bearing in mind that a su-

perconductor is a messy many-body system. In general one obtains coupled equations

252 8. Non-equilibrium superconductivity

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

(here a connected one) appears together with a vacuum ¬‚uctuation diagram. Vacuum

diagrams close onto themselves, no propagator lines end up on the external states,

and they appear as unlinked diagrams. According to the multiplication rule, the

two amplitudes represented by the two sub-parts of the total diagram are multiplied

together to get the total amplitude represented by the diagram. The ¬rst and second

diagrams on the right in Figure 9.4 are of the connected and disconnected type,

respectively. These diagrams, according to the general rule of diagram construction,

can also be accompanied by any vacuum ¬‚uctuations constructable. The symbol + · · ·

in the ¬gure summarizes envisioning all diagrams constructable with the vertices and

propagators de¬ning the theory. The total class of diagrams is thus an in¬nite myriad

with in¬nite repetitions.

The totality of all diagrams can thus (with the help of our most developed sense)

feature that the factorial provided by the expansion of the exponential function is canceled by this

redundancy.

9.1. Diagrammatics 261

be envisioned perturbatively. However, this is of little use unless only trivial lowest

order perturbation theory needs to be invoked. One approach to a more powerful

diagrammatic representation is by using topological arguments to partially re-sum

the diagrammatic perturbation expansion in terms of e¬ective vertices and the full 2-

state propagators, i.e. in terms of so-called skeleton diagrams.12 In the next section,

we shall ¬rst pursue the hierarchal option on our way to this goal, expressing any

N -state amplitude in terms of amplitudes with di¬erent numbers of external states.

Before embarking on deriving the fundamental diagrammatic equation, we intro-

duce the inverse propagator. The inverse of the free or bare propagator is speci¬ed

by the (partial di¬erential) equation satis¬ed by the free propagator

(G’1 )1¯ G(0) = δ12 (9.5)

0 1 ¯

12

or since the propagator is symmetric in its labels

(G’1 )1¯ G(0) = δ12 = G1¯ (G’1 )¯ .

(0)

(9.6)

0 0

1

¯

1 12

12

We have written the equation satis¬ed by the free propagator in matrix notation,

in terms of an integral operator as summation over repeated indices is implied.13

For later use we introduce diagrammatic notation for the inverse free propagators as

depicted in Figure 9.6.

1

1

(G’1 )11 =

0

Figure 9.6 Diagrammatic notation for the inverse free propagator.

Using the basic diagrammatic rule: to interact or not, we shall start obtaining dia-

grammatic identities relating amplitudes, and eventually express these diagrammatic

relations in terms of di¬erential equations.

9.1.3 Fundamental dynamic relation

To get started on a systematic categorization of the plethora of diagrams, let us ¬rst

consider the case where one particle is not allowed to interact and let us separate out

its state to appear on the left in the diagram specifying the amplitude in question as

depicted in Figure 9.7. Since not interacting is an option even for a particle capable of

interacting, this seemingly irrelevant case of a completely non-interacting particle is a

¬rst step in the general deconstruction of an N -state amplitude into amplitudes with

less external states, and allows furthermore a comment on the quantum statistics of

identical particles.

12 This was performed in Section 4.5.2, starting with the canonical formalism.

13 The inverse free contour-ordered Green™s function encountered in Section 4.4.1, or the inverse

free matrix Green™s function of Section 5.2.1, stipulating the additional matrix structure in the

dynamical indices, had integral kernels typically consisting of di¬erential operators operating on the

delta function.

262 9. Diagrammatics and generating functionals

2

1

N

Figure 9.7 General N -state diagram.

Since the particle in state 1 is assumed not to interact, its only option is to

propagate directly to a ¬nal state, and the amplitude A1234...N can in this case be

expressed in terms of the amplitude which has two external states less according to

the basic rule: everything can happen on the way between the (N ’ 2) other ¬nal

states, and the diagrammatic equation displayed in Figure 9.8 is obtained.

2 2

1 2

1

3

=

+···+

N ’1

N N

1 N

Figure 9.8 Diagrams for the non-interacting particle labeled by 1.

The N -state amplitude is in this case represented by the amplitudes speci¬ed by

(N ’ 2) external states, i.e. A23...N without the index M labeling the state where the

propagator starting in state 1 ends up. If sM = s1 , the process is not allowed since

a non-interacting particle can not propagate to a di¬erent species state, and this

feature is faithfully respected by the diagrammatics, since then the corresponding

(0)

propagator according to Eq. (9.3) vanishes, G1M = 0, and the contribution from the

corresponding diagram vanishes since by the multiplication rule the bare propagator

amplitude multiplies the adjacent (N ’ 2)-state amplitude.

The quantum statistics of identical particles introduces minus signs when two

identical fermions interchange states and the amplitudes are symmetric upon inter-

change of bosons, say

A213...N = ± A123...N (9.7)

where the upper (lower) sign is for bosons (fermions), respectively.

For the case of non-interacting identical particles, only free propagation and e¬ects

of the quantum statistics of the particles are involved as displayed in Figure 9.9.

9.1. Diagrammatics 263

3

2

3 3

2 2

3

2

± +

=

1 1

4 4

1 4

1 4

Figure 9.9 The 4-state diagrams for two non-interacting identical particles.

In the following we consider ¬rst bosons, in which case the amplitude functions

are symmetric upon interchange of pairs of external state labels. The features of

antisymmetry for fermions are then added.14 The symmetry property of amplitudes

forces the vertices to be symmetric in their indices, e.g. for the 3-vertex g213 = g123 ,

etc.

Returning to the diagram for the general N -state amplitude and respecting the

other option for the particle in state 1, to interact, gives the additional ¬rst two

diagrams as depicted on the right in Figure 9.10 for the case of a theory with three-

and four-line connector vertices. The equation relating amplitudes as depicted in

Figure 9.10 is the fundamental dynamic equation of motion in the diagrammatic

language (for the case of three- and four-line vertices but trivially generalized).

2 2 2

1 1

1 1 1

= + 3!

2!

N N N

2

1 2

3

+···+

+

N ’1

N

1 N

Figure 9.10 Fundamental dynamic equation for three- and four-vertex interactions.

The option of interaction through the 3-state vertex is for the N -external-state

(N +1)

amplitude expressed in terms of the amplitude A¯¯ with (N + 1) external

233...N +1

states, where two internal propagators are contracted at the vertex. This leads to

the ¬rst diagram on the right in Figure 9.10, representing according to the Feynman

14 In diagrammatics the essential is the topology of a diagram, and the interpretation of diagrams

for the case of fermions is by the end of the day the same as for bosons except for the rule that a

relative minus sign must be assigned to a diagram for each closed loop of fermion propagators.

264 9. Diagrammatics and generating functionals

rules the amplitude as speci¬ed in Figure 9.11.

2

1

(0)

= G1¯ g¯¯¯ A¯¯

1 123 232...N

N

Figure 9.11 Diagram and corresponding analytical expression.

Repeated state labels are summed over in accordance with the superposition

principle. Similarly for diagrams with higher-order vertices in Figure 9.10 displayed

for a theory with an additional 4-attachment vertex.

Although combinatorial prefactors are an abomination in diagrammatics we have

in accordance with custom introduced them in Figure 9.10 by hand, the convention

being: an N -line vertex carries an explicit prefactor 1/(N ’ 1)!, the reason being to

be relieved at a di¬erent junction as immediately to be revealed. Consider a theory

with only a 3-attachment vertex, and follow the further adventures of one of the

particles emanating from the interaction vertex according to its two options, interact

or not, as depicted in Figure 9.12.

2

2

2 2

1

1 1

= + +

1

3 3

3

3

2

1

1

+2

3

2

1

+ disconnected diagram

= 2—

+ higher-order contributions.

3

Figure 9.12 Further adventures of a particle line emanating at a vertex.

The upper row of diagrams on the right in Figure 9.12 corresponds to the option

of not interacting. In lowest order in the interaction, the second and third diagrams

on the right give the same contribution. The inserted combinatorial factor in Figure

9.10 is thus the device to make the bare vertex diagram (here a 3-vertex) appear

9.1. Diagrammatics 265

with no combinatorial factor. In a theory with only a 3-attachment vertex, the

inserted combinatorial factor appearing with the vertex in Figure 9.10, thus makes

the diagrammatic expansion of the 3-state amplitude start out with the lowest-order

connected diagram, the bare vertex 3-state amplitude, carrying no additional factor

as depicted in Figure 9.13.

2 2

1

1

···

=

+

3 3

Figure 9.13 Lowest-order connected 3-state diagram for a 3-vertex theory.

A similar function has the combinatorial factor inserted in front of the 4-vertex

diagram in Figure 9.10.

9.1.4 Low order diagrams

Let us now familiarize ourselves with the Feynman rules and derive the expressions of

lowest-order diagrammatic perturbation theory. The reader not interested in entering

into this in¬nite forest of diagrams can skip the next few pages and go straight to

the next sections where more powerful methods are developed. These will allow us

systematically to generate the jungle of diagrams. However, for the adventurous

reader let us see what kind of diagrams will emerge when we apply the simple law

of dynamics, to interact or not to interact! A lesson to be learned from this is

that although the basic rule is as simple as it possibly can be, in this brute force

generation of diagrams one can easily miss a diagram, something history has proved

over and again. The functional methods we shall consider shortly are able to capture

the complete diagammatics in a simple way and in this way are able to help us in

ensuring against mistakes.

We can now in any diagram follow the further possible options of any particle

line emanating at a vertex, interact or not, and in this way unfold order by order

the in¬nite total canopy in the jungle of diagrams constituting perturbation theory.

For example, consider the 2-state amplitude (or two-point or 2-state propagator

or Green™s function) and a theory with the option of interaction only through the

3-attachment vertex. The two options for dynamics then generate the diagrams

depicted in Figure 9.14.

1

+

= 2

Figure 9.14 Interaction or not option for the 2-state amplitude.

266 9. Diagrammatics and generating functionals

A new diagrammatic entity enters in the ¬rst diagram on the right in Figure

9.14, the sum of all vacuum diagrams. The ¬rst diagram on the right in Figure

9.14 represents the product of two quantities, the bare 2-state amplitude, the bare

propagator, times the amplitude resulting from the sum of all vacuum diagrams:

free propagation accompanied by vacuum ¬‚uctuations, and nothing further is to be

revealed diagrammatically in this part. The second diagram on the right corresponds

to the option of interaction (in QED it could represent photon absorption or emission

by electrons and positrons or pair creation). We note the general structure emerging

in this way for the 2-state amplitude: the appearance of the bare 2-state amplitude

and the appearance of a higher-order amplitude, here the 3-state amplitude.

Next we concentrate on the second diagram on the right in Figure 9.14, and

explore the options, interact or not, of one of the lines emanating from the vertex

and obtain the diagrams depicted in Figure 9.15.

1

= + 2

1

+ 2

Figure 9.15 Diagrams generated by particle emanating at the vertex.

The ¬rst two diagrams on the right in Figure 9.15 correspond to the option of not

interacting, viz. either propagating freely back to the vertex or freely to the external

state. The last diagram encompasses the option of interacting, exposing one more

vertex in our 3-state vertex theory.

The 1-state amplitude appearing in the ¬rst and second diagram on the right

in Figure 9.15 (as a disconnected and connected piece, respectively), the tadpole

diagram, can in a 3-vertex theory be expressed in terms of the 2-state amplitude

contracted at the vertex as depicted in Figure 9.16, since the only option for the

line is to interact (the option of not interacting was already exhausted in the ¬rst

diagram in Figure 9.14).

1

=

2

Figure 9.16 Tadpole or 1-state amplitude in a 3-vertex theory is expressable in

terms of the vertex and the 2-state amplitude contracted at the vertex.

Inserting into the second diagram on the right in Figure 9.16 the expression for

9.1. Diagrammatics 267

the 2-state amplitude speci¬ed by the expression in Figure 9.14 gives in a three-line-

vertex theory the diagrammatic equation for the tadpole depicted in Figure 9.17.

1 1

+

=

2 4

Figure 9.17 Tadpole equation for a three-line-vertex theory.

The 1-state diagram, the tadpole, has thus been expressed in terms of the bare

tadpole times the amplitude representing the sum of all the vacuum diagrams plus

a higher correlation amplitude, here the 3-state amplitude contracted at vertices

according to the second diagram on the right in Figure 9.17.

Exercise 9.1. Obtain the diagrammatic equation for the tadpole if a 4-line vertex

is also included in the theory.

Let us now further expose interactions in the 2-state amplitude in Figure 9.14.

Insert the diagrammatic expansion of the tadpole in Figure 9.17 into the ¬rst diagram

on the right in Figure 9.15, and then substitute the resulting expression for the second

diagram on the right in Figure 9.14, and further explore the options for particle lines

emanating from vertices, interaction or not. This gives the diagrammatic expansion

of the 2-state amplitude depicted in Figure 9.18.

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

Figure 9.23 Diagrammatic notation for the source in the state indicated by the dot.

We have thus introduced the new diagrammatic feature that a particle line, as

dictated by the generating functional, can end up on a source. The propagator dot

and the corresponding source dot are thus shared in accordance with the convention

that the corresponding state label are contracted, i.e. repeated indices are summed,

integrated, over in accordance with the de¬nition in Eq. (9.8).25

If the source is not set to zero after di¬erentiation

δ N Z[J]

A12...N [J] = (9.28)

δJ1 δJ2 · · · δJN

25 The dot was also used in connection with the vertices, and another reason for this is that in

fact a vertex is a generalization of a source, generating multi-particle states.

9.2. Generating functional 275

we generate a new quantity, the amplitude in the presence the source. A source

dependent amplitude is a function of the state labels exposed by the labels of the

sources with respect to which the generating functional is di¬erentiated as well as a

functional of the source.

For the non-interacting theory in the presence of the source, the amplitude A1 [J]

is represented by the diagram depicted in Figure 9.24.

(0)

= G12 J2

1 2

Figure 9.24 Diagrammatic representation of the amplitude A1 for a free theory in

the presence of the source.

We now turn to show how to express in terms of functional di¬erential equations,

all the diagrammatic equations relating amplitudes, as exempli¬ed in Figure 9.10,

and derived by the simple diagrammatic rule: to interact or not. This is achieved by

¬rst expressing the fundamental dynamic diagrammatic equation displayed in Figure

9.10, in terms of a di¬erential equation for the generating functional.

The ¬rst derivative of the generating functional generates according to its de¬ni-

tion the terms

δZ[J] 1 1

= A1 + A1¯ J¯ + A1¯¯ J¯ J¯ + A ¯¯¯ J¯ J¯ J¯ + . . . . (9.29)

22 23 2 3

3! 1234 2 3 4

δJ1 2

Di¬erentiating the generating functional with respect to the source of a certain la-

bel removes this source, corresponding diagrammatically to removing a cross, and

exposes this state in a bare propagator as each source dependent amplitude thus no

longer ends up on this source but in the corresponding particle state, a particle is

launched.26 An external state, no longer contracted with the source, is thus exposed

in each of the diagrams on the right-hand side in Figure 9.22 representing the gener-

ating functional, viz. the state with the label of the source with respect to which we

di¬erentiate. We therefore introduce the diagrammatic notation for the ¬rst deriva-

tive of the generating functional, the 1-state amplitude in the presence of the source,

where a state on a free propagator line extrudes from the generating functional as

depicted in Figure 9.25.

≡ δZ

1

δJ1

Figure 9.25 Diagram representing the ¬rst derivative.

The cross in the diagram in Figure 9.25 is there to remind us that the ¬rst

26 Orterminated as kept track of for convenience by yet an index in the collective index, and not

as in Chapter 4 by an arrow.

276 9. Diagrammatics and generating functionals

derivative, the source dependent 1-state amplitude function, is still a functional of

the source.

The equation for the ¬rst derivative of the generating function, Eq. (9.29), can

therefore be expressed diagrammatically as depicted in Figure 9.26.

= +

1 1

+ ···

+ +

2! 3!

Figure 9.26 Diagrammatic expansion of the 1-state amplitude in the presence of

the source.

Let us consider a 3-vertex theory. The ¬rst diagram on the right in Figure 9.26,

the tadpole, is then given by the diagram in Figure 9.16, i.e. speci¬ed by the vertex

and the 2-state amplitude. The second diagram on the right in Figure 9.26 can

according to the two options of the external state line, interact or not, be split into

the two diagrams on the right-hand side depicted in Figure 9.27. For the latter option

the exposed state propagates directly to the source as depicted in the ¬rst diagram

on the right.