<<

. 11
( 22)



>>

„R = . (8.218)
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.

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