+ 2! +

⎝ +...⎠

1

⎛ ⎞

+ 3! +

⎝ ...⎠

1

+ ...

Figure 9.39 The 1-state amplitude in terms of connected amplitudes.

We have thus been able by simple topological arguments to express the 1-state

amplitude in terms of 1-state connected amplitudes. In fact, the information in

the equation depicted in Figure 9.39 can be further compressed. We recognize that

the diagrams in any of the parenthesis all sum up to the diagrammatic expansion

for the generating functional Z[J]. We have thus achieved expressing the 1-state

amplitude in the presence of the source in terms of the 1-state connected diagrams

in the presence of the source and the generator Z[J] as depicted in Figure 9.40

9.5. Generator of connected amplitudes 289

+

=

+ +

1 1 + ···

2! 3!

Figure 9.40 First derivative diagrammatic equation.

On the right we see the 1-state connected diagrams in the presence of the source.

We shall therefore introduce the generator of connected diagrams, and we introduce

a hatched circle with a cross as the diagrammatic notation for the generator of

connected diagrams as depicted in Figure 9.41.

+ + 2! + 3!

1 1

≡ + ···

Figure 9.41 Generator of connected Green™s functions.

The ¬rst term on the right in Figure 9.41 comprises the sum of all connected

vacuum diagrams. Removing a cross in the connected generator speci¬ed in Figure

9.41 by functional di¬erentiation exposes the sum of 1-state connected diagrams in

the presence of the source, etc.

The diagrammatically derived equation depicted in Figure 9.40 can then be rewrit-

ten in the form depicted in Figure 9.42.

=

Figure 9.42 Relation between the derivatives of the generator and connected gen-

erator.

We introduce the notation G12...N for the amplitude represented by the N -state

connected diagrams, the hatched circle with N external states, and have analytically

290 9. Diagrammatics and generating functionals

for the generator of connected amplitudes

∞

1

G12...N J1 J2 · · · JN ,

W [J] = (9.62)

N!

N =0

the equation that is represented diagrammatically in Figure 9.41.

Since removing a cross corresponds to di¬erentiation with respect to the source,

the relation depicted in Figure 9.42 can then be written in the form

δZ[J] δW [J]

= Z[J] . (9.63)

δJ1 δJ1

We immediately solve equation Eq. (9.63), by the above analysis up to an undeter-

mined multiplicative constant, and obtain

Z[J] = eW [J] . (9.64)

The overall multiplicative factor will in the following subsection be determined to be

the sum of all connected vacuum diagrams in the absence of the source (the connected

vacuum diagrams of the theory). We have already introduced a diagrammatic nota-

tion for this quantity, the ¬rst diagram on the right in Figure 9.41, and the overall

constant is thus accounted for by de¬nition of the N = 0-term in Eq. (9.62). In the

above analysis this term was a source-independent irrelevant constant, not captured

by the argument due to the derivative. The generator of all amplitudes, Z[J], is thus

equal to the exponential of the generator of only connected amplitudes. The simple

structure of the combinatorial factors in the de¬nition of the exponential function

is thus enough at the level of generators to express the relationship between the

connected diagrams and all the diagrams, including disconnected diagrams.

Inversely we have34

W [J] = ln Z[J] . (9.65)

9.5.2 Combinatorial proof

We now give the general combinatorial argument for the relation between the gen-

erator of connected amplitudes W [J] and the generator Z[J], again arguing at the

diagrammatic level, but now for amplitudes in the absence of sources. This will ¬x the

overall multiplicative factor missed in the above argument to be determined to be the

sum of all connected vacuum diagrams of the theory.35 This is achieved by the follow-

ing observation. Any N -state amplitude can be classi¬ed according to its connected

and disconnected sub-diagrammatic topological feature of its external attachments.

34 Inthermodynamics Z[J] represents the partition function and W [J] represents the free energy,

and we have the diagrammatics necessary for a ¬eld theoretic approach to critical phenomena, and

the renormalization group. In probability theory, Z[J] is the characteristic function, the generator

of moments and W [J] is the generator of cumulants. In a quantum ¬eld theory we should restore

the imaginary unit, iW [J] = ln Z[J].

35 The argument also shows that for the time-ordered Green™s function de¬ned in terms of the

¬eld operators, Eq. (4.21) (or the contour-ordered Green™s function, Eq. (4.50)), the denominator

exactly cancels the separated o¬ vacuum diagrams in the numerator.

9.5. Generator of connected amplitudes 291

For example, for the 3-state amplitude we get the topological classi¬cation as de-

picted on the right in Figure 9.43 (skipping for clarity the overall factor representing

the sum of all vacuum diagrams in the absence of the source accompanying each of

the diagrams on the right in Figure 9.43).

= +

+ + +

Figure 9.43 The 3-state diagrams in terms of connected diagrams.

The general combinatorial proof of the relationship between the generator of all

amplitudes, the A1...N s, and the generator of connected amplitudes, the G1...N s, now

proceeds. Any N -state amplitude A1...N is a sum over all the possible products of

connected sub-amplitudes (multiplied by the overall sum of vacuum in the absence of

the source which we keep implicit). Any N -state amplitude can thus be divided into

its 1-state connected sub-amplitude parts (say m1 in all, m1 ≥ 0), multiplying its 2-

state connected amplitudes (say m2 in all),. . . , and its n-state connected amplitudes

(say mn in all), and we have (suppressing on the right the overall multiplicative factor

representing the sum of vacuum diagrams)

(N ) (1) (1) (2) (2)

’ GP1 · · · GPm GPm · · · GPm ···

A1,2,...,N

1 +1 ,Pm 1 +2 1 +m 2 , m 1 +m 2 +1

1

{mn }

(n) (n)

· · · GPm · · · GPN ’n ,...,PN , (9.66)

,...,Pm 1 +. . . +m n ’1 +n

1 +···+(n ’1)m n ’1 +1

where G denotes a connected amplitude, and the arrow indicates that a particular

choice of external state labels has been chosen as indicated by the permutation P of

the N labels. By construction the numbers specifying the sub-amplitudes satisfy a

constraint, the relation m1 + 2m2 + · · · + nmn = N , since we consider the N -state

amplitude, the case of N external states. The prime on the summation indicates that

for each N the sum is over only sets of sub-amplitude labeling values that satisfy the

constrain. Some of these ms are by construction zero; for example for, say, the 4-state

amplitude there is the combination (m1 = 1, m2 = 0, m3 = 1, m4 = 0) describing

the diagram with one 1-state connected diagram multiplying a 3-state connected

292 9. Diagrammatics and generating functionals

diagram. Clearly, mN +n = 0 for n ≥ 1. Introducing the notation mp = 0 to mean

that there is no connected sub-amplitude with p external states we can write the

constrain

∞

p mp = N (9.67)

mp =0

letting the sub-diagram number run freely from zero to in¬nity.

In Eq. (9.66), a particular choice of grouping of terms was made as indicated by

the presence of the permutation P . The number of ways the external states of an N -

state amplitude can be divided into the above topological speci¬ed set of connected

sub-amplitudes is

N!

M≡ (9.68)

m1 !(2m2 )!(3m3 )! · · · (nmn )!

or in the freely running-label notation

N!

M≡ , (9.69)

m1 !(2m2 )!(3m3 )! · · · (∞m∞ )!

where ∞m∞ simply indicates that for high enough external state labeling number,

say beyond L, we have (L + n)mL+n = 0 for any n ≥ 1 owing to the constraint,

Eq. (9.67). In the generating functional where the N -state amplitude is contracted

with N external sources, all of these terms have identical value.

Within each subset of sub-amplitudes, for example the product of 2-state dia-

grams, the labels de¬ning the external states could have been paired di¬erently giv-

ing (2m2 )!/((m2 )!(2!)m2 ) di¬erently chosen sub-amplitudes which when contracted

with the sources give the same value. For the set of 3-state sub-amplitudes there are

analogously (3m3 )!/((m3 )!(3!)m3 ) possible choices giving identical contribution, etc.

For the N -state amplitude contracted with the N sources, we then have

1 (N ) 1 11 11

J1 · · ·JN = ( G12 J1 J2 )m2 · · · ( G1...L J1 · · ·JL )mL .

(G1 J1 )m1

A

N ! 1,...,N m1 ! m2 ! 2! mL ! L!

{mn }

(9.70)

The generator Z can therefore be expressed in terms of connected amplitudes

according to (the N = 0-term, the sum of all vacuum diagrams in the absence of the

source, will be dealt with shortly)

∞

1 (N )

J1 J2 · · · JN

Z[J] = A

N ! 1,2,...,N

N =0

∞

1 11 11

( G12 J1 J2 )m2 · · · ( G12...L J1 J2 · · · JL )mL .

(G1 J1 )m1

=

m1 ! m2 ! 2! mL ! L!

N =0 {mn }

(9.71)

9.5. Generator of connected amplitudes 293

This can be rewritten

∞

1 11 11

( G12 J1 J2 )m2 · · · ( G12...L J1 J2 · · · JL )mL

(G1 J1 )m1

Z[J] =

m1 ! m2 ! 2! mL ! L!

N =0 {mn }

∞

1 11 11

( G12 J1 J2 )m2 · · · ( G12...n J1 J2 · · ·Jn )mn · · ·

(G1 J1 )m1

=

m! m2 ! 2! mn ! n!

,...=0 1

m1 ,m2

(9.72)

where the last summation runs freely over all ml s so clearly any term in the ¬rst sum

is present once in the second sum, and any term in the second sum is unique. Any

term in the double sum is also unique and contains any term in the sum on the right

with the freely running summation and we have argued for the validity of the last

equality sign in Eq. (9.72).

In the discussion we suppressed the multiplicative factor representing the sum of

all the vacuum diagrams. To get the correct formula for Z[J], we should thus in

Eq. (9.72) interpret the term with all mp s equal to zero, m1 = 0 = m2 = m3 , as the

sum of all the vacuum diagrams or rather as unity since we should remember the

overall multiplicative factor we left out of the argument representing the sum of all

the vacuum diagrams, Z[J = 0], connected and disconnected. We shall now obtain

the expression for Z[J = 0] in terms of the sum of connected vacuum diagrams. The

combinatorial argument runs equivalent to the above. A vacuum diagram with dis-

connected parts classi¬es itself into connected vacuum parts characterized according

to the number of vertices in the connected diagrams: a product of products of con-

nected diagrams with one, two, etc., vertices. The constraint and the combinatorics

will then be the same as above, N now characterizing the total number of vertices

in the vacuum diagrams in question, and we end up with the terms on the right-

hand side of Eq. (9.72) except for the absence of the source and the Gs now having

the meaning of connected vacuum diagrams with the possible di¬erent numbers of

vertices. We have thus shown that the sum of all the vacuum diagrams is given by

the exponential of the sum of all connected vacuum diagrams. Note that the term

contributing the unit term to this exponential function is provided by the vacuum

contribution for the option of not interacting, the contribution of the free theory

as discussed at the end of Section 9.2.2. Diagrammatically we have thus identi¬ed

that the ¬rst diagram in Figure 9.41 represents the term W [J = 0], the sum of all

connected vacuum diagrams.

We therefore get

Z[J] = eW [J=0] eG1 J1 e 2! G12 J1 J2 · · · e n ! G12. . . n J1 J2 ···Jn · · · ,

1 1

(9.73)

where W [J = 0] denotes the ¬rst term on the right in the de¬nition of the generator

of connected diagrams in Figure 9.41, and thereby

Z[J] = eW [J] , (9.74)

where W [J] is given by the expression in Eq. (9.62), since the G-amplitudes above

were, by construction, the connected ones.

294 9. Diagrammatics and generating functionals

9.5.3 Functional equation for the generator

By construction W [J] is the generator of connected amplitudes or Green™s functions

δ N W [J]

G12...N = . (9.75)

δJ1 δJ2 · · · δJN

J=0

For the ¬rst derivative of the generator of connected Green™s functions we get

according to the de¬ning equation, Eq. (9.62), the trivial equation

∞

δW [J] 1

J1 J2 · · · JN ,

= G¯ + G¯ (9.76)

1

N ! 112...N

δJ¯

1 N =1

which has the diagrammatical form depicted in Figure 9.44.

+

=

+ 2! + 3!

1 1 +...

Figure 9.44 First derivative of the generator of connected amplitudes.

The equation Eq. (9.76), displayed diagrammatically in Figure 9.44, has no ref-

erence to the content of the theory in question, but expresses only the polynomial

structure of the generator (of connected amplitudes) in terms of the source. To get

the theory into play we shall use the fundamental dynamic equation, Eq. (9.32), and

the established relation, Eq. (9.74). Since Z is related to W by the exponential func-

tion, all equations for Z can be turned into equations for W (the exponential function

is the one which when di¬erentiated brings back itself). Inserting Eq. (9.74) into the

fundamental equation, Eq. (9.32), or rather Eq. (9.34) for the 3- plus 4-vertex theory,

and using Eq. (9.22), we get

δ 2 W [J] δW [J] δW [J]

δW [J] 1

(0)

= G12 J2 + g234 +

δJ1 2 δJ4 δJ3 δJ3 δJ4

δ 3 W [J] δW [J] δ 2 W [J] δW [J] δW [J] δW [J]

1

+ g2345 +3 +

3! δJ5 δJ4 δJ3 δJ3 δJ5 δJ4 δJ3 δJ4 δJ5

(9.77)

9.5. Generator of connected amplitudes 295

the fundamental functional di¬erential equation for the generator of connected am-

plitudes (here for the case of a 3- plus 4-vertex theory).

In diagrammatic notation we therefore have the equation depicted in Figure 9.45.

1 1

+ +

= 2 2

1 1 1

+ + +

3! 2 3!

Figure 9.45 Fundamental equation for the generator of connected Green™s function

for a 3- plus 4-vertex theory.

In deriving the equation depicted diagrammatically in Figure 9.45 we reversed

our previous order of ¬rst deriving equations by the diagrammatic rule, to interact

or not, and instead used the fundamental functional di¬erential equation, Eq. (9.32),

whereby the propagator lines emerging from vertices into connected Green™s functions