G12 = G12 + G13 Σ34 G42 (9.94)

by iterating from the other side. We have obtained the non-equilibrium Dyson equa-

tions.42

We now show how the Dyson equation can be obtained by using the e¬ective

action. More importantly, we show that we can use Eq. (9.84) to obtain the equation

for the connected 2-state amplitude in the presence of the source where it is expressed

in terms of the derivative of the e¬ective action. This will lead to a simple method

whereby all amplitudes can be expressed in terms of the 2-state connected amplitude,

the full propagator, and one-particle irreducible vertices.

The Legendre transformation, according to Eq. (9.78), gives rise to the relation

δ 2 W [J] δ

δ δ•2 δ δ

= = = G12 . (9.95)

δJ1 δJ1 δ•2 δJ1 δJ2 δ•2 δ•2

Taking the derivative of Eq. (9.84) with respect to the source then gives

δ 2 W [J] δ 2 “[•]

’ = δ12 . (9.96)

δJ2 δJ3 δ•3 δ•1

Adding and subtracting in the e¬ective action the so-called free term, “[•] ≡

’ 2 •1 (G(0) )’1 •2 +“i [•],

1

i.e. splitting o¬ again the inverse propagator term we previ-

12

ously included in “12 , sothat “i now denotes the original e¬ective action introduced

in Eq. (9.80), provides the self-energy

δ 2 “i [•]

Σ12 = (9.97)

δ•1 δ•2

in the presence of the source, as expressed through the ¬eld. Inserting into Eq. (9.96)

gives

δ 2 “[•] δ 2 “i [•]

(0) ’1

= ’(G )12 + (9.98)

δ•1 δ•2 δ•1 δ•2

and inserting into Eq. (9.96) gives

δ 2 W [J] δ 2 “i [•]

’(G(0) )’1

’ + = δ12 . (9.99)

21

δJ2 δJ1 δ•2 δ•1

42 Recovering the non-equilibrium Dyson equations thus makes contact with quantum ¬eld theory

studied by canonical means in the previous chapters. For the non-equilibrium states we studied in

the previous chapters, we had valid approximate expressions for the self-energy, and did not need

to go further into the diagrammatic structure of higher-order vertices.

304 9. Diagrammatics and generating functionals

Matrix multiplying by the bare propagator from the right gives43

δ 2 W [J] δ 2 W [J] δ 2 “i [•] (0)

(0)

= G12 + G, (9.100)

δJ1 δJ3 δ•3 δ•4 42

δJ1 δJ2

which in terms of diagrams has the form depicted in Figure 9.55.

•

= +

Figure 9.55 Dyson equation for the 2-state Green™s function in the presence of the

source.

Iterating the equation gives the full propagator

δ 2 W [J] 2 2 2

(0) δ “i [•] (0) δ “i [•] (0) δ “i [•]

(0) (0) (0)

= G11 + G12 G31 + G12 G34 G51 + ...

δJ1 δJ1 δ•2 δ•3 δ•2 δ•3 δ•4 δ•5

(9.101)

the analog of the Dyson equation depicted in Figure 9.53, but now for the case where

the source is present. The second derivative relationship between the generator of

connected Green™s functions and the e¬ective action can compactly be rewritten

suppressing the matrix indices, i.e. the two state labels occurring upon di¬erentiation

are now only indicated by the primes, in the form

1

W [J] = (9.102)

G’1 ’ “i [•]

0

as we recall the formula for a matrix X

1

= 1 + X + X 2 + X 3 + ... . (9.103)

1’X

This is the relationship between the full propagator and the self-energy we arrived

at earlier by topological classi¬cation of diagrams, expressing the connected 2-point

Green™s function in terms of the self-energy. Here we have constructed the functional

analog in terms of a functional di¬erential equation.

By taking further source derivatives of Eq. (9.96), we express the higher-order

connected Green™s functions in terms of the full propagator and the higher-order

one-particle irreducible vertices.

Taking the derivative of Eq. (9.96) with respect to the source and using Eq. (9.84)

gives

δ 3 W [J] δ 2 W [J] δ 2 W [J] δ 2 W [J] δ 3 “[•]

= . (9.104)

δJ1 δJ2 δJ3 δJ1 δJ1 δJ2 δJ2 δJ3 δJ3 δ•1 δ•2 δ•3

In terms of diagrams we have for Eq. (9.104) the relation depicted in Figure 9.56.

43 This equation is of course immediately recognized as the Dyson equation, Eq. (4.141), G12 =

(0) (0)

G12 + G13 Σ34 G42 .

9.6. One-particle irreducible vertices 305

•

=

Figure 9.56 Connected 3-state diagram expressed by the 1P-irreducible 3-vertex.

Exercise 9.10. Show by taking further source derivatives of Eq. (9.96) that the

equation obtained for the 4-state connected Green™s function has the diagrammatic

form (for a theory with 3- and 4-connector vertices) depicted in Figure 9.57.

•

• +

=

•

• • +

+

• •

Figure 9.57 Connected 4-state diagram expressed by 1P-irreducible vertices.

If in the above equation we set the source to zero, and thereby the ¬eld to zero,

instead of encountering quantities depending on the source and ¬eld, we will obtain

expressions for the connected Green™s functions in term of the full 2-point Green™s

function and the irreducible vertex functions. Since the full 2-point Green™s function

is the one into which we can feed our phenomenological knowledge of the mass of

a particle, these equations are basic for the renormalization procedure. The bare

306 9. Diagrammatics and generating functionals

Green™s function with its bare mass, and the bare vertices have thus left the theory

explicitly, leaving room for the trick of renormalization.

In Section 9.8, we shall use the equations, Eq. (9.78) and Eq. (9.84), the Legendre

transformation between source and ¬eld variables, to replace source-derivatives by

¬eld-derivatives and thereby obtain the equations satis¬ed by the e¬ective action

and the diagrammatics of the one-particle irreducible vertices. But ¬rst we turn to

show how equations very e¬ciently relating the connected Green™s function can be

generated.

9.7 Diagrammatics and action

In this section we show how the fundamental di¬erential equation for the dynamics,

Eq. (9.32), can be turned into an equation from which the relationships between the

connected Green™s functions can easily be obtained. This is done by introducing the

action, which is de¬ned in terms of the inverse propagator and the bare vertices of

the theory according to44

1 1

S[φ] ≡ ’ φ1 (G’1 )12 φ2 + g12...N φ1 φ2 · · · φN , (9.105)

0

2 N!

N

here for a theory with vertices of arbitrary high connectivity. The fundamental

equation, Eq. (9.32), expressing the dynamics of a theory can then be written in the

form (for an arbitrary theory speci¬ed by the above action)

δ

δS[ i δJ ]

0= + J1 Z[J] , (9.106)

δφ1

where by de¬nition

δ

δS[ i δJ ] δS[φ]

= (9.107)

δφ1 δφ1

δ

φ’ i δJ

i.e. the action is di¬erentiated and then the source derivative is substituted for the

¬eld. We have written the equation in a form having a quantum ¬eld theory in

mind but shall immediately shift to the Euclidean version, or simply suppressing the

appearance of /i by absorbing the factor in the source derivative.

44 At this junction in the generating functional formulation of a quantum ¬eld theory the solemnity

of the action is scarcely noticed, but as just another formal construction. In the next chapter we

show how the action in the functional integral formulation of a quantum ¬eld theory naturally

appears as the fundamental quantity describing the dynamics. The action can also be given a

fundamental status in the operator formulation of the generating functional technique (recall Section

3.3), if the dynamics is based on Schwinger™s quantum action principle [50]. However, the point of

the presentation in this chapter is to base the dynamics directly on diagrams and then by simple

topological arguments construct the generating functional technique.

9.8. E¬ective action and skeleton diagrams 307

Since the generator of connected diagrams, W , is the logarithm of Z, we have the

relation valid for an arbitrary functional F

1δ δW [J] δ

(Z[J] F [J]) = + F [J] (9.108)

Z[J] δJ1 δJ1 δJ1

and by repetition

δN δW [J] δ δW [J] δ

···

Z[J] = Z[J] + + , (9.109)

δJ1 · · · δJN δJ1 δJ1 δJN δJN

where operator notation has been used, i.e. the operations are supposed to operate

on a functional F .

Since the action is a sum of polynomials we have according to Eq. (9.109)

δ

δS[ δW + δ

δS[ δJ ] δJ ]

δJ

Z[J] = Z[J] . (9.110)

δφ1 δφ1

The fundamental equation, Eq. (9.106), can thus be written in the form

δS[ δW + δ

δJ ]

δJ

0= + J1 . (9.111)

δφ1

Using the explicit form of the action for an arbitrary theory we have

δS[ δW + δ

δJ ] δW δ

’(G(0) )’1

δJ

= +

12

δφ1 δJ2 δJ2

1 δW δ δW δ

···

+ g12...N + +

(N ’ 1)! δJ2 δJ2 δJN δJN

N

(9.112)

and using Eq. (9.111) and performing the di¬erentiations and lastly multiply by the

bare propagator we immediately recover Eq. (9.77) (for the 3- plus 4-vertex theory).

Having the fundamental equation on the form speci¬ed in Eq. (9.111) turns out

in practice to be very useful for generating the relations between the connected full

Green™s functions, and exempli¬es the expediency and powerfulness of the generating

functional formalism.

9.8 E¬ective action and skeleton diagrams

In this section, we shall use the equations, Eq. (9.78) and Eq. (9.84), the Legen-

dre transformation between source and ¬eld variables, to replace source-derivatives

by ¬eld-derivatives and thereby obtain the equations obeyed by the e¬ective action.

Upon setting the ¬eld to zero, • = 0, we then obtain the skeleton diagrammatic

308 9. Diagrammatics and generating functionals

equations satis¬ed by the one-particle irreducible vertices. Instead of using topolog-

ical diagrammatic arguments to obtain the skeleton diagrammatics, we turn to use

the generating functional method to achieve the same goal.

On the right side in Eq. (9.112) we can introduce the average ¬eld and obtain

δS[ δW + δ

δ2W

δJ ] δ

= ’(G(0) )’1 •2 +

δJ

12

δφ1 δJ2 δJ2 δ•2

δ2W δ2W

1 δ δ

· · · •N +

+ g12...N •2 + ,

(N ’ 1)! δJ2 δJ2 δ•2 δJN δJN δ•N

N

(9.113)

where we in addition have used Eq. (9.95) to substitute the ¬eld derivative for the

source derivative.

Inserting Eq. (9.111) into Eq. (9.84) and using Eq. (9.78) thus gives the relation

between the action and the e¬ective action

δ

δS • + W [J]

δ“[•] δ•

=’ , (9.114)

δ•1 δ•1

where the right-hand side is short for the right-hand side in Eq. (9.113).

For a 3- plus 4-vertex theory we obtain

δ 2 W [J]

δ“[•] 1 1

’(G’1 )12 •2 +

= g123 •2 •3 + g123

0

δ•1 2 2 δJ2 δJ3

δ 2 W [J]

1 3

+ g1234 •2 •3 •4 + g1234 •4

3! 3! δJ2 δJ3

δ 2 W [J] δ 2 W [J] δ 2 W [J] δ 3 “[•]

1

+ g1234 (9.115)

3! δJ2 δJ5 δJ3 δJ6 δJ4 δJ7 δ•5 δ•6 δ•7

as the last term emerges upon noting

δ δ 2 W [J] δ 3 W [J] δ 2 “[•] δ 3 W [J]

δJ2

= =

δ•2 δJ3 δJ4 δ•2 δJ2 δJ3 δJ4 •2 •2 δJ2 δJ3 δJ4

δ 2 W [J] δ 2 W [J] δ 3 “[•]

= , (9.116)

δJ3 δJ5 δJ4 δJ6 δ•2 δ•5 δ•6

where in obtaining the last equality we have used Eq. (9.96) and Eq. (9.104). The

relationship expressed in Eq. (9.116) has the diagrammatic representation depicted

in Figure 9.58.

9.8. E¬ective action and skeleton diagrams 309

•

δ =

δφ1

1

Figure 9.58 Average ¬eld dependence of the propagator.

The implicit dependence of the propagator on the average ¬eld, through the source,

is thus such that taking the derivative inserts a one-particle irreducible vertex in

accordance with the relation depicted in Figure 9.58.

The equation for the ¬rst derivative of the e¬ective action, Eq. (9.115), has for a

3- plus 4-vertex theory the diagrammatic representation depicted in Figure 9.59.

• 1

=’ + 2

1 1

+ +

3! 2

•

1

+

1

+ 3!

2

Figure 9.59 Diagrammatic relation for the ¬rst derivative of the e¬ective action for

a 3- plus 4-vertex theory.

310 9. Diagrammatics and generating functionals

The stubs on the bare vertices in Figure 9.59 indicates the uncontracted state label

identical to the state label on the left.

Here we ¬nd the origin for calling “[•] the e¬ective action: if thermal or quantum

¬‚uctuations are neglected, leaving only the ¬rst three terms on the right in Figure

9.59, the (derivative of the) e¬ective action reduces to the (derivative of the) action.

This corresponds to dropping the W -terms in Eq. (9.114). In other words, the