1

+

= 2

Figure 9.27 Interaction or not options for the 1-source term.

The structure of the above equation is: free propagation to the source times the sum

of vacuum diagrams plus exposed vertex diagram.

Similarly for the 2-source diagram on the right in Figure 9.26 we get the options

as depicted in Figure 9.28. The factor of two appearing in front multiplying the ¬rst

term on the right is the result of the option that when non-interacting the external

state line can end up on either of the two sources and the two diagrams specify the

same number.

9.2. Generating functional 277

·2 +1 ·

1 1 1

=

2 2 2 2

Figure 9.28 Interaction or not options for the 2-source term.

The structure of the above equation is: free propagation to the source times the

1-state amplitude contracted on the source (one integer lower-state amplitude than

the one on the left) plus exposed vertex diagram.

Similarly for the 3-source diagram we have the options as depicted in Figure 9.29

(and we have equivalent for the further higher-numbered source diagrams in Figure

9.26).

·3 + 3! ·

1 1 1 1

=

3! 3! 2

Figure 9.29 Interaction or no interaction options for the 3-source term.

If we collect the resulting diagrams into their two di¬erent types: those with

the amplitude factor of free propagation to the source and those with an exposed

vertex, the diagrammatic equation for the ¬rst derivative of the generating functional

becomes the one depicted in Figure 9.30.

⎛

1⎝

+

=

2

⎞

+1 1

+...⎠

+ 3!

2

⎞

⎛

+1 +...⎠

⎝ +

+ 2

Figure 9.30 First derivative equation for a 3-vertex theory.

The sum of the diagrams in the parenthesis in the last line of Figure 9.30 are seen

278 9. Diagrammatics and generating functionals

to be exactly the diagrams constituted by the generating functional and we have the

diagrammatic identity depicted in Figure 9.31.

⎞

⎛

1

+...⎠ =

⎝ +2

+

Figure 9.31 Propagation to the source times generating functional part.

The systematics of the prefactors of the diagrams in the parenthesis in Figure 9.31

are easily identi¬ed through their generation: the term with N sources getting the

prefactor 1/N !.

The diagrams in the ¬rst parenthesis in Figure 9.30 can also be expressed in

terms of the generating functional. They all start out with the launched propagator

entering the 3-vertex whose two other stubs either exposes lines in the sum of vacuum

diagrams, or the 1-state amplitude contracted on the source, or the 2-state amplitude

contracted on the source, etc. These latter parts thus sum up diagrammatically to

the generating functional and we can therefore represent the diagrammatic equation

in Figure 9.30 in the form depicted in Figure 9.32, the fundamental diagrammatic

equation for the dynamics of a 3-vertex theory.

1

= +

2

Figure 9.32 Fundamental diagrammatic equation for the 1-state amplitude, in the

presence of the source, for a 3-vertex theory.

Next we wish to identify the analytical expression corresponding to the ¬rst di-

agram on the right in Figure 9.32, or equivalently, the analytical expression for the

diagrams in the ¬rst parenthesis in Figure 9.30. To this end we consider the second

derivative of the generating function which according to Eq. (9.29) becomes

δ 2 Z[J] δ 1 1

= A2 + A2¯ J¯ + A2¯¯ J¯ J¯ + A ¯¯¯ J¯ J¯ J¯ + ...

22 23 2 3

3! 2234 2 3 4

δJ3 δJ2 δJ3 2

1 1

= A23 + A23¯ J¯ + A23¯¯ J¯ J¯ + A ¯¯¯ J¯ J¯ J¯ + ... . (9.30)

33 34 3 4

3! 23345 3 4 5

2

Di¬erentiating the generating functional exposes the corresponding state labels of

amplitudes, and if we contract these on the vertex function we get27

δ 2 Z[J]

1 (0) 1 (0) 1 (0) 1 (0) 1

G1¯ g¯ = G1¯ g¯ A23 + G1¯ g¯ A23¯ J¯ + G1¯ g¯

123 A23¯¯ J¯ J¯

2 1 123 δJ3 δJ2 123 123 33 34 3 4

21 21 21 2

27 Thevertices can thus be viewed as internal sources for creation and annihilation of particles, a

point we shall exploit later.

9.2. Generating functional 279

1 (0) 1

+ G1¯ g¯ A ¯¯¯ J¯ J¯ J¯ + ... (9.31)

123

3! 23345 3 4 5

21

i.e. exactly the analytical expression corresponding to the diagrams in the ¬rst paren-

thesis on the right in Figure 9.30. The correct factorial prefactors are generated term

by term, the term with N sources getting the prefactor 1/N !, and all terms have an

overall factor 1/2 since they were generated by a 3-line vertex theory. The diagrams

in the ¬rst parenthesis in Figure 9.30 are thus represented in terms of di¬erentiating

the generating functional twice. We have thus derived diagrammatically the funda-

mental analytical equation, the Dyson“Schwinger equation, obeyed by the generating

functional for a 3-vertex theory28

δ2

δZ[J] 1

(0)

= G1¯ g¯ + J¯ Z[J] . (9.32)

2 123 δJ3 δJ2 1

1

δJ1

Just as in diagrammatics, the ingredients here are the bare propagators and vertices,

but now instead of the diagrammatic rule of dynamics, to interact or not, we have

instead free propagation to the source and di¬erentiations with respect to the source.

The two lines protruding out of the generator in the ¬rst diagram on the right

in Figure 9.32 has thus the same operational meaning as in Figure 9.25: it signi-

¬es di¬erentiation with respect to the source, here where the labels with which the

di¬erentiation takes place are contracted at the vertex.29

By introducing the generating functional, the diagrammatic equations for ampli-

tudes in the presence of a source can be represented by a di¬erential equation, so

far we have achieved it for the 1-state amplitude, but the game can be continued by

taking further derivatives. The functional di¬erential equation, Eq. (9.32), will thus

be the fundamental dynamic equation for a 3-vertex theory.

The power of the generating functional technique is that all the relations existing

between the amplitudes in a theory, as expressed by the diagrammatic equation

in Figure 9.10, are contained in the fundamental functional di¬erential equation,

of the type Eq. (9.32) (or Eq. (9.34) the analogous equation for a theory with an

additional four-line vertex, or quite generally for a theory with an arbitrary number

of vertices). This is quite a compression of the information contained in the set

of diagrammatic relations between amplitudes that has been achieved here. From

the fundamental di¬erential equation we can obtain all the diagrammatic equations

relating amplitudes by functional di¬erentiation. All the diagrammatic equations are

thus equivalently representable by di¬erential equations. For example, for the 2-state

amplitude or Green™s function in a 3-vertex theory we obtain by di¬erentiating with

respect to the source on both sides in Eq. (9.32)

δ2

δ 1

(0)

A11 = G1¯ g¯ + J¯ Z[J]

2 123 δJ3 δJ2 1

1

δJ1

J=0

δ3

1

(0)

= G1¯ g¯ + δ¯ Z[J] , (9.33)

2 123 δJ3 δJ2 δJ1 11

1

J=0

28 The generating functional approach to quantum ¬eld theory was championed by Schwinger [50].

29 We note that the vertex in the equation in Figure 9.32 is acting like a 3-particle source .

280 9. Diagrammatics and generating functionals

which is the functional representation of the diagrammatic equation depicted in Fig-

ure 9.14.

We now have two ways of interpreting the two lines entering the generating func-

tional in the ¬rst diagram on the right in Figure 9.32, either in the diagrammatic

language options of interact or not, or as two functional di¬erentiations of the gen-

erating functional.

Exercise 9.6. Obtain by diagrammatic reasoning for a theory with both 3- and

4-vertex interaction the Dyson“Schwinger equation (letting δ/δJ ’ δ/iδJ for proper

quantum ¬eld theory notation, for details see Section 10.2.1)

δ 2 Z[J] δ 3 Z[J]

δZ[J] 1 1

(0)

= G1¯ J¯ + g¯ + g¯ Z[J] (9.34)

1

2 123 δJ3 δJ2 3! 1234 δJ4 δJ3 δJ2

1

δJ1

satis¬ed by the generating functional.

For a non-interacting, free, quantum ¬eld theory we can solve Eq. (9.32) immedi-

ately (with δ/δJ ’ δ/iδJ for proper quantum ¬eld theory notation) and obtain for

the generator of the free theory

(0)

i

Z0 [J] = e 2 J1 G1¯ J¯

. (9.35)

1

1

The overall multiplying constant equals one in accordance with the normalization

Z0 [J = 0] = 1, the sum of all the vacuum diagrams are equal to one. For the free

theory this follows trivially, the only vacuum diagram being the one where the free

propagator closes on itself, and since the equal time propagator by nature of being a

the conditional probability amplitude it satis¬es G0 (x, t; x , t; ) = δ(x ’ x ), leaving

the vacuum diagram equal to one. Since our interest is the real-time description

of non-equilibrium situations, the closed time path formalism guarantees the even

stronger normalization condition of the generator, Z[J] = 1, provided that the source

on the two parts of the closed time path are taken as identical. We note that the

free closed time path generator is unity, Z0 [J] = 1, if the sources on the two contour

parts are identical, J+ = J’ , in view of the identity Eq. (5.39). We have

1 (0)

Z0 [J] = eiW0 [J] , W0 = J1 G1¯ J¯ (9.36)

11

2

and W0 vanishes, W0 [J] = 0, if the sources on the two contour parts are identical,

J+ = J’ .

In the closed time path formalism, the free Green™s function entering Eq. (9.35) is

the free contour-ordered Green™s function. If we introduce the two parts of the closed

contour explicitly and the notation J± for the source on the forward and return parts,

respectively, the components of the matrix Green™s function of Eq. (5.1) appears

multiplied by the respective sources and integrations are over real time.

If we wish to express the generator in the physical or symmetric matrix Green™s

function representation, we should rotate the real-time sources by π/4 to give

’1

1

J1 1 J+

=√ (9.37)

J2 1 1 J’

2

9.3. Connection to operator formalism 281

as well as the Green™s functions, and we obtain, suppressing variables other than the

time,

∞∞

W0 [J] = dt dt (J2 (t) GR (t, t ) J1 (t ) + J1 (t) GA (t, t ) J2 (t ) + J2 (t) GK (t, t ) J2 (t )).

0 0 0

’∞’∞

(9.38)

By choosing properly the real-time dynamical indices of the sources, we can by dif-

ferentiation generate the various real-time propagators, GRAK .

0

9.3 Connection to operator formalism

In Chapter 4 we showed how to derive the Feynman diagrammatics for non-equilibrium

situations starting from the canonical formulation in terms of quantum ¬elds, i.e. we

started from the equations of motion for the contour or real-time Green™s functions,

describing the interactions in the system, and ended up with their diagrammatic

representation in terms of perturbation theory. In this chapter, we have started from

diagrammatics and have obtained the equation of motion for the contour or real-

time Green™s functions in terms of the generating functional. We can also make the

direct connection back to the quantum ¬elds by expressing the contour or real-time

generating functional in terms of them according to30

T c ei dxd„ φ(x,„ ) J(x,„ )

= Tr(ρ(H)Tc ei dxd„ φ(x,„ ) J(x,„ )

Z[J] = ) (9.39)

c c

since for example the two-point Green™s function (modulo the imaginary unit) is then

speci¬ed in terms of the, for simplicity, scalar quantum ¬eld operator, φ(x, „ ), on the

multi-particle space according to

δZ[J]

Tc (φ(x, „ ) φ(x , „ )) = ’ . (9.40)

δJx ,„ δJx,„

J=0

In Eq. (9.39) the contour is the closed time path depicted in Figure 4.5, as we have a

non-equilibrium situation in mind, and we have in Eq. (9.40) generated the contour

ordered 2-state Green™s function. Introducing the two parts of the closed contour, the

matrix Green™s function of Eq. (5.1) emerges. If we wish to generate the components

of the symmetric or physical matrix Green™s function, Eq. (5.41), we should rotate

the ¬elds and sources according to Eq. (9.37) (recall Eq. (9.38)).

Exercise 9.7. Consider the case of a self-coupled bose ¬eld as described by the

potential V (φ). Show that the generating functional can be expressed in terms of

the free generator according to

Z[J] = e’iV ( i δ J (x, „ ) ) Z0 [J] .

δ

(9.41)

30 For the case of zero temperature the generator is the vacuum-to-vacuum amplitude in the

presence of coupling to the source, Z[J] = 0|Tc ei c dxd„ φ(x,„ ) J (x,„ ) |0 .

282 9. Diagrammatics and generating functionals

We have considered, as above, the case of a real or hermitian bose ¬eld. If we con-

sidered spin-less bosons, say sodium atoms at low temperatures where their internal

degrees of freedom can be neglected, we would have for the generating functional

dxdt ψ(x,t) ·(x,t) + ψ † (x,t) · — (x,t)

Z[·, · — ] = T c ei , (9.42)

c

where now the source is not as above a real function, but a doublet of complex c-

number functions. We note the important feature of the closed time path formalism

that the generator equals one if the sources are identical on the upper and lower parts

of the contour.

9.4 Fermions and Grassmann variables

For the case of fermions, the sources must be anti-commuting numbers, so-called

Grassmann variables, in order to respect the antisymmetry property of Green™s func-

tions or amplitudes in general. In quantum ¬eld theory, we shall always be concerned

with a Grassmann algebra consisting of an even number of generators

{·± , ·β } = 0 , (9.43)

where ±, β = 1, 2, . . . , 2n. All possible products (ordered by convention ± < β,

etc.), the set {1, ·± , ·± ·β , . . . , ·± . . . ·ν }, constitute a basis for the Grassmann algebra,

which in addition is a vector space over the complex numbers of dimension 22n since

any generator can either be included or not in a product. Since we consider an

even number of generators, they can be grouped in pairs, so-called conjugates, and

—

renamed, ·± and ·± , i.e. now ± = 1, . . . , n. The conjugation property is endowed

with the properties: (·± )— = ·± , and (c ·± )— = c— ·± and (·± . . . ·β )— = ·β . . . ·± .

— — —

Each of the variables satis¬es its Grassmann or exterior algebra, and owing to

the anti-commutation relation, which implies · 2 = 0, the highest polynomial to be

built is thus linear

f (·) = c0 + c1 · , (9.44)

the monomial, where the coe¬cients c0 and c1 are arbitrary complex numbers. Sim-

ilarly for a pair · and · —

f (·, · — ) = c0 + c1 · + c2 · — + c3 ·· — . (9.45)

The linear space of functions of conjugate variables being four-dimensional.

As a consequence of the anti-commutation relation,

—

= 1 + · + ·— .

e·+· (9.46)

— —

Exercise 9.8. Show that for pairs with di¬erent labels, say ·1 ·1 and ·2 , ·2 they

commute and powers vanish, i.e.

— — —

(·1 ·1 )2 = 0 .

[·1 ·1 , ·2 , ·2 ] = 0 , (9.47)

9.4. Fermions and Grassmann variables 283

Show that

n n n

—

·± ·± —

—

·± ·±

e = e = (1 + ·± ·± ) . (9.48)

± =1

±=1 ±=1

Di¬erentiation, symbolized by the operator ‚/‚·, is the linear operation de¬ned

by ¬rst anti-commuting the variable next to the operation, giving for example

‚— ‚

(’· · — ) = ’· — .

(· ·) = (9.49)

‚· ‚·

For the function in Eq. (9.45) we thus get the derivatives

‚ ‚

f (·, · — ) = c1 + c2 · — —

— f (·, · ) = c2 ’ c3 ·

, (9.50)

‚· ‚·

and

‚‚ ‚‚

f (·, · — ) = c3 = ’ —

— f (·, · ) . (9.51)

—

‚· ‚· ‚· ‚·

Di¬erentiations with respect to a pair of Grassmann variables thus anti-commute.

For the case of fermions, the sources we encounter must satisfy the algebra of

anti-commuting variables, say the anti-commutation relations

·(x, „ ) · — (x , „ ) = ’· — (x , „ ) ·(x, „ ) (9.52)

and we have the generating functional

d„ (ψ(x,„ ) ·(x,„ ) + ψ † (x,„ ) · — (x,„ ))

Z[·, · — ] = T c ei dx

(9.53)

c

generating for example the two-point fermion contour ordered Green™s function, or

propagator, according to

δ 2 Z[·, · — ]

= ’i Tc(ψ(x, „ ) ψ † (x , „ ))

G(x, „ ; x , „ ) = i — (9.54)

δ· (x , „ ) δ·(x, „ )

the anti-commutation of the ¬elds under the contour ordering being respected since

derivatives with respect to Grassmann variables anti-commute.

Instead of the equality

δ

= δ(x ’ x ) δ(„ ’ „ )

, J(x , „ ) (9.55)

δJ(x, „ )

valid for bosonic sources, we thus have for fermions

δ δ

, · — (x , „ )

= δ(x ’ x ) δ(„ ’ „ ) =

, ·(x , „ ) (9.56)

— (x, „ )

δ·(x, „ ) δ·

and the following combinations of di¬erentiations anti-commuting

δ δ δ δ

, =0= , . (9.57)

δ·(x, „ ) δ· — (x , „ ) δ·(x, „ ) δ·(x , „ )

284 9. Diagrammatics and generating functionals

The topological arguments of Section 9.2.2 are unchanged for the case of including

also fermions, and we obtain for example the fundamental dynamical equation for

the case of electrons interacting through Coulomb interaction, V ,

δZ[·, · — ] δ3 — —

(0)

= G1¯ V¯ — δ· — + ·¯ Z[·, · ] (9.58)

1234 1

1

δ·1 δ·4 δ·3 2

or for coupled fermions and bosons for example

δZ[J, ·, · — ] δ3 — —

(0)

= G1¯ g¯ — + ·¯ Z[J, ·, · ] (9.59)

123 1

1

δ·1 δJ3 δ·2

and similar for the other source derivatives. Here we have for once made the species

labeling of the sources explicit.31

For a non-interacting, free, quantum ¬eld theory we can immediately solve the

corresponding Eq. (9.32), and obtain the generator of the free theory for fermions,

recall Eq. (9.48),

— (0)

Z0 [· — , ·] = e 2 ·1

i

G1¯ ·¯

. (9.60)

1

1

9.5 Generator of connected amplitudes

We now show how to express the generator of all amplitudes, connected and discon-

nected, in terms of a less redundant quantity, the generator of connected amplitudes.

Their relation is provided simply by the exponential function

Z[J] = eW [J] , W [J] = ln Z[J] . (9.61)

We shall ¬rst provide an intuitive demonstration arguing only at the diagrammatic

level, and then give the general combinatorial proof.

9.5.1 Source derivative proof

The diagrams collected in the generating functional Z contain redundancy, viz. the

presence of disconnected diagrams.32 Say, for an 8-state amplitude there will a dia-

gram which is the product of the ¬rst diagram on the right in Figure 9.4 multiplied

by itself, describing processes which do not interfere. Furthermore, there is the re-

dundancy of disconnected vacuum diagrams, the blobs of particles in and out of the

vacuum. The disconnected diagrams, we now show, quite generally can be factored

out of any N -state diagram. By this procedure the generator will be expressed in

terms of the generator of only connected diagrams. It turns out that it is the ex-

ponential function which relates these two quantities. The presence of disconnected

31 For any theory whose diagrammatics we derived in Chapter 4, we know the vertices and we

can now immediately write down the fundamental functional di¬erential equation satis¬ed by the

generating functional.

32 For the diagrammatics we encountered in Chapter 4, all physical quantities were ab initio

expressed in terms of connected diagrams owing to using the closed time path or contour formulation.

9.5. Generator of connected amplitudes 285

diagrams is equivalent to processes that do not interfere with each other. The physi-

cal content of expressing the theory only in terms of connected diagrams is profound,

viz. it is possible to describe a subsystem without bothering about the rest of the

Universe with which it does not interact. This is in accordance with all experimental

experience: processes separated far enough in space do not in¬‚uence each other. We

have in the diagrammatic approach stated the laws of Nature in terms of diagram-

matic rules and we now show that the feature of having to deal only with connected

diagrams is built in implicitly.33

Let us go back to the equation for the ¬rst derivative of the generator, the di-

agrammatic equation depicted in Figure 9.26. For the ¬rst diagram on the right,

the tadpole or 1-state amplitude, we have in general the diagrammatic relationship

depicted in Figure 9.33.

=

Figure 9.33 Tadpole and connected tadpole relation.

In Figure 9.33, the hatched circle denotes the sum of all connected tadpole or

1-state amplitude diagrams. The diagrammatic argument for the validity of this

relation is that since the external particle line has no option of ending on an external

state it must enter into a vertex, thereby creating connected diagrams, and any such

can be accompanied by any vacuum side show.

The class of diagrams contained in the second term on the right in Figure 9.26

can be split topologically into the two distinct classes depicted in Figure 9.34.

R

= +

Figure 9.34 One-source road diagrams and disconnected diagrams.

Here the ¬rst diagram contains all the diagrams where we can follow at least one

set of connected lines from the external state to the source, there is a road from the

external state to the source. The second diagram is the sum of diagrams where there

is no road from the initial state to the source, the external state and the source are

disconnected. Then the two propagator lines must enter connected diagrams which

33 In the canonical derivation of quantum ¬eld theory diagrammatics of chapter 4, the cancella-

tion of the disconnected diagrams follows from the same argument as given in this and the next

subsections, or the observation was super¬‚uous in the close time path formulation as they occurred

in multiples with opposite signs.

286 9. Diagrammatics and generating functionals

can be accompanied by any vacuum diagram. In the road diagram, disconnected

diagrams must be vacuum diagrams. In the road diagram the disconnected vacuum

bubbles can therefore be split o¬ and the connected road diagram appears, as depicted

in the ¬rst diagram on the right in Figure 9.35.

= +

Figure 9.35 Splitting o¬ the sum of vacuum diagrams in the road diagram.

To get the second set of diagrams on the right in Figure 9.35, we have used the

relation depicted in Figure 9.33, the sum of connected 1-state diagrams multiplied

by the sum of vacuum diagrams is the sum of 1-state diagrams.

Next we go on to the third diagram on the right in Figure 9.26. It can be split

uniquely into the topologically di¬erent classes speci¬ed on the right in Figure 9.36.

R2

= +

+ +

Figure 9.36 Road diagram classi¬cation.

Here the ¬rst diagram on the right comprises all the diagrams with roads from

the external state to both sources, the second diagram all the diagrams with no roads

from the external state to the sources, and the last two diagrams comprise all the

diagrams with roads to only one of the sources. Clearly, this groups the diagrams

uniquely into topologically di¬erent classes. In the road diagram the vacuum part

splits o¬ from the connected road diagram to both sources and we get the relation

depicted in Figure 9.37.

9.5. Generator of connected amplitudes 287

= +

+ 2

Figure 9.37 Splitting o¬ the vacuum diagrams in the road diagram in Figure 9.36.

Here the factor of two appears in front of the last diagram because the last two

diagrams in Figure 9.36 give identical contributions, and we have again used the

fact that the sum of connected 1-state diagrams multiplied by the sum of vacuum

diagrams is the sum of 1-state diagrams.

For the class of diagrams contained in the third term on the right in Figure 9.26 for

the 1-state amplitude in the presence of the source, we can again split them uniquely

into di¬erent topological classes: the set where the external state is connected to all

the three sources, or to two or only one or none, i.e. the external state is disconnected

from the sources, and we obtain the relation depicted in Figure 9.38.

+3

=

+3 +

Figure 9.38 Road diagram classi¬cation.

288 9. Diagrammatics and generating functionals

Similarly we can proceed for the diagrams in Figure 9.26 with four and more

sources: split them into classes where the external state is connected to 0, 1, 2, 3, 4,

etc., of the sources.

Collecting the results obtained so far, we can re-express the equation for the 1-

state amplitude in the presence of the source, the diagrammatic equation depicted in

Figure 9.26, in the form speci¬ed in Figure 9.39. The higher-order diagrams in the

parenthesis not displayed will all, according to the above construction, appear with

the factorial prefactor speci¬ed by the number of sources.

⎛ ⎞

+ 2! + 3!

⎝ +...⎠

+ 1 1

=

⎛ ⎞

+ + ·2 + 3!

⎝ +...⎠

1 3

2!

⎛ ⎞

+ 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

represents functional di¬erentiations. We could of course also immediately arrive at

the equation in Figure 9.45 diagrammatically, the options for entering into connected

diagrams for say propagators emerging from the 4-vertex is either into a 3-state

diagram, or 2- and 1-state diagrams. The prefactor of the next to last diagram on

the right in Figure 9.45 is caused by the three identical diagrams with the appearance

of the 2-state diagram.

We have thus expressed the 1-state connected Green™s function, in the presence

of the source, in terms of higher-order connected Green™s function and the free prop-

agators and vertices of the theory. By taking further derivatives in Eq. (9.77) we

can obtain the di¬erential equation satis¬ed by any connected Green™s function and

immediately write down its diagrammatic analog. We have thus made the full circle

back to the canonically derived non-equilibrium Feynman diagrammatics of Chap-

ters 4 and and 5, where the diagrams represented averages of the quantum ¬elds, but

now we are armed in addition with the powerful tool of a functional formulation of

non-equilibrium quantum ¬eld theory.

The amplitudes which in this chapter were de¬ned in terms of the diagrams

are thus for the case of quantum ¬eld theory the expectation values of products of

the quantum ¬elds of the theory. For example, the 2-state connected amplitude is

296 9. Diagrammatics and generating functionals

the 2-point Green™s function (normal or anomalous for the superconducting state as

dictated by the Nambu index), etc. The 1-state amplitude is the average value of

the quantum ¬eld. The 1-state amplitude, the tadpole, thus vanishes for a state with

a de¬nite number of particles, but can be non-vanishing for, for example, photons

in a coherent state. However, even when treating a system with a de¬nite number

of bosons, it can be convenient to introduce states where the average value of the

bose ¬eld is non-vanishing. This will be the case when we discuss the Bose“Einstein

condensate in Section 10.6.

Exercise 9.9. Show by taking one more source derivative of Eq. (9.77) that the

equation for the 2-state connected Green™s function in the presence of the source, for

a 3-vertex theory, has the diagrammatic form depicted in Figure 9.46.

= +

1

+ 2

Figure 9.46 Equation for the 2-state connected Green™s function for a 3-vertex

theory.

Then argue instead diagrammatically from the equation in Figure 9.45 to obtain the

above equation. One then learns to appreciate the skill of di¬erentiation.

9.6 One-particle irreducible vertices

In order to get a handle of the totality of connected diagrams we shall further exploit

their topology for classi¬cation. We introduce the concept of one-particle irreducible

diagrams (1PI-diagrams). All diagrams can then be classi¬ed uniquely by the topo-

logical property: they can be cut in two by cutting zero (1PI), one, two, etc., internal

bare lines. This will lead to the appearance of the one-particle irreducible vertices,

and to the important formulation of the theory in terms of the e¬ective action.

Consider the 1-state connected Green™s function in the presence of the source, i.e.

the derivative of the generator of connected Green™s functions

δW [J]

•1 = . (9.78)

δJ1

9.6. One-particle irreducible vertices 297

We shall refer to this function as the ¬eld.36 Besides being a function of the state

exposed by di¬erentiation, the ¬eld is also a functional of the source, •1 = •1 [J]. We

shall leave this feature implicit. However, in the diagrammatic notation we shall keep

the source dependence explicit, through the cross, as we introduce the diagrammatic

notation depicted in Figure 9.47 for the ¬eld.

•1 = 1

Figure 9.47 Diagrammatic representation of the 1-state connected Green™s function

or average ¬eld, the tadpole.

The state label of the ¬eld, exposed by di¬erentiating the generator of connected

Green™s functions, launches a free propagator which in its further propagation has two

options. The trivial one is where it propagates directly to the source in accordance

with the ¬rst diagram on the right in Figure 9.45, this option being represented

by the ¬rst diagram on the right in Figure 9.48. The other option corresponds to

interaction and the corresponding diagrams can be uniquely classi¬ed topologically

into distinct classes as follows: the exposed state where a propagator is launched can

enter into a connected diagrammatic structure which has the property that it can

not be cut in two by cutting only one internal bare propagator line, i.e. excepting

the launched propagator. By de¬nition such diagrams must not end on the source,

and these diagrams are thus a subset of the set of diagrams described by the ¬rst

diagram on the right in Figure 9.44, and are referred to as one-particle irreducible

diagrams, 1PI-diagrams. Diagrammatically this set of diagrams is represented by the

second diagram on the right in Figure 9.48. The next option is that the launched

propagator enters into a one-particle irreducible diagrammatic part and emerges into

a diagrammatic part such that the total diagram can be cut into two parts by cutting

one internal line at exactly one or two or three, etc., places, all of these lines therefore

emerging into the 1-state connected Green™s function in the presence of the source,

the ¬eld.37 Diagrammatically these sets of diagrams are therefore represented by the

third, etc., diagrams on the right in Figure 9.48 (combinatorial factors are inherited

from our convention, here expressed in the starting equation depicted in Figure 9.44).

The 1-state connected Green™s function, the tadpole, is thus represented in terms of

the one-particle irreducible vertices with attached tadpoles as depicted in Figure 9.48.

36 Or average ¬eld or classical ¬eld. The reason for this terminology will become clear in the

next chapter (or by comparison with the diagrammatic representation of the canonical operator

formalism). The diagrammatic structure of the theories considered are identical to those of the

quantum ¬eld theories we studied in Chapter 4. Therefore, interpreting the diagrammatic theory

as a quantum ¬eld theory, the 1-state amplitude is the average value of the quantum ¬eld (in

the presence of the source). For photons in a coherent state it describes the classical state of the

electromagnetic ¬eld.

37 Two (or more) lines can not enter into the same connected diagram, since then it is part of the

one-particle irreducible part.

298 9. Diagrammatics and generating functionals

= + +

1 1 + ···

+ +

2! 3!

Figure 9.48 The 1P-irreducible vertex representation of the 1-state connected

Green™s function in the presence of the source.

We could continue and construct the one-particle irreducible vertex representation

for any N -state amplitude, but we do not pause for that and relegate it to Section

9.6.2.

The 1-state connected amplitude in the presence of the source which by de¬nition

had the diagrammatic expansion depicted in Figure 9.44 and for a 3- plus 4-vertex

theory was shown to satisfy the diagrammatic equation depicted in Figure 9.45 has

now been organized into a di¬erent diagrammatic classi¬cation by introducing the

one-particle irreducible vertex functions, “1,2,...,N , which diagrammatically are rep-

resented by cross-hatched circles with amputated lines protruding and the dots as

usual represent the states where lines can end up or emerge from, as shown in Figure

9.49.

2

1

“12...N =

N

Figure 9.49 One-particle irreducible N -vertex function.

The diagrams on the right in Figure 9.48 correspond to a di¬erent re-grouping

of the diagrams compared to those on the right-hand side in Figure 9.45, the re-

grouping being based on a topological feature easily visually recognizable for any

diagram: its 0, 1, 2, etc., irreducibility with respect to internal cutting. According

to the topological construction, the one-particle irreducible vertices do not depend

on the source J. They are uniquely speci¬ed in diagrammatic perturbation theory

in terms of the bare vertices and bare propagators (and their topological property

of one-particle irreducibility). As we shall see in the next section, they provide yet

9.6. One-particle irreducible vertices 299

another way of capturing the content of the diagrammatic perturbation theory. The

virtue of the diagrammatic relationship expressed in Figure 9.48 is that no loops

appear explicitly, they are all buried in the one-particle irreducible vertices.

The diagrammatic structure of the equation expressed in Figure 9.48 should be

stressed: the tadpole in the presence of the source is expressed in terms of tad-

poles in the presence of the source attached to 1PI-irreducible vertices, i.e. in terms

of so-called tree diagrams, diagrams that become disconnected by cutting just one

propagator line. This observation shall be further developed in Section 10.3.

Analytically, the diagrammatic equation in Figure 9.48 reads

1 1

(0)

•1 = G12 J2 + “2 + “23 •3 + “234 •3 •4 + “2345 •3 •4 •5 + ... . (9.79)

2 3!

To write Eq. (9.79) in a compact form, we collect the one-particle irreducible vertices

into a generator, the generator of the one-particle irreducible vertex functions, the

e¬ective action38

∞

1

“[•] ≡ “12...N •1 •2 · · · •N (9.80)

N!

N

so that the one-particle irreducible vertices, or one-particle irreducible amputated

Green™s functions, are obtained by functional di¬erentiation

δ N “[•]

“12...N = . (9.81)

δ•1 δ•2 · · · δ•N

•=0

Recall that the one-particle irreducible vertices, “12...N , by construction do not de-

pend on the source, and the ¬eld is a function we can vary as it is a functional of the

source which is at our disposal to vary.

We can then rewrite Eq. (9.79) as

δ“[•]

(0)

•1 = G12 J2 + . (9.82)

δ•2

We introduce the diagrammatic notation depicted in Figure 9.50 for the e¬ective

action, the generator of 1PI-vertices.

•

“[•] =

Figure 9.50 Diagrammatic notation for the e¬ective action.

We introduce the diagrammatic notation for the functional derivative of the ef-

fective action depicted in Figure 9.51.

38 The e¬ective action is also referred to as the e¬ective potential for the theory. We shall return

to the reason for the terminology in Section 9.8. In the next chapter we develop the e¬ective action

approach, developing functional integral expressions for the e¬ective action.

300 9. Diagrammatics and generating functionals

•

δ“[φ]

=

δφ1 1

Figure 9.51 Diagrammatic notation for the ¬rst derivative of the e¬ective action.

The dot in Figure 9.51 signi¬es as usual a state label and the functional depen-

dence on the ¬eld is made explicit. Similarly, diagrams containing additional dots

represent additional functional derivatives with respect to the ¬eld, and give, upon

setting the ¬eld equal to zero, • = 0, the one-particle irreducible vertices depicted

diagrammatically in Figure 9.49.

Operating on both sides of Eq. (9.79) with the inverse free propagator according

to Eq. (9.5) thus gives

1 1

0 = J1 + “1 + (’G’1 + “)12 •2 + “123 •2 •3 + “1234 •2 •3 •4 + ... (9.83)

0

2 3!

and we can rewrite Eq. (9.83) in the form (upon absorbing the inverse free propagator

in the de¬nition of the 2-state irreducible vertex (’G’1 + “)12 ’ “12 ):

0

δ“[•]

0 = J1 + . (9.84)

δ•1

Diagrammatically, Eq. (9.84) is represented as depicted in Figure 9.52.

•

0= +

Figure 9.52 Source and e¬ective action relationship.

The content of Eq. (9.78) and Eq. (9.84) is that up to an overall constant, the

e¬ective action is the functional Legendre transform of the generator of connected

Green™s functions39

“[•] = W [J] ’ J • (9.85)

and the Legendre transformation thus determines the overall value, “[• = 0].

We note that in the absence of the source, J = 0, Eq. (9.84) becomes40

δ“[•]

= 0, (9.86)

δ•1

the e¬ective action is stationary with respect to the ¬eld. This is an equation stating

that the possible values of the ¬eld can be sought among the ones which make the

e¬ective action stationary.

39 In equilibrium statistical mechanics, the e¬ective action “ is thus Gibbs potential or free energy,

i.e. the (Helmholtz) potential or free energy in the presence of coupling to an external source, a

J-reservoir.

40 In the applications to non-equilibrium situations we consider in Chapter 12, this option is not

available as part of the source is an external classical force, the classical force that drives the system

out of equilibrium, and we shall employ Eq. (9.84).

9.6. One-particle irreducible vertices 301

9.6.1 Symmetry broken states

Having introduced the e¬ective action according to Eq. (9.80) we are considering the

normal state, i.e. we assume that the ¬eld vanishes in the absence of the source

δW [J]

•1 = = 0. (9.87)

δJ1

J=0

These, however, are not the only type of states existing in nature, there exist states

with spontaneously broken symmetry, i.e. states for which41

δW [J]

≡ •cl = 0 .

•1 = (9.88)

1

δJ1

J=0

We shall consider precisely such a situation and use the formalism presented in this

chapter when we discuss Bose“Einstein condensation in Section 10.6. In Chapter

8 we already encountered the generic symmetry broken state, the superconducting

state. It can be discussed as well in the present formalism by just allowing the ¬eld

or order parameter to be a composite object. We discuss this case in Section 10.5

where we in addition to a one-particle source include a two-particle source.

For a symmetry broken state we shall de¬ne the e¬ective action according to

∞

1

“[•] ≡ “12...N [•cl ] (•1 ’ •cl )(•2 ’ •cl ) · · · (•N ’ •cl ) . (9.89)

1 1 2 N

N!

N

This means that according to Eq. (9.84) we again have

δ“[•]

= ’J1 . (9.90)

δ•1

• =•cl

The e¬ective action vanishes for vanishing source.

By a shift of variables, • ’ •cl ’ •, we can of course rewrite Eq. (9.89) as

∞

1

“12...N •1 •2 · · · •N ,

“[•] = (9.91)

N!

N

where now the vertices “12...N = “12...N [•cl = 0] are evaluated in the normal or

1

so-called disordered or symmetric state where the classical ¬eld vanishes, •cl = 0.

1

We thus realize the fundamental importance of the e¬ective action: it allows us to

41 Such states are well-known in equilibrium statistical mechanics, for example from the existence

of ferro-magnetism, the appearance below a de¬nite critical temperature of an ordered state with

a magnetization in a de¬nite direction despite the rotational invariance of the Hamiltonian. These

spontaneously broken symmetry states were ¬rst studied in the mean ¬eld approximation, the

Landau theory, and the full theory of phase transitions, critical phenomena, were obtained by

Wilson using ¬eld theoretic methods. Super¬‚uid phases are broken symmetry states, and even more

fundamentally, the masses of quarks are the result of the Higgs ¬eld having a nonzero value.

302 9. Diagrammatics and generating functionals

explore the existence of symmetry broken states by searching for extrema of the

e¬ective action, i.e. solutions of

δ“[•]

=0 (9.92)

δ•1

for which the ¬eld is di¬erent from zero, • = 0.

We shall also encounter symmetry broken states created by a simpler mechanism,

viz. owing to the presence of an external classical ¬eld, but again the e¬ective

action approach shall prove useful for such non-equilibrium states, as we elaborate

in Chapter 12.

9.6.2 Green™s functions and one-particle irreducible vertices

In this section we shall show that since the generator of connected Green™s functions

and the e¬ective action are related by a Legendre transform we can, by using the

functional methods, easily obtain the systematic functional di¬erential equations

expressing connected Green™s functions in terms of the one-particle irreducible vertex

functions. But ¬rst let us argue for such equations at the purely diagrammatic level.