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!

⎛ ⎞