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Figure 9.23 Diagrammatic notation for the source in the state indicated by the dot.

We have thus introduced the new diagrammatic feature that a particle line, as
dictated by the generating functional, can end up on a source. The propagator dot
and the corresponding source dot are thus shared in accordance with the convention
that the corresponding state label are contracted, i.e. repeated indices are summed,
integrated, over in accordance with the de¬nition in Eq. (9.8).25
If the source is not set to zero after di¬erentiation
δ N Z[J]
A12...N [J] = (9.28)
δJ1 δJ2 · · · δJN
25 The dot was also used in connection with the vertices, and another reason for this is that in
fact a vertex is a generalization of a source, generating multi-particle states.
9.2. Generating functional 275


we generate a new quantity, the amplitude in the presence the source. A source
dependent amplitude is a function of the state labels exposed by the labels of the
sources with respect to which the generating functional is di¬erentiated as well as a
functional of the source.
For the non-interacting theory in the presence of the source, the amplitude A1 [J]
is represented by the diagram depicted in Figure 9.24.

(0)
= G12 J2
1 2


Figure 9.24 Diagrammatic representation of the amplitude A1 for a free theory in
the presence of the source.

We now turn to show how to express in terms of functional di¬erential equations,
all the diagrammatic equations relating amplitudes, as exempli¬ed in Figure 9.10,
and derived by the simple diagrammatic rule: to interact or not. This is achieved by
¬rst expressing the fundamental dynamic diagrammatic equation displayed in Figure
9.10, in terms of a di¬erential equation for the generating functional.
The ¬rst derivative of the generating functional generates according to its de¬ni-
tion the terms
δZ[J] 1 1
= A1 + A1¯ J¯ + A1¯¯ J¯ J¯ + A ¯¯¯ J¯ J¯ J¯ + . . . . (9.29)
22 23 2 3
3! 1234 2 3 4
δJ1 2
Di¬erentiating the generating functional with respect to the source of a certain la-
bel removes this source, corresponding diagrammatically to removing a cross, and
exposes this state in a bare propagator as each source dependent amplitude thus no
longer ends up on this source but in the corresponding particle state, a particle is
launched.26 An external state, no longer contracted with the source, is thus exposed
in each of the diagrams on the right-hand side in Figure 9.22 representing the gener-
ating functional, viz. the state with the label of the source with respect to which we
di¬erentiate. We therefore introduce the diagrammatic notation for the ¬rst deriva-
tive of the generating functional, the 1-state amplitude in the presence of the source,
where a state on a free propagator line extrudes from the generating functional as
depicted in Figure 9.25.



≡ δZ
1
δJ1


Figure 9.25 Diagram representing the ¬rst derivative.

The cross in the diagram in Figure 9.25 is there to remind us that the ¬rst
26 Orterminated as kept track of for convenience by yet an index in the collective index, and not
as in Chapter 4 by an arrow.
276 9. Diagrammatics and generating functionals


derivative, the source dependent 1-state amplitude function, is still a functional of
the source.
The equation for the ¬rst derivative of the generating function, Eq. (9.29), can
therefore be expressed diagrammatically as depicted in Figure 9.26.



= +



1 1
+ ···
+ +
2! 3!




Figure 9.26 Diagrammatic expansion of the 1-state amplitude in the presence of
the source.

Let us consider a 3-vertex theory. The ¬rst diagram on the right in Figure 9.26,
the tadpole, is then given by the diagram in Figure 9.16, i.e. speci¬ed by the vertex
and the 2-state amplitude. The second diagram on the right in Figure 9.26 can
according to the two options of the external state line, interact or not, be split into
the two diagrams on the right-hand side depicted in Figure 9.27. For the latter option
the exposed state propagates directly to the source as depicted in the ¬rst diagram
on the right.




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

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