With the usual convention of absorbing the functional determinant in the de¬ni-

tion of the functional integral we get, in accordance with Eq. (10.63) and Eq. (10.69),

that the amplitudes of the free theory are obtained according to

’1

Dφ φ1 φ2 · · · φ2N e’ 2 • G0

i

•

A12...2N =

(iG’1 )p1 p2 (iG’1 )p3 p4 · · · (iG’1 )p2N ’1 p2N .

= (10.70)

0 0 0

a.p.p.

By inspecting the path integral expression for the generating functional

’1

Dφ e’ 2 • G0

i

•

eiSi [φ] eiφ J

Z[J] = (10.71)

one can envisage its perturbation expansion and corresponding Feynman diagrams by

this recipe: expand all exponentials except the one containing the inverse free propa-

gator, the Gaussian term, and evaluate the averages according to the above formula,

Eq. (10.70). This recipe for functional integration of products of ¬elds weighted by

their Gaussian form provides Wick™s theorem, but now in the functional or path

integral formulation of the ¬eld theory. From this observation we can immediately

recover the non-equilibrium Feynman diagrammatics of a quantum ¬eld theory by

expanding the exponential containing the interaction in Eq. (10.41).

10.2. Generators as functional integrals 329

The limiting procedure used in Section 10.1 to de¬ne functional integration can

be made rigorous only for the Euclidean case. For the quantum ¬eld theory case, an

alternative now o¬ers itself, viz. to de¬ne the functional integrals in terms of their,

as above, perturbative expansions in the non-Gaussian interaction part.

Exercise 10.4. If the Gaussian part of the integrand in Eq. (10.63) is interpreted as

a probability distribution for the random or stochastic variable x, then Eq. (10.69)

is the statement that any correlation function of a Gaussian random variable, with

zero mean, is expressed in terms of all possible products of the two-point correlation

function.

Show that the generating function, i.e. the Fourier transform of the normalized

probability distribution

’1/2

A

e’ 2 x

T

1

Ax

P (x) = det , (10.72)

2π

is

A’1 k

P (k) = e’ 2 k

T

1

. (10.73)

Exercise 10.5. Consider a set of independent stochastic variables {xn }n=1,...,N ,

each with arbitrary probability distributions except for zero mean and same ¬nite

√

variance, say σ. Show that the stochastic variable X = (x1 + · · · + xN )/ N will then

obey the central limit theorem, i.e. in the limit N ’ ∞, the stochastic variable X

will be Gaussian distributed with variance σ.

Another application of the formula Eq. (10.66), allows us to rewrite Eq. (10.58)

(0) 1δ δ

δ i G0

Z[J] = eiSi [’i δ J ] e 2 JG J

eiSi [φ] + i• J

= e 2 δφ (10.74)

δφ

thereby giving the following functional integral expression

’1 1δ δ

1

Dφ e 2 δ φ G0

Z[J] = e 2 Tr ln G0 eiSi [φ] + i• J . (10.75)

δφ

φ=0

From here we see directly that Z[J] is the sum of all the vacuum diagrams for the

theory in question in the presence of the source J. This observation is again the

equivalent of Wick™s theorem, but here at its most expedient form involving both

functional integration and di¬erentiation.

Introducing the generator of connected Green™s functions

Z[J] = eiW [J] (10.76)

and recalling the combinatorial argument of Section 9.5, the above important ob-

servation gives that iW [J] is the sum of all the connected vacuum diagrams in the

presence of the source J.

For the connected Green™s functions we then obtain the functional integral ex-

pression

Dφ φ1 φ2 · · · φN eiS[φ]

≡ φ1 φ2 · · · φN .

G12...N = (10.77)

Dφ eiS[φ]

330 10. E¬ective action

Often the denominator, which cancels all the disconnected contributions in the nu-

merator, is left implicit as a normalization factor in the de¬nition of the functional

integral.

For the average or classical ¬eld, •1 , considered in Section 9.6, we thus have the

functional integral expression for the Euclidean case

Dφ φ1 eS[φ]

≡ φ1 ≡ φ1 ,

•1 = (10.78)

Dφ eS[φ]

the reason for calling •1 ≡ G1 the average ¬eld now being obvious.

The diagrammatics obtained by the above procedures are of course naive per-

turbation theory, expressed in terms of the bare propagators and vertices. A rep-

resentation which contains the full propagators and the e¬ective vertices is a better

representation since it expresses the physics of a particular situation, viz. the state

under consideration. This representation can be obtained at the diagrammatic level

by topological arguments, leading to the so-called skeleton diagrams as discussed in

Section 4.5.2. In Section 9.8, we followed another way and employed the e¬ective

action to show how easily the skeleton diagrammatics is obtained from the analytical

functional di¬erentiation formalism. In the next section, we show how the partially

re-summed perturbation expansion of Green™s functions, the skeleton diagrammatic

representation, is expressed in the functional integral formalism.

10.3 Generators and 1PI vacuum diagrams

In the previous section we showed that the generating functionals had perturbative

expansions corresponding diagrammatically to the sum of all vacuum diagrams ex-

pressed in naive perturbation theory. In this section we shall exploit the functional

integral representation of a quantum ¬eld theory to relate the various generators to

classes of one-particle irreducible vacuum diagrams.

We therefore turn to show that the generator of connected Green™s functions

can be expressed in terms of the e¬ective action and a restricted functional integral.

A restricted functional integral is a functional integral interpreted in terms of its

perturbative expansion or equivalently the corresponding Feynman diagrams, and

where only certain topological classes of diagrams are retained. First, we recall

the result derived diagrammatically, the relationship displayed in Figure 9.48: that

the tadpole, the ¬rst derivative of the generator of connected amplitudes, has a

diagrammatic expansion in terms of only tree diagrams, tadpoles attached to one-

particle irreducible vertices. This means that the generator of connected amplitudes,

W [J], itself is given by the irreducible vertices attached to tadpoles. This suggests

that the generator of connected amplitudes, W [J], can be speci¬ed in terms of the

e¬ective action, “[φ]. We now turn to show that it is indeed the case and this in

terms of a functional integral where the e¬ective action appears instead of the action

and the functional integral is restricted:

Dφ ei“[φ] +iφ J ,

i W [J] = (10.79)

CTD

10.3. Generators and 1PI vacuum diagrams 331

where CTD indicates that only the Connected Tree Diagrams should be kept of all the

vacuum diagrams generated by the perturbative expansion of the functional integral.

Tree diagrams contain no loops, they are contained within the 1PI vertices, and tree

diagrams can be cut in two by cutting a single line of a tadpole.

To keep track of the number of loops in the diagrams generated by the unrestricted

functional integral in Eq. (10.79), we introduce the parameter a

’1

˜

Dφ eia

eiWa [J] = (“[φ] +φ J)

. (10.80)

The vacuum diagrams generated by this functional integral can be characterized as

follows. Separate out in the e¬ective action the quadratic term, which according

to Eq. (9.83) is the inverse of the full Green™s function of the theory multiplied by

a’1 . Then expand the rest of the exponential and use Wick™s theorem according to

the previous section, or rather the just derived procedure for Gaussian averaging of

products of ¬elds to obtain the perturbative expansion of the functional integral in

Eq. (10.80), and its corresponding Feynman diagrams. A Green™s function has thus

associated a factor a and each of the one-particle irreducible vertices in the rest of the

e¬ective action has associated a factor a’1 as has the source, which in this context we

also refer to as a vertex (on a par with “1 ). A diagram with V vertices (of either kind)

and P propagator lines is thus proportional to aP ’V . Since the diagrams generated

by the path integral in Eq. (10.80) are vacuum diagrams they are loop diagrams, the

tree diagrams being those with zero number of loops. Since it takes two times two

protruding lines from vertices (or one vertex) to form one loop, the number of loops

L is speci¬ed by L = P ’ V + 1, and an L-loop diagram carries an overall factor

proportional to aL’1 .15 The theory de¬ned by the functional integral in Eq. (10.80)

can thus be described at the diagrammatic level in terms of the diagrams for the

theory where a is unity, a = 1, according to

˜ ˜

aL’1 W (L) [J] ,

Wa [J] = (10.81)

L=0

˜

where W (L) [J] comprises the sum of connected vacuum diagrams with L loops for

the theory de¬ned by the action “[•], i.e. the theory speci¬ed by Eq. (10.80) for the

case a = 1. We note that the tree diagrams singled out in Eq. (10.79) correspond to

˜

the zero loop term W (0) [J].

In the limit of vanishing a, the value of the functional integral, Eq. (10.80), is

determined by the ¬eld at which the exponent is stationary, denote it •, according

to ’1

˜

eiWa [J] ∝ eia (“[•] +• J) , (10.82)

where the prefactor (a horrendous determinant term) involves the square root of

a(G’1 ’ Σ) and therefore its lowest power is a0 and will therefore turn out to be

0

15 This

observation gives, for the e¬ective action, a characterization of its diagrammatic structure,

and a controlled approximation scheme, the loop expansion. Say for a quantum ¬eld theory, the

diagrammatic representation of the e¬ective action corresponds to an expansion in L , where L is

the number of loops in a diagram.

332 10. E¬ective action

harmless when a eventually is set to zero. The stationary ¬eld, •, is determined as

the solution of the equation

δ“[•]

+ J1 = 0 (10.83)

δ•1

thus making contact with the original theory, since this is the equation satis¬ed by

the e¬ective action, Eq. (9.84). According to Eq. (10.81), in the limit of vanishing

a we have Wa [J] a’1 W (0) [J], and by taking the logarithm of Eq. (10.82) we get

˜ ˜

(noting that in this limit, the constant prefactor in Eq. (10.82) gives no contribution)

˜

W (0) [J] = “[•] + • J . (10.84)

But according to the Legendre transformation, Eq. (9.85), this implies

˜

W (0) [J] = W [J] (10.85)

and we have shown the validity of Eq. (10.79). That is, we have shown that the gener-

ator of connected Green™s functions can be expressed as the sum of all connected tree

diagrams where the vertices are one-particle irreducible.16 In diagrammatic terms,

the generator of connected Green™s functions, Eq. (10.79), can thus be displayed as

depicted in Figure 10.1.

=

+ +

1 1 + ···

2! 3!

Figure 10.1 The tree diagram expansion of the generator of connected amplitudes

in terms of the one-particle irreducible vertices.

The sum of all connected vacuum diagrams in the presence of the source is thus

captured by keeping only the tree diagrams if at the same time the bare vertices are

exchanged by the one-particle irreducible vertices.

The e¬ective action “[φa ], Eq. (9.80), taken for an arbitrary ¬eld value φa can

also be expressed in terms of a restricted functional integral, viz.

Dφ eiS[φ+φa ] ,

“[φa ] = (10.86)

1PICVD

16 This provides a proof in terms of the functional integral method, that i“ consists of the one-

particle irreducible vertices. We already knew this because of its diagrammatic construction accord-

ing to Section 9.6.

10.4. 1PI loop expansion of the e¬ective action 333

where 1PICVD indicates that in the perturbation expansion, only the connected

one-particle irreducible vacuum diagrams should be kept of the connected diagrams

generated by the perturbative expansion of the path integral, since upon expanding

in φa the prescription on the restricted functional integral generates “[φa ] according

to Eq. (9.80). In particular we have shown that “[0] is the quantity represented by

the sum of all connected one-particle irreducible vacuum diagrams for the theory (in

the absence of the source).

Since W [J] is related to “[•] by a Legendre transformation, the above observation

for “[0] corresponds to the statement that “[0] equals W [J] for the value of the

source for which the ¬eld δW [J]/δJ1 vanishes. Since the vanishing of δW [J]/δJ1 is

equivalently to δZ[J]/δJ1 vanishing, we can state the observation as

W [J] = sum of one-particle irreducible connected (10.87)

vacuum diagrams (1PICVD).

δ Z [J ]

δ J 1 =0

We shall make use of this observation when we consider the loop expansion of the

e¬ective action.

10.4 1PI loop expansion of the e¬ective action

In this section we shall use the path integral representation of the generators to

get a useful path integral expression for the e¬ective action which has an explicit

diagrammatic expansion. We follow Jackiw, and show how to express the e¬ective

action in terms of the one-particle irreducible connected vacuum diagrams for a

theory with a shifted action [52].

Consider a ¬eld theory speci¬ed by the action S[φ] and the corresponding path

integral expression for the generating functional

Dφ eiS[φ]+if φ ,

Z[f ] = (10.88)

where we have used the notation f for the one-particle source. In fact, in the next

chapter, when we consider non-equilibrium phenomena in classical statistical dynam-

ics the source will not be set equal to zero by the end of the day as it will contain

the classical force coupled to the classical degree of freedom of interest.

The path integral is invariant with respect to an arbitrary shift of the ¬eld, recall

Eq. (10.3),

φ ’ φ + φ0 (10.89)

giving for the generating functional

Dφ eiS[φ+φ0 ]+if (φ+φ0 ) = eiS[φ0 ]+if φ0 Z1 [f ] ,

Z[f ] = (10.90)

334 10. E¬ective action

where

Z1 [f ] = Dφ ei(S[φ+φ0 ]’S[φ0 ])+if φ . (10.91)

The subscript on Z1 [f ] is not a state label but just discriminates the generator from

the original generating functional Z[f ]. State labels in the functional di¬erentiations

are in the following suppressed throughout, and matrix multiplication is implied.

The generator of connected Green™s functions then becomes

’i ln Z = S[φ0 ] + f φ0 ’ i ln Z1 [f ]

W [f ] =

= S[φ0 ] + f φ0 + W1 [f ] , (10.92)

where

iW1 [f ] = ln Dφ ei(S[φ+φ0 ]’S[φ0 ])+if φ . (10.93)

To make the so-far arbitrary function φ0 a functional of f , we choose φ0 to be the

average ¬eld which e¬ects the Legendre transformation to the e¬ective action, “[φ],

i.e.

δW [f ]

φ0 ≡ φ = , (10.94)

δf

where a bar now speci¬es the average ¬eld, φ = •, for visual clarity in the following

equations. Recalling that this vice versa gives f implicitly as a functional of φ,

f = f [φ], we have according to Eq. (10.94) and Eq. (10.92)

δS[φ] δW1 δφ

+f + =0 (10.95)

δf

δφ δφ

and thereby, since the second factor on the left is the full Green™s function,

δS[φ] δW1

+f + = 0. (10.96)

δφ δφ

The e¬ective action can, according to Eq. (10.92) and Eq. (10.94), be expressed

as

“[φ] = W [f ] ’ φf = S[φ] + W1 [f ] = S[φ] + W1 [φ] , (10.97)

where in the last equality we have been sloppy, using the same notation for W1 as

a functional of the implicit function of φ as that of f , W1 [f ]. But by employing

Eq. (10.96) in Eq. (10.93) we can in fact eliminate the explicit dependence on f , and

get the expression for W1 as a functional of the average ¬eld, φ, as speci¬ed by the

functional integral

δS[φ] δW1

iW1 [φ] = ln Dφ exp i(S[φ + φ] ’ S[φ]) ’ iφ + . (10.98)

δφ δφ

10.4. 1PI loop expansion of the e¬ective action 335

The aim is now to evaluate W1 [φ], or rather to show that it can be expressed in

terms of one-particle irreducible connected vacuum graphs. We therefore introduce

the generating functional

δS[φ]

˜

Z[φ; J] = Dφ exp i(S[φ + φ] ’ S[φ]) ’ iφ + iJφ (10.99)

δφ

for the theory governed by the action

δS[φ]

˜

S[φ, φ] = S[φ + φ] ’ S[φ] ’ φ , (10.100)

δφ

i.e. the action for the original theory expanded around the average ¬eld but keeping

only second- and higher-order terms. Correspondingly for the generator of connected

Green™s functions in this theory we have

˜ ˜

iW [φ; J] = ln Z[φ; J] (10.101)

and evidently by comparing Eq. (10.99) and Eq. (10.98)

˜

W1 [φ] = W [φ; J] . (10.102)

J=’δW1 /δφ

We shortly turn to show that for this particular choice of the source as speci¬ed

˜

in Eq. (10.102), J = ’δW1 /δφ, the generator Z vanishes

˜

δ Z[φ; J]

=0 (10.103)

δJ

J=’δW1 /δφ

or equivalently for the generator of connected Green™s functions

˜

δ W [φ; J]

=0, (10.104)

δJ

J=’δW1 /δφ

i.e. the average ¬eld

˜

δ W [φ; J]

•= (10.105)

δJ

˜

vanishes for the theory governed by the action S for the value of the source J =

’δW1 /δφ. The statement in Eq. (10.102) thus becomes equivalent to the statement

˜

that W1 [φ] is identical to the e¬ective action for the theory governed by S[φ, φ]

˜

for vanishing average ¬eld, “[φ; • = 0]. We then use the result of Section 10.3,

that in general “[• = 0] is given by the one-particle irreducible connected vacuum

diagrams, or equivalently for the generator of connected Green™s functions the ex-

pression Eq. (10.87), viz. that W [f ]δW/δf =0 consists of the sum of all the one-particle

irreducible vacuum diagrams. The functional W1 thus in diagrammatic terms only

consists of the sum of all the one-particle irreducible vacuum diagrams for the theory

˜

governed by S[φ, φ]. These diagrammatic identi¬cations will be exploited shortly.

336 10. E¬ective action

To establish the validity of Eq. (10.102) we di¬erentiate Eq. (10.98) with respect

to the average ¬eld φ

δ2

δW1 1 δS[φ + φ] δS[φ]

Dφ ’ ’φ

= (S[φ] + W1 [φ])

Z1

δφ δφ δφ δφ δφ

δS[φ] δW1

— exp i S[φ + φ] ’ S[φ] ’ φ ’φ . (10.106)

δφ δφ

The term originating from the ¬rst term in the parenthesis on the right-hand side

can be rewritten as

δS[φ + φ] δS[φ] δW1

Dφ exp i S[φ + φ] ’ S[φ] ’ φ ’φ

δφ δφ δφ

δ δS[φ] δW1

’i Dφ exp i S[φ + φ] ’ S[φ] ’ φ ’φ

=

δφ δφ δφ

δS[φ] δW1 δS[φ] δW1

Dφ exp i S[φ + φ] ’ S[φ] ’ φ ’φ

+ +

δφ δφ δφ δφ

(10.107)

and since the ¬rst term on the right is an integral of a total derivative it vanishes,

giving

δ2

δW1 1 δW1

Dφ ’φ

= (S[φ] + W1 [φ])

Z1

δφ δφ δφδφ

δS[φ] δW1

— exp i S[φ + φ] ’ S[φ] ’ φ ’φ . (10.108)

δφ δφ

The ¬rst term on the right in Eq. (10.108) is equal to the term on the left, giving the

equation

δ 2 (S[φ] + W1 [φ]) δS[φ] δW1

Dφ φ exp i S[φ + φ] ’ S[φ] ’ φ ’φ = 0.

δφ δφ δφ δφ

(10.109)

The ¬rst factor

δ2 δ 2 “[φ]

S[φ] + W1 [φ] = (10.110)

δφ δφ δφ δφ

is according to Eq. (9.95) the inverse Green™s function and therefore nonzero, and we

have the sought after statement of Eq. (10.103)

δS[φ] δW1

Dφ φ exp i S[φ + φ] ’ S[φ] ’ φ ’φ =0. (10.111)

δφ δφ

10.4. 1PI loop expansion of the e¬ective action 337

We have thus according to Eq. (10.102) shown that

W1 [φ] = sum of all one-particle irreducible connected vacuum (10.112)

diagrams (1PICVD) for the theory de¬ned by the

˜

action S[φ, φ].

˜

Dividing the action S[φ, φ] into its quadratic part and the interaction part

˜ ˜ ˜

S[φ; φ] = S0 [φ; φ] + Sint [φ; φ] (10.113)

we have

1 δ 2 S[φ] 1

φ ≡ φD’1 [φ] φ

˜

S0 [φ; φ] = φ (10.114)

2 δφδφ 2

and

∞

1 δ N S[φ]

˜ φ1 · · · φN .

Sint [φ; φ] = (10.115)

N ! δφ1 · · · δφN

N =3

˜

In the path integral expression for the generator Z, Eq. (10.99), the normalization

factor

√

’1

i

Dφ e 2 φD φ = i det D (10.116)

was kept implicit, but by exposing it the expression for the e¬ective action, Eq. (10.98),

can ¬nally be written as

i

“[φ] = S[φ] ’ Tr ln iD’1 [φ] ’ i ln eiSint [φ;φ] 1PICVD ,

˜

(10.117)

2

where the last term should be interpreted as

˜ ˜ ˜

Dφ eiS0 [φ;φ] eiSint [φ;φ]

eiSint [φ;φ] = (10.118)

1PICVD

and the subscript “1PICVD” indicates the restriction to the one-particle irreducible

connected vacuum diagrams resulting from the functional integral. We have explicitly

displayed the one-loop contribution, the second term on the right in Eq. (10.117),

and consequently we have the normalization

1 1PICVD = 1. (10.119)

The ¬rst term on the right in Eq. (10.117), the zero loop or tree approximation,

speci¬es the classical limit, determined by the stationarity of the action, and the

second term gives the contribution from the Gaussian ¬‚uctuations. The last term,

the higher loop contributions, gives the quantum corrections due to interactions,

radiative corrections. Reinstating gives the result that the contribution for a given

loop order is proportional to raised to that power.

For a 3- plus 4-vertex theory, the e¬ective action has the series expansion in

terms of one-particle irreducible vacuum diagrams as depicted (explicitly to three

loop order) in Figure 10.2, where we have reinstated the notation • = φ.

338 10. E¬ective action

“[•] = + +

+

+

+ +

+ +

+ + ···

Figure 10.2 The 1PI vacuum diagram expansion of the e¬ective action.

We note that the one-particle reducible diagram depicted in Figure 10.3, which

contributes in the sum of vacuum diagrams to W [J = 0], is absent in the series

expansion of the e¬ective action.

Figure 10.3 One-particle reducible diagram contributing to W [J = 0].

The above functional evaluation of the e¬ective action generates the one-particle

irreducible loop expansion in terms of skeleton diagrams, and in¬nite partial sum-

mation of naive perturbation theory diagrams is thus already done. A virtue of the

above expansion is that at each loop level for the e¬ective action it contains far fewer

diagrams than the naive perturbation expansion.

In Section 10.6, where the e¬ective action approach is applied to a Bose gas,

and in Chapter 12, where the theory of classical statistical dynamics is applied to

vortex dynamics in a superconductor, we shall need to take the loop expansion to

the next level where only two-particle irreducible vacuum diagrams will appear. We

therefore ¬rst go back to the generating functional technique, but now we will include

a two-particle source.

10.5. Two-particle irreducible e¬ective action 339

10.5 Two-particle irreducible e¬ective action

The e¬ective action can be taken to the next level in irreducibility in which only two-

particle irreducible vertices appear. To construct such a description, we introduce

a two-particle source K12 in addition to the one-particle source J1 , and a generator

of Green™s functions where the connected Green™s functions of the theory now are

contracted on both types of sources, i.e. de¬ned according to the diagrammatic

expansion in terms of the two sources as depicted in Figure 10.4.

+ 1

, + ···

=

2

+ +

1 1

2 2

+ 1

+ 1

+ ···

2 2

+ 1

+ ···

2

Figure 10.4 Diagrammatic expansion of the generator, W [J, K], in the presence of

one- and two-particle sources.

The diagrammatic notation for the two-particle source is thus as displayed in

Figure 10.5.

K12 = 2

1

Figure 10.5 Diagrammatic notation for the two-particle source.

340 10. E¬ective action

The diagrammatic notation for the generator makes explicit the feature that it

depends on both a one- and a two-particle source as stipulated in Figure 10.6.

,

W [J, K] =

Figure 10.6 Diagrammatic notation for the generator in the presence of one- and

two-particle sources.

The generator consists of the same one-particle source terms as the previous

generator of Section 9.5, and therefore generates the connected amplitudes or Green™s

functions of the theory according to

δ N W [J, K]

G12...N = . (10.120)

δJ1 δJ2 · · · δJN

J=0,K=0

In addition the generator contains two-particle source terms and mixed terms.

We notice the new feature of the presence of the two-particle source, that dif-

ferentiating with respect to the two-particle source can lead to the appearance of

disconnected diagrams; for example, see Figure 10.7.

1

1

+

δW [J,K]

2 =

δK12 J=K=0

2

2

Figure 10.7 Removing a two-particle source can create disconnected diagrams.

Taking the derivative with respect to the one-particle source

δW [J, K]

•1 = (10.121)

δJ1

10.5. Two-particle irreducible e¬ective action 341

we can, analogous to the procedure of Section 9.6, exploit the topological features

of diagrams to construct the diagrammatic expansion in terms of two-particle irre-

ducible vertices, as depicted in Figure 10.8. In the following we leave out in the

diagrammatic notation the implicit source dependences of quantities.

= + +

+ +

+ +

1 1

2 3!

+ ···

+ +

Figure 10.8 Two-particle irreducible expansion of the 1-state Green™s function.

The topological arguments for the diagrammatic equation displayed in Figure 10.8

is: the particle state exposed can propagate directly to either a one-particle source

or a two-particle source. In the latter case its other state can end up in anything

342 10. E¬ective action

connected, and these two classes of diagrams are depicted as the two ¬rst diagrams

on the right in Figure 10.8. Or the exposed state can enter into a two-particle

irreducible diagram, giving the class of diagrams represented by the third diagram

on the right, or into a two-particle reducible diagram. A two-particle irreducible

vertex is by de¬nition a vertex diagram which can not be cut in two by cutting only

two lines, otherwise it is two-particle reducible. In the case of entering into a two-

particle reducible diagram, the exposed state can enter into a two-particle irreducible

vertex which emerges by one line into anything connected, accounting for the fourth

diagram on the right containing the self-energy in the skeleton representation where

it is two-particle irreducible (recall the topological discussion of diagrams in Section

4.5.2). Or it can enter into a two-particle irreducible vertex which emerges by two

lines into anything connected, which can be done in the two ways as depicted in the

¬fth and sixth displayed diagrams on the right, or three or four, etc., lines as depicted

in Figure 10.8. We note that, from a two-particle irreducible vertex, three lines can

not emerge into a connected 3-state diagram since such a part is already included in

the vertex owing to its two-particle irreducibility.

Analytically we have, according to the diagrammatic equation depicted in Figure

10.8, the equation

1

(0)

•1 = G12 J2 + K23 •3 + “2 + Σ23 •3 + “2(34) G34 + “234 •3 •4

2

1 (0)

+ “2345 •3 •4 •5 + “23(45) •3 G45 + “2(34)(56) G34 G56 + ... .

3!

(10.122)

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

1

J1 + K12 •2 + “1 + (’G’1 + Σ)12 •2 + “1(23) G23 +

0= “123 •2 •3

0

2

1

“1234 •2 •3 •4 + “12(34) •2 G34 + “1(23)(45) G23 G45 + · · ·

+ ,

3!

(10.123)

which corresponds to the diagrammatic equation depicted in Figure 10.9.

10.5. Two-particle irreducible e¬ective action 343

0 = + +

+ +

+1 +

2

+ ···

+

1 1 δ“[φ,G]

≡ +

+ δφ1

Figure 10.9 Two-particle irreducible vertices and source relation.

The last equality de¬nes the ¬eld-derivative of the two-particle irreducible e¬ective

action, i.e, just as the diagrams in Figure 9.48 lead to the introduction of the one-

particle irreducible e¬ective action, we collect the two-particle irreducible vertex

344 10. E¬ective action

functions into the two-particle irreducible e¬ective action

∞

1

“[•, G] ≡ “12...N •1 •2 · · · •N

N!

N

“1(23) •1 G23 + “1(23)(45) •1 G23 G45 + · · · ,

+ (10.124)

which in addition to the ¬eld is a functional of the full propagators.

In the two-particle irreducible action, we encounter two di¬erent types of vertices,

viz. only ¬eld attachment vertices

δ N “[•, G]

“12...N = (10.125)

δ•1 · · · δ•N

φ=0,G=0

which are two-particle irreducible and for which we introduce the diagrammatic no-

tation depicted in Figure 10.10.

2

1

“12...N =

N

Figure 10.10 The 2PI vertex with only ¬eld attachments.

In addition we encounter vertices with also propagator attachments, for example

δ δ δ δ

“1(23)4(56) = “[•, G] (10.126)

δ•1 δG23 δ•4 δG56

φ=0,G=0

for which we introduce the diagrammatic notation depicted in Figure 10.11.

4

5

3

“1(23)4(56) =

2 6

1

Figure 10.11 Vertex with both ¬eld and propagator attachments.

In terms of the two-particle irreducible e¬ective action, we can diagrammatically

represent the equation depicted in Figure 10.9 as depicted in Figure 10.12 (rede¬ning

“12 ≡ (’G’1 + Σ)12 ).

0

10.5. Two-particle irreducible e¬ective action 345

0 = + +

Figure 10.12 Sources and 2PI e¬ective action relation.

Analytically, the diagrammatic relationship depicted in Figure 10.12 is

δ“[•, G]

= ’J1 ’ K12 •2 . (10.127)

δ•1

By diagrammatic construction we have analogously to the one-particle irreducible

case, G1 ≡ •1 ,

δW [J, K]

G1 = (10.128)

δJ1

but now in addition

δW [J, K] 1

= (G12 + •1 •2 ) (10.129)

K12 2

and these two relationships give implicitly the sources as functions of the ¬eld and

the full Green™s function

J = J[•, G] (10.130)

and

K = K[•, G] . (10.131)

Since the sources are independent, so are • and G. We then have the two generators

being related by the double Legendre transformation, i.e. with respect to two sources,

1 1

W [J, K] ’ • J ’ •K • ’ GK

“[•, G] = (10.132)

2 2

J=J[•,G],K=K[•,G]

and we obtain the second relation for the two-particle irreducible e¬ective action and

the sources

δ“[•, G] 1

= ’ K12 . (10.133)

δG12 2

For K = 0 we encounter the usual Legendre transformation and e¬ective action,

i.e. “[•] = “[•, G(0) ] for the value of the Green™s function for which

δ“[•, G(0) ]

= 0. (10.134)

(0)

δG12

By construction “[•, G] is the generator, in the ¬eld variable •, of the two-

particle irreducible vertices with lines representing the full Green™s function, G, and

346 10. E¬ective action

“[• = 0, G] is thus the sum of all two-particle irreducible connected vacuum diagrams.

Using Eq. (10.132) and Eq. (10.133) we have

δ“[0, G] δ“[0, G]

’ i ln Dφ φ exp i S[φ] + φJ (0) ’ φ

“[0, G] = Tr G φ

δG δG

i ln Dφ φ exp {iS0 [φ]} ,

+ (10.135)

where J (0) is the value of the source for which δW [J, K]/δJ vanishes, i.e. tadpoles

vanish.

By construction “[•, G] is the generator with respect to the ¬eld, •, of two-particle

irreducible vertex functions. For example, δ 2 “[•, G]/δ•1 δ•2 evaluated at vanishing

¬eld, • = 0, is the diagrammatic expansion for the inverse two-state Green™s function

with two-particle reducible diagrams absent and lines representing the full Green™s

function, i.e.

δ 2 “[•, G]

= G’1 = (G(0) ’ Σ[G])’1 . (10.136)

12 12

δ•1 δ•2

•=0

10.5.1 The 2PI loop expansion of the e¬ective action

In this section we shall take the discussion of Section 10.4 to the next level, the two-

particle irreducible (2PI) level and following Cornwall, Jackiw and Tomboulis obtain

the expression for the e¬ective action in terms of two-particle irreducible vacuum

diagrams [53]. We shall use the path integral representations of the generators to

¬rst get a useful path integral expression for the two-particle irreducible e¬ective

action which has an explicit diagrammatic expansion. In the two-particle irreducible

description of the previous section, physical quantities are expressed in terms of the

average ¬eld and the full Green™s functions. The generating functional with one-

and two-particle sources, f and K, corresponding to the diagrammatic expansion in

Figure 10.4 is

i

Dφ exp iS[φ] + iφf + φKφ = eiW [f,K] .

Z[f, K] = (10.137)

2

The normalization constant is chosen so that Z[f = 0, K = 0] = 1.

The derivatives of the generating functional generate the average ¬eld

δW ¯

= φ1 (10.138)

δf1

and the 2-state Green™s function according to

δW 1 ¯

= φ φ2 + iG12 , (10.139)

21

δK12

where

iG12 = φ1 φ2 ’ φ1 φ2 (10.140)

10.5. Two-particle irreducible e¬ective action 347

and we use for short

i

Dφ φ1 φ2 exp iS[φ] + iφf + φKφ

φ1 φ2 = (10.141)

2

for the amplitude A12 .

The two-particle irreducible e¬ective action, the double Legendre transform of

the generating functional of connected Green™s functions, Eq. (10.132)

1 i

“[φ, G] = W [f, K] ’ f φ ’ φKφ ’ GK (10.142)

2 2

ful¬lls

δ“

= ’f ’ Kφ (10.143)

δφ

and

δ“ i

= ’ K. (10.144)

δG 2

The double Legendre transformation can be performed sequentially, i.e. we ¬rst

de¬ne for ¬xed K

“K [φ] = (W [f, K] ’ φf ) δW [f,K]/δf =φ (10.145)

and then de¬ne G according to

δ“K [φ] 1

= (φ φ + iG) (10.146)

δK 2

and the e¬ective action according to

1 i

“[φ, G] = “K [φ] ’ φKφ ’ GK . (10.147)

2 2

That the two de¬nitions of the Green™s function and the e¬ective action are identical

follows from the identity

δ“K [φ] δW [f, K] δf δW [f, K] δf

’φ

= +

δK δf δK δK δK δW [f,K]/δf =φ

δW [f, K]

= . (10.148)

δK δW [f,K]/δf =φ

Considering K as ¬xed, “K [φ] is the e¬ective action for the theory governed by

the action

1

S K [φ] = S[φ] + φKφ. (10.149)

2

We therefore consider the generating functional

K

Dφ eiS

Z K [f ] = [φ]+iφf

(10.150)

348 10. E¬ective action

and observe

Z K [f ] = Z[f, K] . (10.151)

The generating functional of connected Green™s functions, for ¬xed K, is

W K [f ] = ’i ln Z K [f ] (10.152)

with the corresponding e¬ective action

“K [φ] = W K [f ] ’ φf . (10.153)

We can now use the method of functional evaluation of the e¬ective action of Section

10.4 and obtain

“K [φ] = S K [φ] + W1 [φ] ,

K

(10.154)

where

δS K [φ] K

δW1 [φ]

= ’i ln Dφ exp i S [φ + φ] ’ S [φ] ’ φ ’φ

K K K

W1 .

δφ δφ

(10.155)

Introducing the functional “2 according to the equation

i i i

“[φ, G] = S[φ] + Tr ln G’1 + TrD’1 [φ]G + “2 [φ, G] ’ Tr1 (10.156)

2 2 2

with the inverse of the propagator D de¬ned as

δ 2 S[φ]

’1

D [φ] ≡ (10.157)

δφ δφ

and using Eqs. (10.147) and (10.154) we have

1 i i

“2 [φ, G] = ’ Tr iD’1 [φ] + K G ’ Tr ln G’1 + W1 [φ] + Tr1 .

K

(10.158)

2 2 2

Lastly, we want to show that “2 is the sum of all the two-particle irreducible

vacuum graphs in a theory with vertices determined by the action

∞

δ N S[φ]

1

φ1 · · · φN ,

Sint [φ; φ] = (10.159)

N ! δφ1 · · · δφN

N =3

and propagator lines by the full Green™s function G. In order to do so we ¬rst

eliminate the two-particle source K

δ“[φ, G] δ“2 [φ, G]

= G’1 ’ D’1 [φ] + 2i

K = 2i . (10.160)

δG δG

Using Eqs. (10.147), (10.154) and (10.155), the e¬ective action, “[φ, G], can be

rewritten as a functional integral

δS K [φ] K

δW1 [φ]

Dφ exp i S [φ + φ] ’ S [φ] ’ φ ’φ

i“[φ,G] K K

e =

δφ δφ

K K

[φ]’ 1 φKφ’ 2 GK) [φ]’ 1 φKφ’ 2 GK)

i i

— ≡ ei(S

ei(S K

Z1 [φ] . (10.161)

2 2

10.5. Two-particle irreducible e¬ective action 349

Introducing the generator

δS K [φ]

˜ Dφ exp i (S K [φ + φ] ’ S K [φ] ’ φ

Z K [φ, J] = + φJ (10.162)

δφ

a calculation similar to the one of Section 10.4 gives

˜

δ Z K [φ, J]

= 0. (10.163)

δJ K

J=’δW1 /δφ

The average value of φ has thus been shown to vanish in the theory governed by the

action

δS K

S [φ, φ] = S [φ + φ] ’ S [φ] ’ φ

K K K

(10.164)

δφ

˜

when the source takes the value J = ’δW1 /δφ. If the generating functional Z K [φ, J]

K

is multiplied by a factor depending on G and φ the average value of φ is still zero.

Using Eqs. (10.143), (10.147) and (10.154) we therefore have

δS K [φ] δW1 [φ]

K

f+ + =0 (10.165)

δφ δφ

and obtain the following functional integral expression for the two-particle irreducible

e¬ective action

ei“[φ,G] = e’ 2 φKφ+ 2 GK Dφ ei(S

K

1

i

[φ+φ]+f φ)

. (10.166)

Using Eqs. (10.143) and (10.144) to eliminate the source f

δ“ δ“

f=’ ’ 2i φ (10.167)

δG

δφ

we obtain

δ“[φ, G]

“[φ, G] ’ G = ’i ln Dφ eiS[φ,G;φ] (10.168)

δG

where

δ“[φ, G] δ“[φ, G]

S[φ, G; φ] = S[φ + φ] ’ φ + iφ φ. (10.169)

δG

δφ

Di¬erentiating Eq. (10.168) with respect to G we obtain

δ 2 “[φ, G] δ 2 “[φ, G] δ 2 “[φ, G]

G’ φ’

0= i φφ , (10.170)

δG δG δG δG

δG δφ

where the angle brackets denote the average with respect to the action S[φ, G; φ].

The action in Eq. (10.164) with the source term ’δW1 /δφ added and the action

K

appearing in Eq. (10.169) di¬er only by an irrelevant constant, ’S[φ], and we can

350 10. E¬ective action

conclude that the average value of the ¬eld is zero for the action S[φ, G; φ], i.e.

φ = 0, and we obtain that

G = ’i φφ (10.171)

i.e. G is the full Green™s function for the theory governed by the action S[φ, G; φ].

Finally we rewrite Eq. (10.160)

G’1 = D’1 [φ] + K ’ Σ[φ, G] , (10.172)

where

δ“2 [φ, G]

Σ[φ, G] = 2i . (10.173)

δG

Since D’1 [φ] + K is the free inverse Green™s function and G’1 is the inverse full

Green™s function for the theory governed by the action in Eq. (10.169), we conclude

that Σ is the self-energy, and Eq. (10.172) thereby the Dyson equation. Since the

self-energy, Σ, is the sum of one-particle irreducible connected vacuum diagrams,

we therefore ¬nally conclude that “2 is given by the sum of two-particle irreducible

connected vacuum diagrams.

We have thus shown that the e¬ective action can be written in the form

i i i

“[φ, G] = S[φ] + Tr ln G’1 + TrD’1 [φ] G + “2 [φ, G] ’ Tr1, (10.174)

2 2 2

where “2 [φ, G] is the sum of all two-particle irreducible connected vacuum diagrams

in the theory with action φG’1 φ/2 + Sint [φ : φ], i.e.

“2 [φ, G] = ’i ln eiSint [φ;φ] 2PI

G, (10.175)

where the superscript and subscript on the angle bracket indicate that the func-

tional integral is restricted to the two-particle irreducible vacuum diagrams and the

propagator lines are the full Green™s function.

In general amplitudes or physical quantities can not be calculated exactly, and

an approximation scheme must be invoked. If no small dimensionless expansion

parameter is available we are at a loss. Furthermore, if non-perturbative e¬ects

are prevalent we are left without a general tool to obtain information. To cope

with such situations, approximate self-consistent or mean ¬eld theories have been

useful, although they are uncontrollable as not easily analytically characterized by a

small parameter. The e¬ective action approach can be used to systematically study

correlations order by order in the loop expansion. It is thus the general starting

point for constructing self-consistent approximations. An important feature of the

loop expansion is that it is capable of capturing important nonlinearities of a theory.

In practice one must at a certain order break the chain of correlations described by the

e¬ective action by brute force, a felony we are quite used to in kinetic theory. The

rationale behind this scheme working quite well for calculating average properties

such as densities and currents is that higher-order correlations average out when

interest is in such low-correlation probes. We shall use the e¬ective action approach

to study classical statistical dynamics in Chapter 12, but ¬rst we apply it in the

quantum context, viz. for the study of Bose gases.

10.6. E¬ective action approach to Bose gases 351

10.6 E¬ective action approach to Bose gases

In this section, the e¬ective action formalism is applied to a gas of bosons.17 The

equations describing the condensate and the excitations are obtained by using the

loop expansion for the e¬ective action. For a homogeneous gas, the expansion in

terms of the diluteness parameter is identi¬ed in terms of the loop expansion. The

loop expansion and the limits of validity of the well-known Bogoliubov and Popov

equations are examined analytically for a homogeneous dilute Bose gas and numeri-

cally for a gas trapped in a harmonic-oscillator potential. The expansion to one-loop

order, and hence the Bogoliubov equation, we shall show to be valid for the zero-

temperature trapped gas as long as the characteristic length of the trapping potential

exceeds the s-wave scattering length.

10.6.1 Dilute Bose gases

The dilute Bose gas has been subject to extensive study for more than half a century,

originally in an attempt to understand liquid Helium II, but also as an interesting

many-body system in its own right. In 1947, Bogoliubov showed how to describe

Bose“Einstein condensation as a state of broken symmetry, in which the expecta-

tion values of the ¬eld operators are non-vanishing due to the single-particle state

of lowest energy being macroscopically occupied, i.e. the annihilation and creation

operators for the lowest-energy mode can be treated as c-numbers [55]. In modern

terminology, the expectation value of the ¬eld operator is the order parameter and

describes the density of the condensed bosons. In Bogoliubov™s treatment, the phys-

√

ical quantities were expanded in the diluteness parameter n0 a3 , where n0 denotes

the density of bosons occupying the lowest single-particle energy state, and a is the

s-wave scattering length, and Bogoliubov™s theory is therefore valid only for homo-

geneous dilute Bose gases. The inhomogeneous Bose gas was studied by Gross and

Pitaevskii, who independently derived a nonlinear equation determining the conden-

sate density [56] [57]. A ¬eld-theoretic diagrammatic treatment was applied by Beli-

aev to the zero-temperature homogeneous dilute Bose gas, showing how to go beyond

Bogoliubov™s approximation in a systematic expansion in the diluteness parameter

√

n0 a3 ; and also showing how repeated scattering leads to a renormalization of the

interaction between the bosons [58, 59]. This renormalization in Beliaev™s treatment

was a cumbersome issue, where diagrams expressed in terms of the propagator for

the non-interacting particles are intermixed with diagrams where the propagator con-

tains the interaction potential. Beliaev™s diagrammatic scheme was extended to ¬nite

temperatures by Popov and Faddeev [60], and was subsequently employed to extend

the Bogoliubov theory to ¬nite temperatures by incorporating terms containing the

excited-state operators to lowest order in the interaction potential [61, 62].

A surge of interest in the dilute Bose gas due to the experimental creation of

gaseous Bose“Einstein condensates occurred in the mid-1990s [63]. The atomic con-

densates in the experiments are con¬ned in external potentials, which poses new

theoretical challenges; especially, the Beliaev expansion in the diluteness parameter

√

n0 a3 is questionable when the density is inhomogeneous. Experiments on trapped

17 In this section we essentially follow reference [54].

352 10. E¬ective action

Bose gases employ Feshbach resonances to probe the regime of large scattering length,

and hence large values of the diluteness parameter. It is therefore of importance to

understand the low-density approximations to the exact equations of motion and the

corrections thereto. In the following, we shall employ the two-particle irreducible ef-

fective action approach, and show that it provides an e¬cient systematic scheme for

dealing with both homogeneous Bose gases and trapped Bose gases. We demonstrate

how the e¬ective action formalism can be used to derive the equations of motion for

the dilute Bose gas, and more importantly, that the loop expansion can be used to

determine the limits of validity of approximations to the exact equations of motion

in the trapped case.

10.6.2 E¬ective action formalism for bosons

A system of spinless non-relativistic bosons is according to Eq. (3.68) and Eq. (10.37)

described by the action

S[ψ, ψ † ] drdt ψ † (r, t) [i‚t ’ h(r) + μ] ψ(r, t)

=

1

drdr dt ψ † (r, t)ψ † (r , t) U (r ’ r ) ψ(r , t)ψ(r, t) , (10.176)

’

2

where ψ is the scalar ¬eld describing the bosons, and μ the chemical potential. The

one-particle Hamiltonian, h = p2 /2m + V (r), consists of the kinetic term and an

external potential, and U (r) is the potential describing the interaction between the

bosons. As usual we introduce a matrix notation whereby the ¬eld and its complex

conjugate are combined into a two-component ¬eld φ = (ψ, ψ † ) = (φ1 , φ2 ).

The correlation functions of the bose ¬eld are obtained from the generating func-

tional

i

Z[·, K] = Dφ exp iS[φ] + i· † φ + φ† Kφ (10.177)

2

by di¬erentiating with respect to the source · † = (·, · — ) = (·1 , ·2 ). Here ·(r, t)

denotes a complex scalar ¬eld, not a Grassmannn variable, as we are considering

bosons. A two-particle source term, K, has been added to the action in the generating

functional in order to obtain equations involving the two-point Green™s function in a

two-particle irreducible fashion as discussed in Section 10.5.1.

The generator of the connected Green™s functions is

W [·, K] = ’i ln Z[·, K], (10.178)

and the derivative

δW ¯

= φi (r, t) (10.179)

δ·i (r, t)

gives the average ¬eld, φ, with respect to the action S[φ] + · † φ + φ† Kφ/2,

¯

i

¦(r, t)

Dφ φ(r, t) exp iS[φ] + i· † φ + φ† Kφ

¯

φ(r, t) = = = φ(r, t) .

¦— (r, t) 2

(10.180)

10.6. E¬ective action approach to Bose gases 353

The average ¬eld ¦ is seen to specify the condensate density and is referred to as the

condensate wave function.18

The derivative of W with respect to the two-particle source is (recall Figure 10.7)

δW 1¯ i

¯

= φi (r, t) φj (r , t ) + Gij (r, t, r , t ) , (10.181)

δKij (r, t; r , t ) 2 2

where G is the full connected two-point matrix Green™s function describing the bosons

not in the condensate

δ2W

’

Gij (r, t, r , t ) =

δ·i (r, t) δ·j (r , t )

δψ(r, t)δψ † (r , t ) δψ(r, t)δψ(r , t )

’i

= , (10.182)

δψ † (r, t)δψ † (r , t ) δψ † (r, t)δψ(r , t )

where δψ(r, t) is the deviation of the ¬eld from its mean value, δψ = ψ ’ ¦. Likewise,

¯

we shall write φ = φ + δφ for the two-component ¬eld. We recall that in the path

integral representation, averages over ¬elds, such as in Eq. (10.182), are automatically

time ordered.

We then introduce the e¬ective action for the bosons, “, the generator of the two-

particle irreducible vertex functions, through the Legendre transform of the generator

of connected Green™s functions, W ,

¯ 1¯ ¯ i

“[φ, G] = W [·, K] ’ · † φ ’ φ† K φ ’ TrGK.

¯ (10.183)

2 2

The e¬ective action satis¬es according to section 10.5.1 the equations

δ“ δ“ i

¯

¯ = ’· ’ K φ , = ’ K. (10.184)

δG 2

δφ

In a physical state where the external sources vanish, · = 0 = K, the variations of

¯

the e¬ective action with respect to the ¬eld averages φ and G vanish, yielding the

equations of motion

δ“

¯=0 (10.185)

δφ

and

δ“

= 0. (10.186)

δG

18 Indeed, as pointed out by Penrose and Onsager, Bose“Einstein condensation is associated

with o¬-diagonal long-range order in the two-point correlation function limr’∞ ψ† (r) ψ(0) =

ψ† (r) ψ(0) = 0 [64]. For a conventional description of bosons in terms of ¬eld operators we refer

to reference [15]. We note that, in the presented e¬ective-action approach, the inherent additional

necessary considerations associated with the macroscopic occupation of the ground state in the

conventional description is conveniently absent.

354 10. E¬ective action

According to Section 10.5.1, the e¬ective action can be written in the form

i i i

“[φ, G] = S[φ] + Tr ln G0 G’1 + Tr(G’1 ’ Σ(1) )G ’ Tr1 + “2 [φ, G] , (10.187)

¯ ¯ ¯

0

2 2 2

where G0 is the non-interacting matrix Green™s function,

i‚t ’ h + μ 0

G’1 (r, t, r , t ) = ’ δ(r ’ r )δ(t ’ t ) (10.188)

’i‚t ’ h + μ

0 0

and the matrix

δ2S

+ G’1 (r, t, r , t )

(r, t, r , t ) = ’ †

(1)

Σ (10.189)

0

δφ (r, t)δφ(r , t ) ¯

φ=φ

will turn out to be the self-energy to one-loop order (see Eq. (10.204)). Using the

action describing the bosons, Eq. (10.176), we obtain for the components

(1) (1)

Σij (r, t, r , t ) = δ(t ’ t ) Σij (r, r ) (10.190)

where

δ(r ’ r) dr U (r ’ r )|¦(r , t)|2 + U (r ’ r )¦— (r , t)¦(r, t)

(1)

Σ11 (r, r ) =

(10.191)

and

(1)

Σ12 (r, r ) = U (r ’ r )¦(r, t)¦(r , t) (10.192)