consider brie¬‚y equilibrium theory, in particular the general property characterizing

equilibrium.15

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

variables. We elaborate on this point in Chapter 9.

15 In Chapter 9 we proceed the other way around, and the reader inclined to take diagrammatics

as a starting point of a physical theory can thus start from there.

70 3. Quantum dynamics and Green™s functions

3.4 Equilibrium Green™s functions

In this section we shall consider a system in thermal equilibrium. In that case the

state of the system is speci¬ed by the Boltzmann statistical operator, Eq. (2.87),

characterized by its macroscopic parameter, the temperature T .

In thermal equilibrium, the correlation functions of a system are subdued to a

boundary condition in imaginary time as speci¬ed by the ¬‚uctuation“dissipation

theorem.16 In the canonical ensemble, for example the relation

† †

ψH (x, t) ψH (x , t ) = ψH (x , t ) ψH (x, t + iβ) (3.100)

is valid, where β = /kT , as a consequence of the cyclic invariance of the trace as

the bracket denotes the average

e’H/kT

. . . ≡ Tr ... . (3.101)

Tr(e’H/kT )

The relationship in Eq. (3.100) can, for example, be stated in terms of the Green™s

functions as

G< (x, t + iβ, x , t ) = ± G> (x, t, x , t ) (3.102)

valid for arbitrary interactions among the particles in the system.

Exercise 3.15. Show that in the grand canonical ensemble, for example the following

relation, is valid

† †

βμ

ψH (x, t ’ iβ) ψH (x , t ) = e ψH (x , t ) ψH (x, t) (3.103)

in which case the average is

e’(H’μN )/kT

. . . ≡ Tr ... (3.104)

Tr(e’(H’μN )/kT

and the Hamiltonian and the total number operator commute if the chemical poten-

tial is nonzero (recall Eq. (2.36)). Stated in terms of Green™s functions we have

βμ

G< (x, t, x , t ) = ± e G> (x, t ’ iβ, x , t ) , (3.105)

where in the grand canonical ensemble for example

†

“ i ψH (x , t ) ψH (x, t)

G< (x, t, x , t ) =

“i †

Tr(e’(H’μN )/kT ψH (x , t ) ψH (x, t)).

=

Tr(e’(H’μN )/kT )

(3.106)

16 Additional discussion of the ¬‚uctuation“dissipation theorem and its importance in linear re-

sponse theory are continued in Chapter 6. That the operators in Eq. (3.100) are the ¬eld operators

is immaterial; the relationship is valid for arbitrary operators.

3.4. Equilibrium Green™s functions 71

The importance of the canonical ensembles should be stressed for the validity of

these ¬‚uctuation“dissipation relations or so-called Kubo“Martin“Schwinger bound-

ary conditions. They state that the Green™s functions are anti-periodic or periodic

in imaginary time depending on the particles being fermions or bosons, the interval

of periodicity being set by the inverse temperature. This is the crucial observation

for the Euclidean or imaginary-time formulation of quantum statistical mechanics,

as further discussed in Section 5.7.

In thermal equilibrium, correlation functions only depend on the di¬erence be-

tween the times, t ’ t , i.e. they are invariant with respect to displacements in time.

If in addition the equilibrium state is translationally invariant, then all Green™s func-

tions are speci¬ed according to

∞

dp dE i (p·(x’x )’E(t’t ))

G(x, t, x , t ) = e G(p, E) (3.107)

(2π )3 2π

’∞

or equivalently

G(p, E, p , E ) = 2π(2π )3 δ(p ’ p ) δ(E ’ E ) G(p, E) . (3.108)

For example,

∞

[ap (t) , a† (t )]± .

i

GK (p, E) = ’i d(t ’ t ) e (t’t )E

(3.109)

p

’∞

The relationship in Eq. (3.105) then takes the form of the detailed balancing

condition E ’μ

G< (p, E) = ± e’ k T G> (p, E) . (3.110)

Exercise 3.16. Show that, for free bosons or fermions speci¬ed by the Hamiltonian

in Eq. (2.21), we have

p2

’ = ’2πi δ(E ’

GR (p, E) GA (p, E) p) , = . (3.111)

p

0 0

2m

Exercise 3.17. Show that, for free longitudinal phonons,, speci¬ed by the Hamilto-

nian in Eq. (1.123), we have

D0 (k, ω) ’ D0 (k, ω) = ’2πi ωk sign(ω) δ(ω 2 ’ ωk ) θ(ωD ’ |ω|)

R A 2 2

(3.112)

where the sign-function, sign(x) = θ(x) ’ θ(’x) = x/|x|, is plus or minus one,

depending on the sign of the argument.

72 3. Quantum dynamics and Green™s functions

Instead of de¬ning the unitary transformation to the Heisenberg picture according

to Eq. (3.26), we can let it be governed by the grand canonical Hamiltonian, H ’ μN ,

and we have, according to Eq. (2.36),17

ψμ (x, t) = e’

i i i

μN t μN t μt

ψH (x, t) e =e ψH (x, t) . (3.113)

De¬ning the grand canonical Green™s functions in terms of these ¬elds, we observe

that they are related to those de¬ned according to Eq. (3.106), or those in the canon-

ical ensemble in the thermodynamic limit, according to

i

μ(t’t )

Gμ (x, t, x , t ) = G(x, t, x , t ) e . (3.114)

Since average densities and currents are expressed in terms of the equal time Green™s

function, formulas have the same appearance in both ensembles.

For the Fourier transformed Green™s function with respect to time, the transition

to the grand canonical ensemble thus corresponds to the substitution E ’ E + μ, as

energies will appear measured from the chemical potential. The detailed balancing

condition, Eq. (3.110), can therefore, for a translationally invariant state, equivalently

be stated in the form

G< (E, p) = ± e’E/kT G> (E, p) (3.115)

and we have dropped the chemical potential index as these are the Green™s functions

we shall use in the following. The absence of the chemical potential in the exponential

shows that the relationships are speci¬ed in the grand canonical ensemble, where

energies are measured relative to the chemical potential (upper and lower signs refer

as usual to bosons and fermions, respectively).

In thermal equilibrium, the kinetic Green™s function and the retarded and ad-

vanced Green™s functions, or rather the spectral weight function, are thus related for

the case of fermions according to

E

GK (E, p) = (GR (E, p) ’ GA (E, p)) tanh . (3.116)

2kT

In thermal equilibrium, all Green™s functions can thus be speci¬ed once a single of

them is known, say the retarded Green™s functions, and the quantum statistics of the

particles is then re¬‚ected in relations governed by the ¬‚uctuation“dissipation type

relationship such as in Eq. (3.116). Occasionally we keep Boltzmann™s constant, k,

explicitly, the non-essential converter between energy and temperature scales.

Exercise 3.18. Show that, for bosons in equilibrium at temperature T , the ¬‚uctuation“

dissipation theorem reads

E

GK (E, p) = (GR (E, p) ’ GA (E, p)) coth . (3.117)

2kT

17 The number operator is assumed to commute with the Hamiltonian. If the number operator

does not commute with the Hamiltonian, such as for phonons, the description is of course in the

grand canonical ensemble and the chemical potential vanishes.

3.4. Equilibrium Green™s functions 73

Exercise 3.19. Show that

i G> (E, p) = (1 ± f“ (E)) A(E, p) (3.118)

and

± i G< (E, p) = f“ (E) A(E, p) , (3.119)

where the functions

1

f“ (E) = (3.120)

eE/kT “ 1

denote either the Bose“Einstein distribution or Fermi“Dirac distribution, for bosons

and fermions respectively.

Exercise 3.20. Show that the average energy in the thermal equilibrium state for

the case of two-body interaction between fermions (recall Exercise 3.7 on page 60),

for example, can be expressed as

1 dap (t)

a† (t) ap (t) + i a† (t)

H = p p p

2 dt

p

t=0

1 d

’i G< (p, t)

= i + p

2 dt

p

t=0

∞

1 dE

’i G< (p, E)

= (E + p)

2 2π

p ’∞

∞

1 dE

= (E + p) A(p, E) f (E) , (3.121)

2 2π

p ’∞

where f is the Fermi function, and thereby for the energy density

∞

H 1 dp dE

= (E + p) A(p, E) f (E) . (3.122)

(2π )3

V 2 2π

’∞

For a system in thermal equilibrium, the correlation function

† †

ψμ (x, t) ψμ (x , t ) = Tr(e(©’(H’μN ))/kT ψμ (x, t) ψμ (x , t )) (3.123)

can be spectrally decomposed by inserting a complete set of energy states in the

multi-particle space

(H ’ μN )|En , N = (En ’ μN )|En , N (3.124)

74 3. Quantum dynamics and Green™s functions

giving

†

e(©’En +μN )/kT ei(t’t )(En ’Em +μ) —

ψμ (x, t) ψμ (x , t ) =

N,n,m

N, En |ψ(x, t = 0))|Em , N + 1 —

N + 1, Em |ψ † (x , t = 0)|En , N . (3.125)

From this expression we observe that the G-greater Green™s function G> (x, t, x , t )

considered as a function of imaginary times is an analytic function in the region,

’1/kT < m(t ’ t ) < 0, if the exponential exp{’En (1/kT + i(t ’ t ))} dominates

the convergence of the sum.

Exercise 3.21. Show similarly that G< (x, t, x , t ) is an analytic function in the

region of imaginary times, 0 < m(t ’ t ) < 1/kT .

Assuming a translational invariant system and using Eq. (2.16) we have

†

e(©’En +μN )/kT ei(t’t )(En ’Em +μ)

ψμ (x, t) ψμ (x , t ) =

N,n,m

— e’i(x’x )pn m N, En |ψ(0, 0)|Em , N + 1

— N + 1, Em |ψ † (0, 0)|En , N , (3.126)

where pnm = Pn ’ Pm is the di¬erence between the total momentum eigenvalues

for the two states in question. For the Fourier transform we then have

ap (t) a† (t ) = (2π)3 e(©’En +μN )/kT ei(t’t )(En ’Em +μ)

p

N,n,m

— δ(p ’ pmn )| N, En |ψ(0, 0))|Em , N + 1 |2 .

(3.127)

Noting the analyticity in the upper-half ω-plane of the following function, we have

for real values of ω

∞

1

dt θ(t) eiωt = , (3.128)

ω + iδ

’∞

where δ = 0+ , or equivalently

∞

dω e’iωt

θ(t) = . (3.129)

’2πi ω + iδ

’∞

3.4. Equilibrium Green™s functions 75

Therefore for the retarded and advanced Green™s functions we have the spectral

representations

e’(En ’μN )/kT —

GR(A) (p, E) = (2π)3 e©/kT

Nn ,n,m

| Nn , En |ψ(0, 0))|Em , Nm |2

δ(p ’ pmn ) “ (n ” m) , (3.130)

+

E + Emn (’) iδ

where Emn = Em ’En +μ(Nm ’Nn ), and we recall that Nm = Nn ±1. The retarded

(advanced) Green™s function is thus analytic in the upper (lower) half-plane for the

energy variable E. The simple poles for the retarded (advanced) Green™s function

are thus spread densely just below (above) the real axis, and in the thermodynamic

limit this spectrum of simple poles coalesces into a continuum creating a branch cut

for the functions along the real axis.

In equilibrium all propagators are thus speci¬ed in terms of a single Green™s

function, say the retarded or equivalently the spectral function, as by analyticity the

retarded and advanced Green™s functions satisfy the causality or Kramers“Kronig

relations for their real and imaginary parts, or compactly

∞

dE GR (E , p) ’ GA (E , p)

R(A)

G (E, p) =

’2πi E ’ E (’) i0

+

’∞

∞

dE A(E , p)

= . (3.131)

2π E ’ E (’) i0

+

’∞

The spectral weight function, has according to Eq. (3.130), the spectral decom-

position

e(©’(En ’μNn ))/kT

’(2π)4

A(p, E) =

Nn ,n,m

— δ(p ’ pmn ) δ(E ’ Emn )| Nn , En |ψ(0, 0))|Em , Nm |2

“ (n ” m)) (3.132)

or equivalently

Emn

e(©’(En ’μNn ))/kT 1 “ e’

(2π)4

A(p, E) = kT

Nn ,n,m

— δ(p ’ pmn ) δ(E ’ Emn )| Nn , En |ψ(0, 0))|Em , Nm |2 , (3.133)

where the upper and lower sign is for bosons and fermions, respectively.

76 3. Quantum dynamics and Green™s functions

The analytic properties of the retarded (or advanced) Green™s function determines

the analytic properties of all the other introduced Green™s functions, and are further

studied in Section 5.6.

The three Green™s functions, GR,A,K , thus carry di¬erent information about the

many-body system: GR,A the spectral properties and GK in addition the quantum

statistics of the concerned particles. In Chapter 5 we will construct the diagrammatic

perturbation theory that, even for non-equilibrium states, keeps these important

features explicit.

Exercise 3.22. Show that for large energy variable, E ’ ∞, the retarded and

advanced Green™s functions always have the asymptotic behavior

1

GR(A) (E, p) . (3.134)

E

In the absence of interactions, i.e. for free bosons or fermions speci¬ed by the

Hamiltonian in Eq. (2.21), one readily obtains for the spectral weight function,

Eq. (3.85),

p2

2π δ(E ’ ξp ) , ’μ= ’ μ,

A0 (p, E) = ξp = (3.135)

p

2m

and according to the ¬‚uctuation“dissipation theorem, all one-particle Green™s func-

tion are then immediately obtained.

In the presence of interactions, the delta-spike in the spectral weight function

will be broadened and a tail appears, however, subject to the general sum-rule of

Eq. (3.89) which for the equilibrium state reads

∞

dE

A(E, p) = 1 . (3.136)

2π

’∞