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from the canonical formalism as presented here and in the ¬rst two chapters, we
consider brie¬‚y equilibrium theory, in particular the general property characterizing
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

. . . ≡ 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
† †
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
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 )
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 )]± .
GK (p, E) = ’i d(t ’ t ) e (t’t )E

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

’ = ’2πi δ(E ’
GR (p, E) GA (p, E) p) , = . (3.111)
0 0

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

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
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
GK (E, p) = (GR (E, p) ’ GA (E, p)) tanh . (3.116)
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
GK (E, p) = (GR (E, p) ’ GA (E, p)) coth . (3.117)
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)

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

1 d
’i G< (p, t)
= i + p
2 dt

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


e(©’En +μN )/kT ei(t’t )(En ’Em +μ) —
ψμ (x, t) ψμ (x , t ) =
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 ) =

— 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)

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

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

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

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)
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-

e(©’(En ’μNn ))/kT
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
e(©’(En ’μNn ))/kT 1 “ e’
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
GR(A) (E, p) . (3.134)

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),

2π δ(E ’ ξp ) , ’μ= ’ μ,
A0 (p, E) = ξp = (3.135)
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

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


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