<< . .

. 21
( : 78)

. . >>

then the sum of all the skeleton self-energy diagrams, i.e. the diagrams which can not
136 5. Real-time formalism

be cut in two be cutting only two full propagator lines. It is the latter representation
that is useful in the Dyson equation.
Equivalently, by iterating from the left gives the matrix Dyson equation

G = G0 + G — Σ — G0 . (5.69)

For an equilibrium state the two equations are redundant, since time convolutions
by Fourier transformation become simple products for which the order of factors is
irrelevant. However, in a non-equilibrium state, the two matrix equations contain
di¬erent information and subtracting them is a useful way of expressing the non-
equilibrium dynamics, and we shall exploit this in Chapters 7 and 8.
Since the transformation of the real-time matrix self-energy is identical to the one
for the matrix Green™s function we get, analogously to the equations from Eq. (5.36)
to Eq. (5.39), and therefore for the components of the self-energy matrix,
ˆ ˆ ˆ ˆ
ΣR = Σ11 ’ Σ12 = Σ21 ’ Σ22 (5.70)
ˆ ˆ ˆ ˆ
ΣA = Σ11 ’ Σ21 = Σ12 ’ Σ22 (5.71)
ˆ ˆ ˆ ˆ
ΣK = Σ11 + Σ22 = Σ12 + Σ21 (5.72)
ˆ ˆ ˆ ˆ
0 = Σ11 ’ Σ12 + ’ Σ21 + Σ22 . (5.73)
By construction
Σ12 (x1 , t1 , x1 , t1 ) t1 > t1
Σ11 (x1 , t1 , x1 , t1 ) = (5.74)
Σ21 (x1 , t1 , x1 , t1 ) t1 > t1

Σ21 (x1 , t1 , x1 , t1 ) t1 > t1
Σ22 (x1 , t1 , x1 , t1 ) = (5.75)
Σ12 (x1 , t1 , x1 , t1 ) t1 > t1
and the matrix self-energy has in the triagonal representation the same triagonal
form as the matrix Green™s function
Σ= . (5.76)

Exercise 5.6. Introducing
Σ< (x1 , t1 , x1 , t1 ) = Σ12 (x1 , t1 , x1 , t1 ) (5.77)

Σ> (x1 , t1 , x1 , t1 ) = Σ21 (x1 , t1 , x1 , t1 ) (5.78)
show that we have, identically to the relationships for the Green™s functions, the
relation for the retarded self-energy

ΣR (x, t, x , t ) = θ(t ’ t ) Σ> (x, t, x , t ) ’ Σ< (x, t, x , t ) (5.79)

and advanced self-energy

ΣA (x, t, x , t ) = ’θ(t ’ t) Σ> (x, t, x , t ) ’ Σ< (x, t, x , t ) (5.80)
5.5. Non-equilibrium Dyson equations 137

and for the kinetic component

ΣK (x, t, x , t ) = Σ> (x, t, x , t ) + Σ< (x, t, x , t ) . (5.81)

Show that the components of the self-energy matrix satis¬es

ΣA (x, t, x , t ) = ΣR (x , t , x, t) (5.82)


ΣK (x, t, x , t ) = ΣK (x , t , x, t) . (5.83)

Exercise 5.7. Show that in the case where the matrix Green™s function is represented
in symmetric form

G= (5.84)

the matrix self-energy has the form

Σ= . (5.85)
ΣA 0

We shall not at present take the diagrammatics beyond the self-energy to higher-
order vertices, since in the following chapters only the Dyson equation is needed. In
Chapter 9 we shall study diagrammatics in their full glory.
From the equation of motion for the free Green™s function (or ¬elds) we then get,
for the matrix Green™s function, the equations of motion

(i‚t1 ’ h(1))G(1, 1 ) = δ(1 ’ 1 ) + (Σ — G)(1, 1 ) (5.86)

(i‚t1 ’ h— (1 ))G(1, 1 ) = δ(1 ’ 1 ) + (G — Σ)(1, 1 ) (5.87)
or introducing the inverse free Green™s function

G’1 (1, 1 ) = (i‚t1 ’ h(1)) δ(1 ’ 1 ) (5.88)

the two equations can be expressed through operating with the inverse free matrix
Green™s function from the left

(G’1 ’ Σ) — G = δ(1 ’ 1 ) (5.89)

and from the right
G — (G’1 ’ Σ) = δ(1 ’ 1 ) . (5.90)

These two non-equilibrium Dyson equations will prove useful in Chapter 7 where
quantum kinetic equations are considered.
The matrix equation, Eq. (5.89), comprises the three coupled equations for GR,A,K

(G’1 ’ ΣR(A) ) — GR(A) = δ(1 ’ 1 ) (5.91)
138 5. Real-time formalism

G’1 — GK = ΣR — GK + ΣK — GA . (5.92)

Analogously, from Eq. (5.90), we obtain

GR(A) — (G’1 ’ ΣR(A) ) = δ(1 ’ 1 ) (5.93)

GK — G’1 = GR — ΣK + GK — ΣA . (5.94)

Exercise 5.8. Show that, subtracting the left and right Dyson equations for GK ,
the resulting equation can be written in the form
1R 1
[G’1 — GK ]’ [Σ + ΣA — GK ]’ ’ [ΣK — GR + GA ]’

, , ,
2 2

1 1
’ [ΣK — (GR ’ GA )]+ + [(ΣR ’ ΣA ) — GK ]+ . (5.95)
= , ,
2 2

If at the end of the day, one makes the lowest-order approximation for the self-
energy (as often done!), introducing the Green™s function formalism and diagrammat-
ics is of course ridiculous as ¬nal results follow from Fermi™s Golden Rule.9 A virtue
of the real-time formalism and its associated Feynman diagrams is that nontrivial
approximations can be established using the diagrammatic estimation technique, and
higher-order correlations studied systematically, as we shall consider in the following
chapters, not least in chapter 10.10
Before studying applications of the real-time technique we shall make obsolete
one version of the imaginary-time formalism, viz. the too pervasive Matsubara tech-
nique. The general imaginary-time formalism has virtues for special Euclidean ¬eld
theory purposes as well as for expedient proofs establishing conserving approxima-
tions. After the discussion of the equilibrium Dyson equation in the next section,
we demonstrate the equivalence of the imaginary-time formalism to the closed time
path formulation and the real-time technique introduced in this chapter.

5.6 Equilibrium Dyson equation
In equilibrium all quantities depend only on time di¬erences, and for translational
invariant situations also only on spatial di¬erences, and convolutions are by Fourier
transformation turned into products. In terms of the self-energy we therefore have
for the retarded Green™s function the equilibrium Dyson equation11

GR (p, E) = GR (p, E) + GR (p, E) ΣR (E, p) GR (p, E) (5.96)
0 0
9 In the same vein, if one employs a mean-¬eld approximation, introducing the formalism of
quantum ¬eld theory seems excessive. This point of view was taken in references [1] and [13].
10 For a discussion of the diagrammatic estimation technique see chapter 3 of reference [1].
11 We recall the result of Section 3.4, that in thermal equilibrium all the various Green™s functions

can be expressed in terms of, for example, the (imaginary part of the) retarded Green™s function.
5.6. Equilibrium Dyson equation 139

which we immediately solve to get
1 1
GR (p, E) = = . (5.97)
G’1 (p, E) ’ ΣR (E, p) E’ p ’ Σ (E, p)

The retarded self-energy determines the analytic structure of the retarded Green™s
function, i.e. the location of the poles of the analytically continued retarded Green™s
function onto the second Riemann sheet through the branch cut along the real axis
(recall Section 3.4), the generic situation being that of a simple pole. For given
momentum value the simple pole is located at E = E1 + iE2 , determined by E1 =
p + e Σ(E, p ) and E2 = m Σ(E, p ), and as

dE e’iEt GR (p, E)
G (p, t) = (5.98)

the imaginary part of the self-energy thereby determines the temporal exponential
decay of the Green™s function, i.e. the lifetime of (in the present case) momentum
states. The e¬ect of interactions are clearly to give momentum states a ¬nite lifetime.
For the Fourier transform of Eq. (5.97) we get (in three spatial dimensions for
the prefactor to be correct)

|x’x | 2m(E’ΣR (E, pE p))
’m e ˆ
GE (x ’ x ) =
|x ’ x |
2π 2

where pE is the solution of the equation pE = 2m(E ’ ΣR (E, pE p)). Interactions
will thus provide a ¬nite spatial and temporal range of the Green™s function.
For the case of electrons, say in a metal, the advanced Green™s function likewise
describes the attenuation of the holes.
Exercise 5.9. Show that the spectral function in equilibrium is given by (using now
the grand canonical ensemble)

“(E, p)
A(E, p) = (5.100)
2 2
E ’ ξp ’ eΣR (E, p) + 2

eΣ(E, p) ≡ Σ (E, p) + ΣA (E, p) (5.101)
“(E, p) ≡ i ΣR (E, p) ’ ΣA (E, p) . (5.102)

We note that the sum-rule satis¬ed by the spectral weight function, Eq. (3.89), sets
limitation on the dependence of the self-energy on the energy variable. The general
features of interaction is to broaden the peak in the spectral weight function and to
shift, renormalize, energies.
140 5. Real-time formalism

Exercise 5.10. Show that for bosons in equilibrium at temperature T , their self-
energy components satisfy the ¬‚uctuation“dissipation relations
ΣR (E, p) ’ ΣA (E, p)
ΣK (E, p) = coth (5.103)
and for fermions
ΣR (E, p) ’ ΣA (E, p)
ΣK (E, p) = tanh . (5.104)

5.7 Real-time versus imaginary-time formalism
Although we shall mainly use the real-time technique presented in this chapter
throughout, it is useful to be familiar with the equivalent imaginary-time formalism
in view of the vast amount of literature where this method has been employed. Or
more importantly to realize the link between the imaginary-time formalism and the
Martin“Schwinger“Abrikosov“Gorkov“Dzyaloshinski“Eliashberg“Kadano¬ “Baym“
Langreth analytical continuation procedure. In the classic textbooks of Kadano¬
and Baym [14] and Abrikosov, Gorkov and Dzyaloshinski [15] on non-equilibrium
statistical mechanics, the imaginary-time formalism introduced by Matsubara [16]
and Fradkin [17] and Martin and Schwinger [18] was used. Being then a Euclidean
¬eld theory it possesses nice convergence properties. However, it lacks appeal to

5.7.1 Imaginary-time formalism
The workings of the imaginary-time formalism are based on the mathematical formal
resemblance of the Boltzmann statistical weighting factor in the equilibrium statis-
tical operator ρ ∝ e’H/kT and the evolution operator U ∝ e’iHt/ for an isolated
system. The imaginary time Green™s function
H ’μ N
G(x, „ ; x , „ ) ≡ ’Tr e’ ˜
T„ (ψ(x, „ ) ψ(x , „ )) (5.105)

is de¬ned in terms of ¬eld operators depending on imaginary time according to (we
suppress all other degrees of freedom than space)

ψ(x) e’
1 1
„ (H’μN ) „ (H’μN )
ψ(x, „ ) = e (5.106)

ψ(x, „ ) = e „ (H’μN ) ψ † (x) e’ „ (H’μN ) ,
1 1
˜ (5.107)
where ψ(x) is the ¬eld operator in the Schr¨dinger picture, and T„ provides the
imaginary time ordering (with the usual minus sign involved for an odd number
of interchanges of fermi ¬elds). The „ s involved are real variables, the use of the
word imaginary refers to the transformation t ’ ’i„ in which case the time-ordered
real-time Green™s function, Eq. (4.10), transforms into the imaginary-time Green™s
function (more about this shortly). Note that ψ(x, „ ) and ψ(x, „ ) are not each others
5.7. Real-time versus imaginary-time formalism 141

adjoints. Knowledge of the imaginary-time Green™s function allows the calculation
of thermodynamic average values.
The imaginary-time single-particle Green™s function respects the Kubo“Martin“
Schwinger boundary conditions, for example

G(x, „ ; x , 0) = ± G(x, „ ; x , β) , (5.108)

owing to the cyclic invariance property of the trace (the notation β = /kT is used).
The periodic boundary condition is for bosons, and the anti-periodic boundary con-
dition is for fermions (the identical consideration in connection with the ¬‚uctuation“
dissipation theorem was discussed in Section 3.4, and is further discussed in Section
6.5). We note the crucial role of the (grand) canonical ensemble as elaborated in
Section 3.4.
In its simple equilibrium applications in statistical mechanics, thermodynamics, or
in linear response theory, the involved imaginary-time Green™s functions are expressed
in terms of a single so-called Matsubara frequency
e’iωn („ ’„ ) G(x, x ; ωn ) ,
G(x, „ ; x , „ ) = (5.109)
β ωn

where ωn = 2nπ/β for bosons and ωn = (2n + 1)π/β for fermions, respectively,
n = 0, ±1, ±2, . . . . Equilibrium or thermodynamic properties and linear transport
coe¬cients can therefore be expressed in terms of only one Matsubara frequency,
and the analytical continuation to obtain them from the imaginary-time Green™s
functions is trivial, say the retarded Green™s function is obtained by GR (x, x ; ω) =
G(x, x ; iωn ’ ω + i0+ ) as the two functions coincide according to GR (iωn ) = G(ωn )
for ωn > 0.
The imaginary-time Green™s functions can also be used to study non-equilibrium
states by letting the external potential depend on the imaginary time. The Matsub-
ara technique is then a bit cumbersome, but can be used to derive exact equations,
say, the Dyson equation for real-time Green™s functions. In fact this was the method
used originally to study non-equilibrium superconductivity in the quasi-classical ap-
proximation [19].12 However, for general non-equilibrium situations, the necessary
analytical continuation in arbitrarily many Matsubara frequencies becomes nontrivial
(and are usually left out of textbooks), and are more involved than using the real-
time technique. Furthermore, when approximations are made, the real-time results
obtained upon analytical continuation can be spurious. However, the main disad-
vantage of the imaginary-time formalism is that it lacks physical transparency. We
shall therefore not discuss it further in the way it is usually done in textbooks, but
use a contour formulation to show its equivalence to the real-time formalism.13
12 Amazingly, the non-equilibrium theory of superconductivity was originally obtained using the
Matsubara technique [19], as, I guess, the imaginary-time formalism was in rule at the Landau
Institute. A plethora of papers and textbooks have perpetrated the use of the imaginary-time
formalism. It is the contestant to be the most important frozen accident in the evolution of non-
equilibrium theory. Let™s iron out unfortunate ¬‚uctuations of the past! Its proliferation also testi¬es
to the fact that idiosyncratically written papers, such as the seminal paper of Schwinger [5], can be
a long time in germination.
13 The imaginary-time formalism can be useful for special purpose applications such as diagram-
142 5. Real-time formalism

5.7.2 Imaginary-time Green™s functions
The imaginary-time Green™s functions are pro¬tably interpreted as contour-ordered
Green™s function, viz. on an imaginary-time contour. First we note, that the times
entering the imaginary-time Green™s function can be interpreted as contour times.
Choosing the times in the time ordered Green™s function in Eq. (3.61), instead to lie
on the contour starting at, say, t0 and ending down in the lower complex time plane
at t0 ’ iβ, the appendix contour ca in Figure 4.4, turns the expression Eq. (3.61)
into the equation for the imaginary-time Green™s function, Eq. (5.105). This observa-
tion, by the way, gives the standard Feynman diagrammatics for the imaginary-time
Green™s function since Wick™s theorem involving the appendix contour is a trivial
corollary of the general Wick™s theorem of Section 4.3.3. We can thus, for example,
immediately write down the non-equilibrium Dyson equation for the imaginary-time
Green™s function, t1 and t1 lying on the appendix contour ca . Considering the case
where the non-equilibrium situation is the result of a time-dependent potential, V ,
the Dyson equation for the imaginary-time Green™s functions or appendix contour-
ordered Green™s function is

G(1, 1 ) = G0 (1, 1 ) + dx3 d„3 dx2 d„2 G0 (1, 3) Σ(3, 2) G(2, 1 )
ca ca
σ3 σ2

+ dx2 d„2 G0 (1, 2) V (2) G(2, 1 ) . (5.110)

The appendix contour interpretation of the imaginary-time Green™s function is in
certain situations more expedient than the real-time formulation when it for example
comes to diagrammatically proving exact relationships, since it has fewer diagrams
than the real-time approach if unfolding its matrix structure is needed. It should
thus be used in such situations, but then it is preferable not to use the Matsubara
frequency technique, but instead stick to the appendix contour formalism.
In practice we need to know how to analytically continue imaginary-time quanti-
ties to real-time functions, say for the imaginary-time Dyson equation, or for terms
appearing in perturbative expansions of imaginary-time quantities. Instead of turn-
ing to standard textbook imaginary-time formalism, the Matsubara technique, it is in
view of the above preferable to go to the appendix contour ordered Green™s functions,
and perform the analytical continuation from there. In fact, this analytical contin-
uation procedure becomes equivalent to the analytical continuation of the contour
ordered functions in the general contour formalism, for example the transition from
the contour-ordered Green™s function to real-time Green™s functions, which we now
turn to discuss. We illustrate this in the next section by performing the proceduce
for the Dyson equation.
The Boltzmann factor can guarantee analyticity of the Green™s function for times

<< . .

. 21
( : 78)

. . >>