Kadanoff-Baym equations and non-Markovian Boltzmann equation in generalized T-matrix approximation

9 Pages·0.2 MB·English

by  D. Semkat

Kadanoff–Baym equations and non-Markovian Boltzmann equation in generalized
T–matrix approximation
D. Semkat, D. Kremp, and M. Bonitz
Universit¨at Rostock, Fachbereich Physik, Universit¨atsplatz 3, D–18051 Rostock
(February 1, 2008)
1
0
0
Arecentlydevelopedmethod[1,2]forincorporatinginitialbinarycorrelationsintotheKadanoff–
2
Baymequations(KBE)isusedtoderiveageneralizedT–matrixapproximationfortheself-energies.

n ItisshownthattheT–matrixobtainsadditionalcontributionsarisingfrominitialcorrelations. Using
a these results and taking the time-diagonal limit of the KBE, a generalized quantum kinetic equa-
J
tion in binary collision approximation is derived. This equation is a far-reaching generalization of

5 Boltzmann–typekineticequations: itselfconsistently includesmemoryeffects(retardation,off-shell

T–matrices) as well as many-particle effects (damping, in-medium T–matrices) and spin-statistics
]
h effects (Pauli–blocking).
c
e
m
-
t
a
I. INTRODUCTION
t
s
.
t
a Nonequilibrium properties of many-particle systems have traditionally been described by kinetic equations of the
m Boltzmann type. Despite their fundamental character, these equations have well-known principal shortcomings, e.g.
(i) the short-time behavior (t < τ - the correlation time) cannot be described correctly, (ii) the kinetic or the
- cor
d quasiparticle energy is conserved instead of the total (sum of kinetic and potential) energy, (iii) no bound states
n are contained, and (iv) in the long-time limit, they yield the equilibrium distribution and thermodynamics of ideal
o
particles.
c
Animportantgeneralizationarethe well-knownKadanoff–Baymequationsderivedby KadanoffandBaym[3],and
[
Keldysh [4]. However, the original KBE contain no contribution from initial correlations. Therefore, the KBE are

2
unable to describe the initial stage of the evolution (t ≤t≤τ ) and the influence of initial correlations which can
0 cor
v
be important for ultrafast relaxation processes.
4
To include initial correlations into the KBE, various methods have been used, including analytical continuation of
8
3 the equilibrium KBE to real times [3,5–8] and perturbation theory with initial correlations [9,10,5]. A convincing
2 solution has been presented by Danielewicz [5], who developed a perturbation theory for a general initial state and
1 derived generalized KBE which take into account arbitrary initial correlations. Finally, a straightforward and very
9 intuitive method which does not make use of perturbation theory but uses the equations of motion for the Green’s
9
functions instead, has been developed in [1,2]. While perturbative approaches are restricted to situations where the
/
t couplingisweak,ourmethodisvalidforarbitrarycouplingstrength. Inparticular,itallowstoconsidersystemswith
a
strong coupling, such as Coulomb systems at low temperatures and/or high density (e.g. metals and dense plasmas)
m
and nuclear matter, and to include bound states. In Sec. II we briefly recall the main ideas of our method. After
-
d this, Sec. III is devoted to the application of our approach to the T–matrix approximation. In Sec. IV we derive a
n non-MarkovianBoltzmann equation in binary collision approximation.
o
c
:
v II. INITIAL CORRELATIONS IN THE KADANOFF–BAYM EQUATIONS
i
X
Starting point of our approach is the first equation of the Martin–Schwinger hierarchy [11],
r
a
(S −U )G =δ ±i~V G , (1)
ac ac cb ab ad,ce ce,bd
∂ ~2∇2
with S = i~ + a δ , (2)
ac ac
∂t 2m
(cid:18) a a (cid:19)
together with an initial condition for G ,
ce,bd
G | =G (t )G (t )±G (t )G (t )+C (t ). (3)
ce,bd tc=te=tb=td=t0 cb 0 ed 0 cd 0 eb 0 ce,bd 0
1
Summation/integration over repeated indices is implied. Here, C denotes initial binary correlations in the system,
and U is an external potential. The self-energy is defined by
δG
Σ G =±i~V G =±i~V G G ± cb . (4)
ac cb ad,ce ce,bd ad,ce cb ed
δU
(cid:26) de(cid:27)
Considering Eq. (4) in the limit t=t′ →t , we get explicitly
0
dt¯Σ (t ,t¯)G (t¯,t )=±i~V {G (t )G (t )±G (t )G (t )+C (t )}. (5)
ac 0 cb 0 ad,ce cb 0 ed 0 cd 0 eb 0 ce,bd 0
Z
Since the time integrationis performedalongthe Keldysh–Schwingercontour,only time-localcontributions ofΣ sur-
viveonthel.h.s. Thelasttermonther.h.sshowsthattheremustexist,inadditiontotheHartree–Fockcontributions
(first two terms), another time-localpart, which is related to initial correlations. That means, the self-energy has the
structure (Σˆ denotes the self-energy in the adjoint equation)
Σ =ΣHF +ΣC +ΣIN, (6)
ab ab ab ab
Σˆ =ΣHF +ΣC +ΣˆIN, (7)
ab ab ab ab
with the time-local terms (here, we give the time arguments explicitly)
ΣIN(t,t′)=ΣIN(t,t )δ(t −t′), (8)
ab ab 0 0
ΣˆIN(t,t′)=ΣˆIN(t ,t′)δ(t−t ). (9)
ab ab 0 0
The further steps aim at the determination of these initial correlation terms and are sketched here, for details, we
refer to Refs. [1,2]. Inserting (4) into (1), one obtains a Dyson–Schwinger equation for t,t′ >t ,
0
(S −U −Σ )G =δ , (10)
ac ac ac cb ab
whichcanbe castintotheformG−1G =δ . Functionaldifferentiationofthis equationwithrespecttotheexternal
ac cb ab
potential U yields a Bethe–Salpeter equation for δG/δU. Performing the same steps for the adjoint equation to (1)
as well, a solution for δG/δU, which incorporates initial binary correlations,is obtained,
δ ΣC +ΣIN +ΣˆIN
δGab ef ef ef
=G G +G G ±G G C G G , (11)
δU ad cb ae h δU i fb ae cf ef,gh gb hd
dc dc
where C has the time structure
C (t t ,t t )=C (t )δ(t −t )δ(t −t )δ(t −t )δ(t −t ). (12)
ab,cd a b c d ab,cd 0 a 0 b 0 c 0 d 0
III. GENERALIZED T–MATRIX APPROXIMATION
In the previous section we have obtained a formal decoupling of the Martin–Schwinger hierarchy by introduction
of the self-energy. Furthermore, our approach shows, that initial correlations can, in principle, be straightforwardly
included into this quantity. The next step on the way to a quantum kinetic equation is to choose a suitable approx-
imation for the self-energy. Among the standard schemes are the random phase approximation (RPA), describing
dynamicalscreening,andthe T–matrix(orbinarycollision)approximation. ThedeterminationofΣ inthese schemes
without inclusion of initial correlations is well-known. For example, the T–matrix approximation leads to a non-
Markovian Boltzmann equation. In Ref. [12], this equation has been derived within the density operator technique.
The nonequilibrium Green’s functions approach, however, opens the possibility to derive two-time quantum kinetic
equations with their well-known advantages (e.g. they fully include the kinetic and spectral one-particle properties).
One-time equations are obtained by taking the time-diagonal limit of the two-time equations in a much simpler way
than within the density operator technique.
In the following, we will use the nonequilibrium Green’s functions theory to derive a generalization of the usual
T–matrix approximation,which includes initial binary correlations.
According to Eqs. (4,11), the self-energy is determined by the functional equations [13]
2
δ Σ +ΣˆIN
Σ =±i~V δ G ±δ G +G G C G ±G fb fb , (13)
ab ad,ce cb ed eb cd cf eg fg,bh hd cf h δU i
de
 
Σˆ =±i~ δ G ±δ G +G C G G ± δ Σˆaf +ΣIaNf G V , (14)
ab  ae cd ac de dg ag,fh fc he h δU i fc ce,bd
ed
 
δΣIN
ΣIN =±i~V G G C G ±G fb ,  (15)
ab ad,ce cf eg fg,bh hd cf δU
( de )
δΣˆIN
ΣˆIN =±i~ G C G G ± af G V . (16)
ab dg ag,fh fc he δU fc ce,bd
( ed )
Notice especially that, due to the structure of the self-energy, the arguments of the functional derivative in the
equations for Σ and Σˆ are the same in both cases,
Σ+ΣˆIN =Σˆ +ΣIN =ΣC +ΣIN +ΣˆIN =Σ˜. (17)
We now introduce an effective two-particle potential Ξ by
δΣ˜ δΣ˜ δG δG
ab = ab ef =±i~Ξ ef, (18)
af,be
δU δG δU δU
cd ef cd cd
and define a generalized T–matrix [16],
T =Ξ ±i~Ξ G G T ±Ξ G G C . (19)
ab,cd ab,cd ae,cf fg he gb,hd ae,cf fg he gb,hd
In terms of Feynman diagrams, Eq. (19) reads (the shaded block denotes the initial correlationC)
Comparing Eq. (19) with the solution (11) for δG/δU, one obtains the relation
δΣ˜
ab =±i~G T G . (20)
de ae,bf fc
δU
cd
So we could identify T with the correlated part of the two-particle function without the bare initial correlation C.
The equation for the self-energy now takes the form
Σ =±i~V {δ G ±δ G +G G C G +i~G G T G }. (21)
ab ad,ce cb ed eb cd cf eg fg,bh hd cf eg fg,bh hd
FunctionaldifferentiationofthisequationyieldsarelationforΞ,whichdependsonT andonthequantityδΣˆIN/δG≡
±i~Φ. Inserting this relation into Eq. (19), and evaluating the functional derivative δΣˆIN/δG, one arrives at two
coupled equations for T and Φ, where self-energiesand Ξ have been eliminated. Keeping only the ladder-type terms,
these equations can be written as
T =V +Φ +V G G C +i~V G G T , (22)
ab,cd ab,cd ab,cd ab,ef eg fh gh,cd ab,ef eg fh gh,cd
Φ =C G G V +i~Φ G G V . (23)
ab,cd ab,ef eg fh gh,cd ab,ef eg fh gh,cd
Eqs. (22,23) can be solved easily (see Appendix A), yielding an explicit expression for T,
T =T +i~T G G C G G T
ab,cd ab,cd ab,ef eg fh gh,ij ik jl kl,cd
+T G G C +C G G T , (24)
ab,ef eg fh gh,cd ab,ef eg fh gh,cd
3
or, in terms of Feynman diagrams,
T T T
T T
.
Here, T denotes the well-known “ladder T–matrix” which obeys
T =V +i~V G G T , (25)
ab,cd ab,cd ab,ef eg fh gh,cd
T V V T
.
The system (22,23) can be regarded as a generalization of the usual T–matrix equation (25), where Eq. (24) shows
explicitly the corrections which are due to initial correlations.
If we now insert Eq. (24) into the equation for the self-energy (21), we obtain Σ in T–matrix (binary collision)
approximation,
Σ =±i~T G ±i~T G G C G
ac ab,cd db ab,ef eg fh gh,cd db
±(i~)2T G G C G G T G , (26)
ab,ef eg fh gh,ij ik jl kl,cd db
T T T T
,
and analogously Σˆ,
Σˆ =±i~T G ±i~C G G T G
ac ab,cd db ab,ef eg fh gh,cd db
±(i~)2T G G C G G T G , (27)
ab,ef eg fh gh,ij ik jl kl,cd db
T T T T
.
Comparing these results with the predicted structure of the self-energies, Eqs. (6,8), the time-local contributions
are identified as
ΣIN =±i~T G G C G , (28)
ac ab,ef eg fh gh,cd db
ΣˆIN =±i~C G G T G , (29)
ac ab,ef eg fh gh,cd db
or, diagrammatically,
IN T
,
IN T
.
4
Interestingly, the correlationpart ΣC of the self-energy contains an initial correlation contribution, too,
ΣC =±i~T G ±(i~)2T G G C G G T G , (30)
ac ab,cd db ab,ef eg fh gh,ij ik jl kl,cd db
and, again in diagrams,
C T T T
.
With Eqs. (26–30) we have found a generalization of the T–matrix approximation. In addition to the usual ladder
term, the self-energies contain explicitly contributions of initial correlations.
All relations derived so far are valid on the Keldysh–Schwinger contour. In order to obtain the Kadanoff–Baym
equations and kinetic equations for the Wigner function, it is now necessary to specify the position of the time
arguments of Green’s functions on the contour. Then we obtain from the Dyson equation (10) the well-known
Kadanoff–Baym equations for the correlation functions g≷ (in the following, small letters denote quantities on the
physical time axis, and the time arguments will be shown explicitly),
t
dt¯ s (t,t¯)−σHF(t,t¯) g≷(t¯,t′)= dt¯ σ>(t,t¯)−σ<(t,t¯) g≷(t¯,t′)
ac ac cb ac ac cb
Z Z
(cid:8) (cid:9) t0 (cid:8) (cid:9)
t′
+ σ≷(t,t¯) g<(t¯,t′)−g>(t¯,t′) , (31)
ac cb cb
Z
t0 (cid:8) (cid:9)
t
dt¯g≷(t,t¯) s† (t¯,t′)−σHF(t¯,t′) = dt¯ g>(t,t¯)−g<(t,t¯) σˆ≷(t¯,t′)
ac cb cb ac ac cb
Z n o tZ0 (cid:8) (cid:9)
t′
+ g≷(t,t¯) σˆ<(t¯,t′)−σˆ>(t¯,t′) . (32)
ac cb cb
Z
t0 (cid:8) (cid:9)
The self-energies read in T–matrix approximation
σ≷(t,t′)=±i~t≷ (t,t′)g≶(t′,t)±i~tC;IN(t,t′)g≶(t′,t)±i~tIN (t,t′)gA(t ,t), (33)
ac ab,cd db ab,cd db ab,cd db 0
σˆ≷(t,t′)=±i~t≷ (t,t′)g≶(t′,t)±i~tC;IN(t,t′)g≶(t′,t)±i~ˆtIN (t,t′)gR(t′,t ), (34)
ac ab,cd db ab,cd db ab,cd db 0
σHF(t,t′)=±i~ (v ±v )g≷(t,t′)δ(t−t′), (35)
ac ab,cd ab,dc db
where the initial correlationcontributions are given by
tIN (t,t′)= dt¯tR (t,t¯)GR (t¯,t )c (t )δ(t −t′), (36)
ab,cd ab,ef ef,gh 0 gh,cd 0 0
Z
tˆIN (t,t′)= dt¯c (t )GA (t ,t¯)tA (t¯,t′)δ(t −t), (37)
ab,cd ab,ef 0 ef,gh 0 gh,cd 0
Z
tC;IN(t,t′)=i~ dt¯dttR (t,t¯)GR (t¯,t )c (t )GA (t ,t)tA (t,t′), (38)
ab,cd ab,ef ef,gh 0 gh,ij 0 ij,kl 0 kl,cd
Z
while the greater/less and the retarded/advancedT–Matrices obey the equations
t≷ (t,t′)=i~ dt¯v G˜R (t,t¯)t≷ (t¯,t′)+i~ dt¯v G≷ (t,t¯)tA (t¯,t′), (39)
ab,cd ab,ef ef,gh gh,cd ab,ef ef,gh gh,cd
Z Z
tR/A(t,t′)=v δ(t−t′)+i~ dt¯v G˜R/A (t,t¯)tR/A (t¯,t′), (40)
ab,cd ab,cd ab,ef ef,gh gh,cd
Z
where we introduced the abbreviations
5
GR/A (t,t′)=gR/A(t,t′)gR/A(t,t′), G≷ (t,t′)=g≷(t,t′)g≷ (t,t′), (41)
ef,gh eg fh ef,gh eg fh
G˜R/A (t,t′)=±Θ[±(t−t′)] G> (t,t′)−G< (t,t′) . (42)
ef,gh ef,gh ef,gh
n o
A further important relation is the optical theorem, which follows from Eqs. (39) and (40),
t≷ (t,t′)=i~ dt¯dttR (t,t¯)G≷ (t¯,t)tA (t,t′). (43)
ab,cd ab,ef ef,gh gh,cd
Z
Equations(31–43)representtheKadanoff–Baymequationsinthegeneralizedbinarycollisionapproximation. Here,
the T–matrix contains contributions which are due to initial binary correlations. These additional terms can be
separated from the “usual” T–matrix, and, in particular, do not influence the structure of the Lippmann–Schwinger
equation (40).
IV. NON-MARKOVIAN BOLTZMANN EQUATION
In the previous section, we presented a far-reaching generalization of the usual T–matrix approximation by incor-
poratinginitialcorrelations. Thisway,theKadanoff–Baymequationshavebecomesufficientlygeneraltodescribethe
evolution of a many-particle system on arbitrary time scales, in particular on ultra-short times after an excitation.
Theirsolutions,thetwo-timecorrelationfunctions,containatremendousamountofinformationonthestatisticaland
dynamicalproperties ofa stronglycorrelatedmany-particlesystem, fully including damping (lifetime) of the one and
two-particle states [14]. However, in many cases the information contained in the Wigner distribution is sufficient.
Therefore, in the following, we will derive an equation for this function, i.e., a kinetic equation in a narrow sense.
For this purpose, we consider the Kadanoff–Baymequations (31,32) in the limit of equal times t=t′ and subtract
themfromeachother. Theresultisanequationforthedistributionfunctionwhichreads,inmomentumrepresentation
(we consider a homogeneous system without external forces),
t
∂
f(p,t)=± dt¯ σ>(p,t,t¯)g<(p,t¯,t)−σ<(p,t,t¯)g>(p,t¯,t)
∂t
Z
t0 (cid:8)
+g<(p,t,t¯)σˆ>(p,t¯,t)−g>(p,t,t¯)σˆ<(p,t¯,t)
=I(p,t)+IIC(p,t). (cid:9) (44)
This so-called time-diagonal equation is a very general representation of a kinetic equation. The r.h.s. describes the
influence of collisions as well as initial correlationson the Wigner distribution and is, in principle, determined by the
exact self-energy and the two-time correlationfunctions.
In order to obtain a closed kinetic equation, two major tasks remain: (i) an approximation for the self-energies
has to be chosen, and (ii) the reconstruction problem, i.e., the determination of g≷ as a functional of the Wigner
distribution,hastobesolved. Thefirsttaskhasalreadybeendealtwithinthe previoussection,withtheresultbeing
the generalizedT–matrix approximation, given by Eqs. (33–43). Let us now consider the reconstruction problem. In
order to obtain the functional relation g≷ =g≷[f], we use the generalized Kadanoff–Baymansatz (GKBA) proposed
by Lipavsky´ et al. [15],
g≷(p,t,t′)=± gR(p,t,t′)f≷(p,t′)−f≷(p,t)gA(p,t,t′) , (45)
n o
with f< =f and f> =1±f. For the products G≷ then follows
G≷(t,t′)=GR(t,t′)F≷(t′)+F≷(t)GA(t,t′), (46)
12 12 12 12 12
where we used the abbreviations F≷ = f≷(p )f≷(p ) and G = G(p ,p ). From Eq. (46) follow relations between
12 1 2 12 1 2
the functions GR/A and G˜R/A which were defined in Eqs. (41,42),
G˜R(t,t′)=GR(t,t′)N (t′), (47)
12 12 12
G˜A(t,t′)=−N (t)GA(t,t′), (48)
12 12 12
where we introduced the Pauli blocking factor N =1±f(p )±f(p ).
12 1 2
6
Now we insert the self-energies in T–matrix approximation, Eqs. (33–38), into the time diagonal equation (44),
replacingt≷ with the help ofthe opticaltheorem(43)andG≷ by means ofthe reconstructionansatz (46). The result
is the collision integral I,
dp dp¯ dp¯
I(p ,t)=(i~)2 2 1 2 dt¯dtdt
1 (2π~)3(2π~)3(2π~)3
Z Z
× tR(p p t,p¯ p¯ t¯)G¯R(t¯,t)tA(p¯ p¯ t,p p t)GA(t,t) F¯>(t)F<(t)−F¯<(t)F>(t)
1 2 1 2 12 1 2 1 2 12 12 12 12 12
n h i
+tR(p p t,p¯ p¯ t¯)G¯A(t¯,t)tA(p¯ p¯ t,p p t)GA(t,t) F¯>(t¯)F<(t)−F¯<(t¯)F>(t)
1 2 1 2 12 1 2 1 2 12 12 12 12 12
h i
−GR(t,t¯)tR(p p t¯,p¯ p¯ t)G¯A(t,t)tA(p¯ p¯ t,p p t) F>(t¯)F¯<(t)−F<(t¯)F¯>(t)
12 1 2 1 2 12 1 2 1 2 12 12 12 12
h i
−GR(t,t¯)tR(p p t¯,p¯ p¯ t)G¯R(t,t)tA(p¯ p¯ t,p p t) F>(t¯)F¯<(t)−F<(t¯)F¯>(t) , (49)
12 1 2 1 2 12 1 2 1 2 12 12 12 12
h io
with G¯R/A =GR/A(p¯ ,p¯ ) and tR/A(p p t,p¯ p¯ t¯)= p p tR/A(t,t¯) p¯ p¯ , and the collision integral arising from
12 1 2 1 2 1 2 1 2 1 2
initial correlations IIC,
(cid:10) (cid:12) (cid:12) (cid:11)
(cid:12) (cid:12)
dp dp¯ dp¯
IIC(p ,t)=i~ 2 1 2 dt¯
1 (2π~)3(2π~)3(2π~)3
Z Z
× tR(p p t,p¯ p¯ t¯)K(p¯ p¯ t¯,p p t)−K(p p t,p¯ p¯ t¯)tA(p¯ p¯ t¯,p p t)
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
dp dp¯ dp¯ dp dp
−(i~(cid:8))2 2 1 2 1 2 dt¯dtdt (cid:9)
(2π~)3(2π~)3(2π~)3(2π~)3(2π~)3
Z Z
× tR(p p t,p¯ p¯ t¯)K(p¯ p¯ t¯,p p t)tA(p p t,p p t)N (t)GA(t,t)
1 2 1 2 1 2 1 2 1 2 1 2 12 12
n
+GR(t,t¯)N (t¯)tR(p p t¯,p¯ p¯ t)K(p¯ p¯ t,p p t)tA(p p t,p p t) , (50)
12 12 1 2 1 2 1 2 1 2 1 2 1 2
with K(p p t,p¯ p¯ t¯)=GR(t,t )c(p p ,p¯ p¯ ;t )G¯A(t ,t¯). o (51)
1 2 1 2 12 0 1 2 1 2 0 12 0
WithEqs.(44,49,50)wehaveobtainedaverygeneralquantumkineticequation. Thecharacterofitsapproximations
goes far beyond that of the usual Boltzmann equation. The collision integral I(p ,t) was derived without any
1
approximation with respect to the times and thus fully includes retardation and memory effects which is usually
referred to as non-Markovianbehavior. Many-particle effects, as for instance self-energy and damping [14], and spin
statistics effects (Pauliblocking)areincluded. So far, norestrictionhasbeen introducedwith respectto the retarded
and advanced propagators GR/A. In principle, they are to be determined self-consistently from their KBE which
followeasilyfromEq.(10). However,toavoidthisessentialcomplication,inmostcasesapproximationsareused. For
example, in the quasiparticle approximation, the propagators are given explicitly by
GR/A(t,t′)= 1 Θ[±(t−t′)]e~i[E12+iΓ12](t−t′) (52)
12 (i~)2
with E = p21 + p22 +ReσR+ReσR and Γ =ImσR+ImσR.
12 2m 2m 1 2 12 1 2
Furthermore, the retarded and advanced T–Matrices are many-particle generalizations of the familiar T–Matrices
of quantum scattering theory. They have to be determined fromthe Lippmann–Schwinger equation(40) which reads
in momentum representation
tR/A(p p t,p′p′ t′)=v(p −p′)(2π~)3δ(p +p −p′ −p′)δ(t−t′)
1 2 1 2 1 1 1 2 1 2
dp¯ dp¯
+i~ 1 2 dt¯v(p −p¯ )(2π~)3δ(p +p −p¯ −p¯ )
(2π~)3(2π~)3 1 1 1 2 1 2
Z Z
×G˜R/A(p¯ ,p¯ ;t,t¯)tR/A(p¯ p¯ t¯,p′p′ t′). (53)
1 2 1 2 1 2
The collisionintegralIIC(p ,t)containsthe termsarisingfrombinarycorrelations,existingin the systeminitially.
1
Itshouldbe stressedexplicitly thatthe structureofthesecontributionsiscompletely generalanddoes notdependon
parameters characterizing the system, such as coupling strength or degree of degeneracy. Furthermore, the inclusion
of initial correlations does not depend on their actual form, i.e. the form of the function c. The damping of the
two-particle propagators leads to a decay of this collision term, i.e. the initial correlations die out on a time scale
which is determined by the one-particle damping rates [14].
Finally, we want to remark here that our result for the non-Markovian Boltzmann equation is in agreement with
the result derived within the framework of the density operator technique, see [12].
7
ACKNOWLEDGMENTS
The authors acknowledge discussions with Th. Bornath (Rostock). This work is supported by the Deutsche
Forschungsgemeinschaft(SFB 198 and Schwerpunkt “Quantenkoh¨arenz in Halbleitern”).
APPENDIX A: SOLUTION OF THE GENERALIZED T–MATRIX EQUATIONS
We rewrite Eqs. (22,23), which contain the ladder-type terms of the generalized T–matrix (19),
T =V +Φ +V G G C +i~V G G T , (A1)
ab,cd ab,cd ab,cd ab,ef eg fh gh,cd ab,ef eg fh gh,cd
Φ =C G G V +i~Φ G G V . (A2)
ab,cd ab,ef eg fh gh,cd ab,ef eg fh gh,cd
Due to the structure of Eq. (A1), T can be split into three parts,
T =T(A) +T(B) +T(C) , (A3)
ab,cd ab,cd ab,cd ab,cd
T(A) =V +i~V G G T(A) , (A4)
ab,cd ab,cd ab,ef eg fh gh,cd
T(B) =V G G C +i~V G G T(B) , (A5)
ab,cd ab,ef eg fh gh,cd ab,ef eg fh gh,cd
T(C) =Φ +i~V G G T(C) . (A6)
ab,cd ab,cd ab,ef eg fh gh,cd
Obviously, Eq. (A4) coincides with the well-known ladder equation of the T–matrix approximation. Thus, we can
identify T(A) with the usual T–matrix T. The ladder equation (A4) now serves as a basis for the solution of (A5)
and (A6). If one assumes for T(B) the form
T(B) =T G G C , (A7)
ab,cd ab,ef eg fh gh,cd
Eq. (A5) is valid if (A4) holds. In order to determine T(C), Eq. (A2) has to be considered. This equation is fulfilled
if Φ is of the structure
Φ =C G G T , (A8)
ab,cd ab,ef eg fh gh,cd
ifthe adjointequationto (A4)is valid. Due to the symmetry propertiesofT,Eq.(A4)andits adjointareequivalent.
Inserting (A8) into Eq. (A6) and assuming T(C) to be of the structure
T(C) =C G G T +i~T G G C G G T , (A9)
ab,cd ab,ef eg fh gh,cd ab,ef eg fh gh,ij ik jl kl,cd
Eq. (A6) is fulfilled, again under the assumption (A4). Collecting all parts together, T can be represented as
T =T +i~T G G C G G T
ab,cd ab,cd ab,ef eg fh gh,ij ik jl kl,cd
+T G G C +C G G T , (A10)
ab,ef eg fh gh,cd ab,ef eg fh gh,cd
together with the equation for the well-known “ladder T–matrix”,
T =V +i~V G G T . (A11)
ab,cd ab,cd ab,ef eg fh gh,cd
[1] D.Semkat, D.Kremp, and M. Bonitz, Phys. Rev.E 59, 1557 (1999)
[2] D.Kremp,D.Semkat,andM. Bonitz, Kadanoff–Baym Equations with Initial Correlations, in Progress in Nonequilibrium
Green’s Functions, M. Bonitz (Ed.), World Scientific, Singapore 2000, p. 34 (archive: cond-mat/9911429)
[3] L. P.Kadanoff and G. Baym, Quantum Statistical Mechanics (W. A. Benjamin, NewYork 1962)
[4] L. V.Keldysh,ZhETF 47, 1515 (1964) [Sov. Phys.-JETP 20, 235 (1965)]
[5] P.Danielewicz, Ann.Phys. (N.Y.) 152, 239 (1984)
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[6] M. Wagner, Phys. Rev.B 44, 6104 (1991)
[7] V.G. Morozov and G. R¨opke,Ann.Phys.(N.Y.) 278, 127 (1999)
[8] V.G.MorozovandG.R¨opke,inProgressinNonequilibriumGreen’sFunctions,M.Bonitz(Ed.),WorldScientific,Singapore
2000, p. 56
[9] S.Fujita, J. Math. Phys. 6, 1877 (1965)
[10] A.G. Hall, J. Phys. 8, 214 (1975)
[11] In order to present the equations in a compact way, we will use a representationless index notation with summation
convention [16]. The indices include thetime arguments, too, unless they are written explicitly.
[12] D.Kremp, M. Bonitz, W. D.Kraeft, and M. Schlanges, Ann. Phys.(N.Y.) 258, 320 (1997)
[13] In [1], ΣˆIN in the argument of the functional derivative on the r.h.s. of (13) is not written explicitly. However, it should
bementioned here, that thereis a coupling between the equation for Σ and its adjoint.
[14] We should mention here that, in the frame of the T–matrix approximation with bare interaction, the damping of the
two-particle states is included on the one-particle level. A real two-particle damping would require to include into the
T–matrix equation three-particle interactions, or thedynamical screening, respectively.
[15] P.Lipavsky´, V.Sˇpiˇcka,and B. Velicky´,Phys. Rev.B 34, 6933 (1986)
[16] W. Brenig and H. Wagner, Z. Phys. 173, 484 (1963)
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