Level shift operators for open quantum systems

0.28 MB·English

by  Marco Merkli

Level shift operators for open quantum systems
Marco Merkli ∗
Dept. of Mathematics and Statistics, McGill University
805 Sherbrooke W., Montreal
Canada, QC, H3A 2K6
6
and
0
0 Centre de Recherches Math´ematiques, Universit´e de Montr´eal
2 Succursale centre-ville, Montr´eal

n Canada, QC, H3C 3J7
a
J
February 7, 2008
7
Abstract

1
Levelshift operators describe thesecond order displacement of eigenvalues underper-
v
3 turbation. Theyplayacentralroleinresonancetheoryandergodictheoryofopenquantum
1 systems at positive temperatures.
0 Weexhibitintrinsic propertiesof levelshift operators, properties which stem from the
1 structure of open quantum systems at positive temperatures and which are common to
0 allsuchsystems. Theydeterminethegeometryofresonances bifurcatingfromeigenvalues
6 of positive temperature Hamiltonians and they relate the Gibbs state, the kernel of level
0
shift operators, and zero energy resonances.
/
h Weshowthatdegeneracyofenergylevelsofthesmallpartoftheopenquantumsystem
p causes theFermiGolden RuleCondition tobeviolated andweanalyzeergodic properties
- of such systems.
h
t
a
m 1 Introduction and main results
:
v
Level shift operators emerge naturally in the context of perturbation theory of (embedded)
i
X eigenvalues,wheretheygovernthe shifts of levels(resonances)atsecondorderinperturbation.
They play a central role in many recent works on ergodic properties of open quantum systems
r
a atpositivetemperature[20,21,9,14,26,17,18,19,23,28,2,1]. Thedynamicsofsuchsystems
isanautomorphismgroupofthealgebraofobservablesgeneratedbyanoperatorL =L +λI,
λ 0
where the selfadjoint L describes the free dynamics of two (or more) uncoupled subsystems,
0
λ R is a coupling constant, and I is an interaction operator.
∈
Ergodic properties are encoded in the spectrum of L . If the system has an equilibrium
λ
state Ω atpositivetemperature 1/β one showsthatL Ω =0 andthatifKerL =CΩ
β,λ λ β,λ λ β,λ
then any state initially close to equilibrium approaches the equilibrium state in the limit of
large times. (We do not address the question of mode or speed of the return to equilibrium in
this outline).
Itfollowsfromthealgebraicstructureofquantumsystemsatpositivetemperaturesthatthe
operatorL hasnecessarilyadegeneratekernelwhoseelementsareinone-to-onecorrespondence
0
withinvariantstatesoftheuncoupledsystem. Inordertoprovereturntoequilibriumoneneeds
toshowthatthedegeneracyoftheeigenvaluezero,whichisembeddedincontinuousspectrum,
is lifted under perturbation: dimKerL = 1 for λ = 0. This has been proven for several
λ
6
∗Supported by a CRM-ISM postdoctoral fellowship and by McGill University; [email protected];
http://www.math.mcgill.ca/∼merkli/
1
concretemodels[20,21,9,26,14,18,2]. The fateofembeddedeigenvaluesunderperturbation
canbe describedbyspectralresonancetheoryifthesystemhascertaindeformationanalyticity
properties, [20, 21, 9, 2], or, for less regular systems, by a Mourre theory [14] or by a positive
commutator theory, [26, 18].
A core strategycommonto these methods is to reduce the spectralanalysisof the operator
L around the origin to that of a reduced operator acting on a smaller Hilbert space (which
λ
is finite-dimensional in all works cited above). This procedure is sometimes called ‘integrating
out degrees of freedom’. For deformation analytic systems it can be implemented by applying
the so-called Feshbach map F [8] to a suitably deformed operator L (τ), where τ C is the
λ
∈
deformationparameter1 . TheFeshbachmaphasanisospectralityproperty,implyingthatthe
kernels of F(L (τ)) and of L are isomorphic. An expansion in the coupling constant λ gives
λ λ
F(L (τ))= λ2Λ +O(λ3), (1)
λ 0
−
where the operator Λ is independent of the deformation parameter τ. The property of return
0
to equilibrium follows if Λ has simple kernel because then (1) and the isospectrality of the
0
Feshbach map imply that for small λ = 0, dimKerL 1, and since L Ω = 0 one must
λ λ β,λ
have KerL = CΩ . The operator Λ6 is called the l≤evel shift operator (associated to the
λ β,λ 0
eigenvaluezeroofL ). Foramoredetaileddescriptionoftheemergenceoflevelshiftoperators
λ
in perturbation theory we refer to the works cited above, and also to [12].
Level shift operators are equally important in the study of systems far from equilibrium,
where the system does not have an equilibrium state, for instance when several thermal reser-
voirs at different temperatures are coupled, [23, 28, 1], or when the small system does not
admit an equilibrium state, [17, 19]. In what follows we discuss the former case. The role of
the equilibrium state is now playedby a reference state ψ , e.g. the product state of the small
λ
system and the reservoirs in equilibria at different temperatures. An interaction operator W
can be chosen such that the Heisenberg dynamics of the system is generated by the operator
K =L +λW,satisfying K ψ =0. Unlike I inthe situationofsystems closeto equilibrium,
λ 0 λ λ
theoperatorW cannotbechosentobenormal. Adetailedspectralanalysisofoperatorsofthis
type (called ‘C-Liouville operators’) is carried out in [28]. One obtains the level shift operator
Λ associatedtotheeigenvaluezeroofK byanexpansionofF(K (τ))inλ,asinthesituation
0 λ λ
above. In the context of systems far from equilibrium, a dynamical resonance theory shows
thatifΛ hassimplekernelthenthesystempossessesauniquetime-asymptoticlimitstate(for
0
small λ), which is a non-equilibrium stationary state.
We should also point out that level shift operators have a dynamical interpretation as
“Davies generators” of the reduced dynamics in the van Hove limit, [14, 13].
In the presentpaper we examine properties of levelshift operatorswhichdo not depend on
particularities of the system in question, but which originate from the structurecommon to all
openquantum systems atpositive temperatures. One of the main ingredientsdetermining this
structure is the Tomita–Takesakitheory of von Neumann algebras.
We describe the geometry of resonancesbifurcating from real eigenvalues of Liouville oper-
ators in Theorem 1.1. Part (c) of that theorem examines the role of the degeneracy of energy
levels of the small system. The interplay between the Gibbs state of the small system and the
kernel of the level shift operator is described in Theorem 1.2. Some of these intrinsic proper-
ties we exhibit here have been observed in the analysis of specific systems carried out in the
references given above.
Apart from an analysis of the structure of level shift operators we study in Section 3 the
dynamics ofsystems where a Bosonicheatreservoiris coupledto a smallsystemwhose Hamil-
tonianhasdegenerateeigenvalues. Forsuchsystemstheso-calledFermiGoldenRuleCondition
1The operator Lλ itself is typically not in the domain of the map F but Lλ(τ) is for τ 6∈ R. Arguments
similar to the ones we give here also work for systems which are not deformation analytic, but they need a
technicallymoreelaboratepresentation.
2
isviolatedduetothefactthattheinteraction“cannotcoupletheBohrfrequencyzero”(arising
from the degenerate levels) to the zero energy reservoir modes in an effective way. We prove
return to equilibrium for such systems by taking into account higher order corrections in the
perturbative spectral analysis of the operator L . A consequence of the degeneracy of energy
λ
levelsofthesmallsystemisthattheapproachtoequilibriumisstillexponentiallyfastbutwith
relaxation time of O(λ−4) as opposed to the shorter relaxation time O(λ−2) for systems with
non-degenerate spectrum.
The organizationof this paper is as follows. In Section 1.1 we summarize some facts about
open quantum systems. Our main results concerning the structure of level shift operators are
Theorems 1.1, 1.2 and 1.4, given in Section 1.2. In Section 2 we present examples of concrete
models to which our results apply. We analyze systems with Hamiltonians having degenerate
eigenvalues in Section 3. Section 4 contains proofs.
1.1 Open quantum systems
A detailed description of open quantum systems can be found in [10], in the above-mentioned
works, and also in [22, 27].
Consideraquantumsystempossessingfinitelymanydegreesoffreedom,likeasingleparticle
or a molecule, or a system with finitely many energy levels. We denote by the Hilbert space
K
of pure states of this “small system” and by H its Hamiltonian. We allow the case dim =
K ∞
but require that the Gibbs state Ψ exists, i.e., that e−βH is trace class on , for some inverse
β
K
temperature 0<β < . In the aboveexample the particle ormolecule must thus be confined,
∞
e.g. by a potential.
We view the Gibbs state Ψ as a vector in the Hilbert space = . The algebra of
β S
H K⊗K
all bounded operators, ( ), contains the observables of the small system. It is represented
B K
on as the von Neumann algebra M = ( ) 1l ( ), and the dynamics on M is
S S K S S
H B K ⊗ ⊂ B H
implemented by t eitLSAe−itLS, where
7→
L =H 1l 1l H (2)
S K K
⊗ − ⊗
is called the standard Liouville operator of the small system. The Gibbs vector Ψ is cyclic
β
andseparatingforM . Wedenotethemodularconjugationassociatedtothepair(M ,Ψ )by
S S β
J . The standard Liouville operator L satisfies the relations L Ψ =0 and J L J = L .
S S S β S S S S
−
In models of systems close to equilibrium, the small system is placed in a (single) envi-
ronment (reservoir) modeled by a “large” quantum system having infinitely many degrees of
freedom. Acommonexampleisa spatiallyinfinitely extendedidealquantumgas(ofBosonsor
Fermions). Weassumethatthereservoirhasanequilibriumstate(forsomeinversetemperature
0<β < ) which is represented by the vector Φ in the reservoir Hilbert space .
β R
∞ H
Observables of the reservoir are operators belonging to (or affiliated with) a von Neumann
algebra M ( ). Their dynamics is given by a group of automorphisms of M , t
R R R
⊂ B H 7→
eitLRAe−itLR, generated by a selfadjoint standard Liouville operator LR. Being a KMS vector
w.r.t. this dynamics, Φ is cyclic and separating for M . We denote the modular conjugation
β R
associated to (M ,Φ ) by J . The operator L has the properties J L J = L and
R β R R R R R R
−
L Φ =0.
R β
The von Neumann algebra M = M M , acting on the Hilbert space = ,
S R S R
⊗ H H ⊗H
contains the observables of the combined system. Elements in this algebra evolve according to
the group of automorphisms of M generated by the selfadjoint operator
L =L +L . (3)
0 S R
The vector
Ω =Ψ Φ (4)
β,0 β β
⊗
3
defines the equilibrium state w.r.t. this dynamics, at inverse temperature β.
To describe the interacting dynamics we introduce the map π which sends linear operators
on to linear operators on according to
R R
K⊗H K⊗K⊗H
π(A B)=A 1l B. (5)
K
⊗ ⊗ ⊗
Theinteractionbetweenthesmallsystemandthereservoirisspecifiedbyaselfadjointoperator
v on such that
R
K⊗H
V =π(v) (6)
is a selfadjoint operator affiliated with M. Let J =J J . We assume that
S R
⊗
B1 L +λV is essentially selfadjoint on (L ) (V) and L +λV λJVJ is essentially
0 0 0
D ∩D −
selfadjoint on (L ) (V) (JVJ), where λ is a real coupling constant.
0
D ∩D ∩D
If assumption B1 holds then the selfadjoint operator
L =L +λI, (7)
λ 0
where
I =V JVJ, (8)
−
generates a group of automorphisms αt =eitLλ e−itLλ of M (see e.g. [16], Theorem 3.5). We
λ ·
are interested in interactions for which the coupled dynamics αt admits an equilibrium state.
λ
It is known that the condition
B2 Ωβ,0 (e−β(L0+λV)/2)
∈D
implies that
e−β(L0+λV)/2Ωβ,0
Ω = (9)
β,λ e−β(L0+λV)/2Ωβ,0
k k
isa(β,αt)–KMSstate,andthatthefollowingpropertieshold: JL J = L ,L Ω =0,and
λ λ − λ λ β,λ
lim Ω Ω =0 (see e.g. [16], Theorem 5.5).
λ→0 β,λ β,0
k − k
Inmodelsforsystemsfarfromequilibriumthe smallsystemiscoupledtoseveralreservoirs.
TheHilbertspaceofthesmallsystemplusR 2reservoirsis = andthe
S R R
≥ H H ⊗H ⊗···⊗H
non-interactingdynamicsofthealgebraM=M M M isgeneratedbytheselfadjoint
S R R
⊗ ⊗···
standard free Liouville operator
R
L =L + L , (10)
0 S R,r
Xr=1
where L acts non-trivially only on the r-th reservoir space. The interaction is determined
R,r
by an operator
R
V = π(v ), (11)
r
Xr=1
where v is the selfadjoint operator on representing the coupling between the small
r R
K⊗H
system and reservoir r. We understand that π(v ) acts trivially on all reservoir Hilbert spaces
r
except the r-th one. As mentioned in the introduction, the role of the equilibrium state is now
played by a reference state ψ . We assume the following.
λ
∈H
C The interacting dynamics of M is generatedby the operatorK =L +λW, where W is
λ 0
a bounded, linear (generally non-symmetric) operator on , s.t. K ψ =0 for all λ in a
λ λ
H
neighbourhood of zero.
Remark. For systems with bosonic heat reservoirs the operator W is not bounded. Such
systems are considered in [28]. The development of a general theory for unbounded non-
symmetric W is a technically intricate affair, we restrict our attention in this note to systems
satisfying condition C (although we give a more general result in Theorem 4.2).
4
1.2 Main results
1.2.1 Systems close to equilibrium
WeassumethatthebasicassumptionsB1andB2aresatisfied. Thespectralprojectionontoan
eigenvalue e of L is denoted by χ and P = Φ Φ denotes the orthogonal projection
onto CΦ . Set PS = χ P aLnSd=eP = 1lR P|.βWihe βa|llow the case dimP = . Denote
e β e LS=e ⊗ R e −e e e ∞
by L the restriction of L to RanP , i.e., L := P L P ↾ . Consider the following
λ λ e λ e λ e RanPe
condition:
A1 P I and IP are bounded operators.
e e e
If condition A1 holds then we define the family of bounded operators
e
Λ (ǫ)=P IP (Le e iǫ)−1P IP , (12)
e e e 0− − e e
for real ǫ= 0. Note that for ǫ >0 we have ImΛ (ǫ) =ǫP IP [(Le e)2+ǫ2]−1P IP 0, so
6 e e e 0− e e ≥
the numericalrange,and hence the spectrumofΛ (ǫ), lie in the closedlowercomplex plane. If
e
(12) has a limit as ǫ 0 (in the weak sense on a dense domain) then we call this limit the level
↓
shift operator associated to the eigenvalue e, and write it as
Λ =P IP (Le e i0 )−1P IP . (13)
e e e 0− − + e e
The projection P has rank one so it is natural to identify RanP with Ranχ . In this
R e LS=e
sense we view the operators (12) and (13) as operators acting on Ranχ .
LS=e ⊂K⊗K
Theorem 1.1
(a) Let e be an eigenvalue of L . A1 holds if and only if A1 holds. If the assumptions
S e −e
A1 hold then J Λ (ǫ)J = Λ (ǫ).
±e S e S −e
−
(b) Assume A1 . Suppose the spectrum of the Hamiltonian H consists of simple eigenvalues
0
and denote the eigenvectors by ϕ . Then Λ (ǫ) = iΓ (ǫ), where Γ (ǫ) is a selfadjoint
i 0 0 0
positive definite operator on RanP which has real matrix elements in thebasis ϕ ϕ .
0 i i
{ ⊗ }
If Λ exists then the same statements are true for that operator.
0
(c) If H has degenerate eigenvalues then neither the real nor the imaginary part of Λ (ǫ)
0
vanish, in general. The matrix elements of Λ (ǫ) in the basis ϕ ϕ are not purely
0 i j
{ ⊗ }
real nor are they purely imaginary, in general. If Λ exists then the same statements are
0
true for that operator.
Part (a) of the theorem shows that σ(Λ (ǫ))= σ(Λ (ǫ)). In particular, the spectrum of
e −e
−
Λ (ǫ) is invariant under reflection at the imaginary axis. Part (b) shows that σ(Λ (ǫ)) lies on
0 0
the negative imaginary axis if H has simple spectrum. Part (c) says that if H has degenerate
spectrum then σ(Λ (ǫ)) can have nonzero real part. In the context of a translation-analytic
0
modelofanN-levelsystemcoupledtoabosonic(orfermionic)heatreservoir[20](seealso[28])
Theorem 1.1 shows that the set of all resonances is invariant under reflection at the imaginary
axis,andthatresonancesbifurcatingfromtheoriginstayontheimaginaryaxiswhilewandering
into the lower complex plane if H is non-degenerate (see also (1), (61) and (62)).
Wepresentaproofofassertions(a)and(b)ofTheorem1.1inSection4.1below. InSection
3 we give examples illustrating statement (c).
Our next result concernsthe interplay betweenthe KMS stateΩ , (9), andthe levelshift
β,λ
operator for e=0, Λ . We introduce the following assumptions.
0
A2 λ P IP (L iǫ)−1P is continuous at λ = 0 as a map from R to the bounded
0 0 λ 0
7→ −
0
operators on , for nonzero (small) ǫ. (We write here L instead of L .)
H λ λ
5
A3 λ Ω is differentiable at λ=0 as a map from R to .
β,λ
7→ H
Theorem 1.2 Assume that Conditions A1 , A2 and A3 hold. Then we have
0
limP IP (L iǫ)−1P IP Ω =P IP ∂ Ω . (14)
0 0 0 0 0 β,0 0 0 λ λ=0 β,λ
ǫ→0 − |
In particular, if Λ exists and if P IP =0 then Λ Ψ =0.
0 0 0 0 β
Even though we do not assume that Λ (ǫ) converges, equation (14) shows that Λ (ǫ)Ω
e e β,0
has a limit as ǫ 0, regardless of the sign of ǫ. By subtracting from (14) that same equation
→
with ǫ replaced by ǫ we get
−
P IP δ(L )P IP Ω =0, (15)
0 0 0 0 0 β,0
and by adding the two equations we obtain
P IP P.V.(L )−1 P IP ψ =P IP ∂ Ω , (16)
0 0 0 0 0 0 0 0 λ λ=0 β,λ
|
where δ(x) = lim 1 ǫ is the Dirac delta distribution and P.V.x−1 = lim x is the
ǫ↓0 πx2+ǫ2 ǫ↓0 x2+ǫ2
principal value distribution. (The limits are understood in the strong sense.)
Remarks. 1) If Λ exists then (15) shows that the Gibbs state Ψ of the small system
0 β
belongs to the kernel of ImΛ , c.f. (4). If P IP = 0, then (14) implies the second statement
0 0 0
of the theorem.
2) If dim < , P IP = 0, and if Λ exists, a characterization of Ker(ImΛ ) which
0 0 0 0
K ∞
implies that (ImΛ )Ψ =0 has recently been given in [13].
0 β
Theorem 1.3 Let V M. Then Conditions B1, B2, A1 and A2, A3 hold for all eigenvalues
e
∈
e of L . In fact, the maps λ P (L iǫ)−1P and λ Ω , appearing in Conditions A2
S 0 λ 0 β,λ
7→ − 7→
and A3, extend analytically (as ( )-valued and -valued maps) to a complex neighbourhood
B H H
of λ=0.
1.2.2 Systems far from equilibrium
WeassumethatconditionCissatisfied. ThefollowingresultisanalogoustotheoneofTheorem
1.2. Let P be the orthogonal projection onto the kernel of L , P = 1l P, and denote by L
0 0
−
the restriction of L to RanP.
0
Theorem 1.4 Suppose λ ψ is differentiable at λ = 0 as a map from R to and denote
λ
7→ H
its derivative at zero by ψ′. Then we have
0
limPWP(L iǫ)−1PWPψ =PWPψ′. (17)
ǫ→0 0− 0 0
Aspecialcase(whichisofinterestinconcreteapplications)isgivenbyareferencestateψ =ψ
λ 0
which does not depend on λ, or by interactions satisfying PWP =0. In either case (17)shows
that PΨ KerΛ .
0 0
∈
2 Examples
A detailed description of the theory of ideal quantum gases is given in [10, 25, 27, 22].
6
2.1 Reservoirs of thermal Fermions
The Hilbert space of states of an infinitely extended ideal Fermi gas which are normal w.r.t.
the equilibrium state at inverse temperature 0<β < is
∞
= (18)
R − −
H F ⊗F
where = n H is the antisymmetric Fock space over the one-particle Hilbert
F− n≥0⊗antisym
space H=L2(LR3,d3k) (momentum representation). The thermal creation operators (distribu-
tions) are given by
a∗β(k)= 1−µβa∗(k)⊗1lF− +(−1)N ⊗√µβa(k), k ∈R3, (19)
p
whith µ (k)=(1+eβω(k))−1 and where
β
k ω(k) 0 (20)
7→ ≥
is the dispersion relation of the fermions considered. A typical example are non-relativistic
fermions for which ω(k) = k 2. The a and a∗ on the r.h.s. of (19) are the ordinary fermionic
| |
Fockannihilationandcreationoperatorswhichsatisfythecanonicalanti-commutationrelations
a(k),a∗(l) =δ(k l). (21)
{ } −
The number operator N in (19) is given by N = a∗(k)a(k)d3k. Relations (18) and (19)
R3
constitute the so-calledAraki-Wyss representation oRf the canonicalanti-commutationrelations
[5]. Smeared-out creation and annihilation operators are defined by a∗(f) = f(k)a∗(k)d3k
R3
and a(f) = f(k)a(k)d3k, for f H, and where f stands for the complexRconjugate. One
R3 ∈
showsthatthRefermioniccreationandannihilationoperatorsarebounded,satisfying a#(f) =
k k
f H. The von Neumann algebra MR is generated by the thermal creation and annihilation
k k
operators a#(f) f H . The vector
{ β | ∈ }
Φ =Ω Ω , (22)
β R
⊗ ∈H
where Ω is the (Fock) vacuum vector in , represents the KMS state w.r.t. the dynamics
−
F
t eitLRa#(f)e−itLR =a#(eitωf), where
7→ β β
L =dΓ(ω) 1l 1l dΓ(ω), (23)
R ⊗ F− − F− ⊗
and dΓ(ω) = ω(k)a∗(k)a(k)d3k is the second quantization of multiplication by ω(k). The
R3
action of the mRodular conjugation JR on creation operators is
JRa∗β(f)JR = (−1)N ⊗a∗( 1−µβf)+a(√µβf)⊗1lF− (−1)N ⊗(−1)N. (24)
(cid:2) p (cid:3)
Now we describe the interaction with the small system. The latter is described at the
beginning of Section 1.1. For m,n 0, m+n 1, let h be maps from R3m R3n to
m,n
≥ ≥ ×
the bounded operators on . For simplicity of exposition, we assume that those maps are
continuous and have compaKct support. We define k(m) =(k ,...,k ) R3m and put
1 m
∈
a∗(k(m))=a∗(k ) a∗(k ), (25)
β β 1 ··· β m
and similarly for a (l(n)). Set
β
v = dk(m)dl(n) h (k(m),l(n)) a∗(k(m))a (l(n))+ adjoint. (26)
m,n ZR3m×R3n m,n ⊗ β β
It is well known that
V =π(v ) M, (27)
m,n m,n
∈
see (6). The following result follows from Theorem 1.3.
Theorem 2.1 Suppose V is a linear combination of terms (27). Then all conditions B1, B2,
A1 , A2, A3 hold. In particular, the conclusions of Theorems 1.1 and 1.2 are valid.
e
7
2.2 Reservoirs of thermal Bosons
The Hilbert space describing states of an infinitely extended ideal Bose gas which are normal
totheequilibriumstateatinversetemperature0<β < (withoutBose-Einsteincondensate)
∞
is
= (28)
R + +
H F ⊗F
where = n H is the symmetric Fock space over the one-particle Hilbert space
F+ n≥0⊗sym
H = L2(R3,d3Lk) (momentum representation). The thermal creation operators (distributions)
are given by
a∗β(k)= 1+ρβa∗(k)⊗1lF+ +1lF+ ⊗√ρβa(k), k ∈R3, (29)
p
where ρ (k)=(eβω(k) 1)−1 is the momentum density distribution, and
β
−
k ω(k) 0 (30)
7→ ≥
is the dispersionrelation. We shallconsiderforsimplicity ofthe expositionω’ssuchthatρ (k)
β
is locally integrable in R3. A typical example is ω(k)= k (massless relativistic Bosons). The
| |
a and a∗ on the r.h.s. of (29) are the ordinary Fock annihilation and creation operators which
satisfy the canonical commutation relations
[a(k),a∗(l)]=δ(k l). (31)
−
Therepresentation(28),(29)istheso-calledAraki-Woods representation ofthecanonicalcom-
mutationrelations[4]. Smeared-outcreationandannihilationoperatorsaredefinedbya∗(f)=
f(k)a∗(k)d3k and a(f)= f(k)a(k)d3k, for f H, where f stands for the complex con-
R3 R3 ∈
Rjugate. The von Neumann alRgebra MR is generated by the Weyl operators Wβ(f) = eiφβ(f),
f H, where
∈ 1
φ (f)= a∗(f)+a (f) (32)
β √2 β β
(cid:0) (cid:1)
istheselfadjointfieldoperator. Theunboundedoperatorsa∗(f),a (f)andφ (f)areaffiliated
β β β
with M . The vector
R
Φ =Ω Ω , (33)
β R
⊗ ∈H
where Ω is the (Fock) vacuum vector in , represents the KMS state w.r.t. the dynamics
+
F
t eitLRWβ(f)e−itLR =Wβ(eitωf), where
7→
L =dΓ(ω) 1l 1l dΓ(ω), (34)
R ⊗ F+ − F+ ⊗
and dΓ(ω) = ω(k)a∗(k)a(k)d3k is the second quantization of multiplication by ω(k). The
R3
modular conjuRgation JR acts on creation operators as
JRa∗β(f)JR =1lF+ ⊗a∗( 1+ρβf)+a(√ρβf)⊗1lF+. (35)
p
Wenowturntotheinteractionbetweenthis reservoirandthesmallsystem(whichisdescribed
at the beginning of Section 1.1). It is given in a similar way as for Fermions, but, for technical
reasons, it has to be restricted to at most quadratic expressions in the thermal creation and
annihilation operators (“two–body interactions”). Given h and h , continuous maps from R3
1 2
and from R3 R3 to the bounded operators on respectively, s.t. h (k,l)=h (l,k)∗, we set
2 2
× K
v = h (k) a∗(k)+h (k)∗ a (k) d3k+ h (k,l) a∗(k)a (l) d3kd3l. (36)
ZR3(cid:2) 1 ⊗ β 1 ⊗ β (cid:3) ZR3×R3 2 ⊗ β β
8
For the purpose of exposition we assume that h have compact support. We introduce
1,2
C(s) = e2sω(k) e−sHh (k)esH 2d3k
1
ZR3 k k
+ e−2s(ω(k)−ω(l)) e−sHh (k,l)esH 2 d3kd3l, (37)
2
ZR3×R3 k k
which measures the regularity of the integral kernels h relative to the Hamiltonian H.
1,2
Theorem 2.2 Suppose the functions h and h are such that
1 2
sup C(s)=C < . (38)
∞
−β/2≤s≤β/2
There is a constant λ (depending on β) s.t. if λ <λ then all the conditions B1, B2, A1–A3
0 0
| |
are met.
Remarks. 1) If dim < then (38) is satisfied.
K ∞
2) We canreplace the conditionof h havingcompactsupport by a suitable weakerdecay
1,2
condition. Moreover, assuming h to satisfy certain regularity properties at k,l = 0 we can
1,2
treat discontinuous h .
1,2
3) One may add to v, (36), integrals involving a (k)a (l), a (k)a∗(l), a∗(k)a∗(l).
β β β β β β
4) If h =0 then the theorem holds for all λ R (see the remark at the end of the proof of
2
∈
Theorem 2.2, Section 4.4).
2.3 Systems with several reservoirs
We consider R 2 fermionic reservoirs coupled through a small system. The Hilbert space
≥
is given by = , with an uncoupled dynamics generated by L =
H HS ⊗ HR1 ⊗ ···⊗HRR 0
L + L + + L . We may choose the reference state to be a product of equilibrium
S R1 ··· RR
states, ψ =Ψ Φ Φ . Denote by J and ∆ the modular conjugations and
0 βS⊗ β1⊗···⊗ βR Ri,S Ri,S
the modular operators associated to the pairs (M ,Φ ) and (M ,Ψ ), respectively. Take
Ri βi S βS
V ,...,V to be linear combinations of the form (27), as in Theorem 2.1, representing the
1 R
couplings to the reservoirs,and suppose that
(∆1/2 ∆1/2) V (∆−1/2 ∆−1/2) M M .
S ⊗ Ri i S ⊗ Ri ∈ S⊗ R
We set
W = V J ∆1/2 J ∆1/2 V J ∆1/2 J ∆1/2 . (39)
Xi n i−(cid:0) S S ⊗ Ri Ri (cid:1) i (cid:0) S S ⊗ Ri Ri (cid:1)o
It is easily verified that Wψ = 0 and thus K ψ = (L +λW)ψ = 0. Since the second
0 λ 0 0 0
part in the sum belongs to the commutant M′, K defines the same dynamics on M as does
λ
L +λ(V + +V ). Thus condition C is satisfied and Theorem 1.4 applies.
0 1 R
···
2.4 Klein–Gordon field for accelerated observer
Letx=(x0,x)beapointinMinkowskispace-timeR R3 (withmetricsignature(+, , , )).
× − − −
The field operator satisfying the Klein–Gordon equation
((cid:3)+m2)ϕ(x)=0, (40)
where (cid:3)=∂2 ∆, ∆ is the Laplacian, m 0, is given by
x0 − ≥
ϕ(x)= dk eikx−iω(k)x0a(k)+e−ikx+iω(k)x0a∗(k) . (41)
ZR3 2ω(k)(cid:2) (cid:3)
p
9
The a#(k) in (41) are the usual bosonic creation and annihilation operators satisfying (31),
and
ω(k)= k2+m2 1/2. (42)
Let (R4;R) and (R4;C) denote the real v(cid:0)alued and(cid:1) the complex valued Schwartz functions
on RS4, respectivelyS. For f (R4;C) we define the smeared field operators, acting on bosonic
∈S
Fock space (see after (28)), by
+
F
ϕ[f]= f(x)ϕ(x)d4x. (43)
ZR4
The adjoint of ϕ[f] is ϕ[f], where f stands for the complex conjugate of f. For f (R4,R)
∈S
we define the unitary Weyl operator W[f]=eiϕ[f].
We introduce the “right wedge” = x R4 x0 < x1 . Let M ( ) be the von
R R +
W { ∈ | | | } ⊂B F
Neumann algebra generated by
W[f] f (R4,R), supp(f) , (44)
R
{ | ∈S ⊂W }
wheresupp(f)denotesthesupportoff. ElementsofM areinterpretedtobethoseobservables
R
which can be measured in the space-time region .
R
W
Itiswellknown(Reeh–Schlieder)thatthevacuumvectorΩ iscyclicforthe algebra
R +
of all polynomials in ϕ[f], f (R4;C). We may thus define an∈aFntiunitary involutio∗n J on
R
∈ S
by
+
F J ϕ[f(x)]J =ϕ[f( x0, x1,x2,x3)], (45)
R R
− −
for all f (R4,C). Relativistic boosts in the x1-direction are given by
∈S
B :(x0,x1,x2,x3) (x0coshτ +x1sinhτ,x0sinhτ +x1coshτ,x2,x3),
τ
7→
for τ R. The action of B lifts to Fock space according to
τ
∈
ϕ[f B ]=eiτLRϕ[f]e−iτLR, (46)
−τ
◦
where L is the selfadjoint Liouville operator on given by
R +
F
L =dΓ [ ∆+m2]1/4x1[ ∆+m2]1/4 . (47)
R
− −
(cid:0) (cid:1)
The map B leaves the wedge invariant so
τ R
W
ατ(A)=eiτLRAe−iτLR (48)
R
defines a group of automorphisms of M .
R
∗
Theorem 2.3 ([7]) ThestateonM determinedbythevacuumvectorΩ isa(2π,ατ)-
R R ∈F+ R
KMS state. The modular operator associated to (MR,ΩR) is ∆R = e−2πLR and the modular
conjugation is J .
R
An observer accelerating in the x1-direction with constant acceleration a > 0, with position
(1/a,x2,x3) at the instant x0 =0, describes the curve [24]
x0 0
 x1   1/a 
=B , (49)
x2 τ(x0) x2
   
 x3   x3 
   
where eτ(x0) =ax0+ a2(x0)2+1. The (x′)1-axis of the accelerated observer’s instantaneous
rest frame is given bypthe half line emanating from the origin and passing through the point
10