An Infinite Level Atom coupled to a Heat Bath
Martin Koenenberg1
Received: January 13, 2011
Communicated by Heinz Siedentop
Abstract. We consider a W∗-dynamical system (M
β, τ), which
models finitely many particles coupled to an infinitely extended heat bath. The energy of the particles can be described by an unbounded operator, which has infinitely many energy levels. We show existence of the dynamics τ and existence of a (β, τ) -KMS state under very explicit conditions on the strength of the interaction and on the inverse temperatureβ.
2010 Mathematics Subject Classification: Primary 81V10; Secondary 37N20, 47N50.
Keywords and Phrases: KMS-state, thermal equilibrium, W∗
-dynamical system, open quantum system.
1 Introduction
In this paper, we study a W∗-dynamical system (Mβ, τ) which describes a
system of finitely many particles interacting with an infinitely extended bosonic reservoir or heat bath at inverse temperature β. Here, Mβ denotes theW∗
-algebra of observables andτis an automorphism-group onMβ, which is defined
by
τt(X) :=eitLQX e−itLQ, X∈Mβ, t∈R. (1) In this context,tis the time parameter. LQis the Liouvillean of the dynamical
system at inverse temperatureβ,Qdescribes the interaction between particles and heat bath. On the one hand the choice of LQ is motivated by heuristic
arguments, which allow to derive the LiouvilleanLQfrom the HamiltonianHof
the joint system of particles and bosons at temperature zero. On the other hand we ensure thatLQanti-commutes with a certain anti-linear conjugationJ, that
will be introduced later on. The Hamiltonian, which represents the interaction
with a bosonic gas at temperature zero, can be the Standard Hamiltonian of the non-relativistic QED, (see or instance [2]), or the Pauli-Fierz operator, which is defined in [7, 2], or the Hamiltonian of Nelson’s Model. We give the definition of these Hamiltonians in the sequel of Definition 11.
Our first result is the following:
Theorem 1.1. LQ, defined in (16), has a unique self-adjoint realization and
τt(X)∈Mβ for allt∈Rand all X∈Mβ.
The proof follows from Theorem 4.2 and Lemma 5.2. The main difficulty in the proof is, thatLQ is not semi-bounded, and that one has to define a suitable
auxiliary operator in order to apply Nelson’s commutator theorem.
Partly, we assume that the isolated system of finitely many particles is con-fined in space. This is reflected in Hypothesis 1, where we assume that the particle Hamiltonian Hel possesses a Gibbs state. In the case where Hel is a
Schrödinger-operator, we give in Remark 2.1 a sufficient condition on the ex-ternal potentialV to ensure the existence of a Gibbs state forHel. Our second
theorem is
Theorem1.2. Assume Hypothesis 1 and thatΩβ0 ∈dom(e−(β/2)(L0+Q)). Then
there exists a(β, τ)-KMS stateωβ on M β.
This theorem ensures the existence of an equilibrium state on Mβ for the
dynamical system (Mβ, τ). Its proof is part of Theorem 5.3 below. Here,L0 denotes the Liouvillean for the joint system of particles and bosons, where the interaction part is omitted. Ωβ0 is the vector representative of the(β, τ)-KMS state for the system without interaction. In a third theorem we study the conditionΩβ0 ∈dom(e−(β/2)(L0+Q)):
Theorem 1.3. Assume Hypothesis 1 is fulfilled. Then there are two cases, 1. If0 6 γ < 1/2 andη1(1 +β) ≪ 1, thenΩβ0 ∈dom(e−β/2 (L0+Q)).
2. Ifγ= 1/2and (1 +β)(η1+η2) ≪ 1, thenΩβ0 ∈dom(e−β/2 (L0+Q)).
Here,γ∈[0,1/2)is a parameter of the model, see (32) andη1, η2 are parame-ters, which describe the strength of the interaction, see (32). In a last theorem we consider the case whereHel=−∆q+ Θ2q2and the interaction Hamiltonian
isλ qΦ(f)at temperature zero forλ6= 0. Then,
Theorem 1.4. Ωβ0 is indom(e−β/2 (L0+Q))for allβ∈(0,∞), whenever |2Θ−1λ| k|k|−1/2f
kHph < 1.
Furthermore, we show that our strategy can not be improved to obtain a result, which ensures existence for all values ofλ, see (60).
some of them [3, 6, 8, 9, 10, 13, 14, 15], which consider a related model, in which the particle Hamilton Hel is represented as a finite symmetric matrix
and the interaction part of the Hamiltonian is linear in annihilation and cre-ation operators. In this case one can prove existence of aβ- KMS without any restriction to the strength of the coupling. (In this case we can apply Theorem 1.3 with γ= 0 andη1= 0). We can show existence of KMS-states for an infi-nite level atom coupled to a heat bath. Furthermore, in [6] there is a general theorem, which ensures existence of a(β, τ)-KMS state under the assumption, thatΩβ0 ∈dom(e−(β/2)Q), which impliesΩ
β
0 ∈dom(e−(β/2)(L0+Q)). In Remark 7.3 we verify that this condition implies the existence of a(β, τ)-KMS state in the case of a harmonic oscillator with dipole interactionλ q · Φ(f), whenever (1 +β)λk(1 +|k|−1/2)f
k ≪1.
2 Mathematical Preliminaries
2.1 Fock Space, Field- Operators and Second Quantization
We start our mathematical introduction with the description of the joint system of particles and bosons at temperature zero. The Hilbert space describing bosons at temperature zero is the bosonic Fock space Fb, where
Fb := Fb[Hph] := C⊕
∞
M
n=1
H(phn), H
(n)
ph := n
O
sym
Hph.
Hphis either a closed subspace ofL2(R
3)orL2( R
3
× {±}), being invariant un-der complex conjugation. If phonons are consiun-dered we chooseHph=L2(R
3), if photons are considered we choose Hph =L2(R
3
× {±}). In the latter case "+" or "-" labels the polarization of the photon. However, we will write hf|giHph :=
R
f(k)g(k)dk for the scalar product in both cases. This is an abbreviation forPp=±R f(k, p)g(k, p)dkin the case of photons.
H(phn) is the n-fold symmetric tensor product of Hph, that is, it contains all
square integrable functions fn being invariant under permutations π of the
variables, i.e., fn(k1, . . . , kn) = fn(kπ(1), . . . , kπ(n)). For phonons we have
kj∈R 3andk
j∈R 3
× {±}for photons. The wave functions inHn
phare states
ofnbosons.
The vectorΩ := (1,0, . . .) ∈ Fb is called thevacuum. Furthermore we denote
annihilation operators,a∗(h)and a(h), are defined forh
∈ Hph by
(a∗(h)fn)(k1, . . . , kn+1) (2)
= (n+ 1)−1/2
n+1
X
i=1
h(ki)fn(k1, . . . , ki−1, ki+1, . . . , kn+1),
(a(h)fn+1)(k1, . . . , kn) (3)
= (n+ 1)1/2
Z
h(kn+1)fn+1(k1, . . . , kn+1)dkn+1,
and a∗(h) Ω = h, a(h) Ω = 0. Sincea∗(h) ⊂ (a(h))∗ and a(h) ⊂ (a∗(h))∗,
the operatorsa∗(h)anda(h)are closable. Moreover, the canonical
commuta-tion relacommuta-tions (CCR) hold true, i.e.,
[a(h), a(eh)] = [a∗(h), a∗(eh)] = 0, [a(h), a∗(eh)] = hh|ehiHph.
Furthermore we define field operator by
Φ(h) := 2−1/2(a(h) + a∗(h)), h∈ Hph.
It is a straightforward calculation to check that the vectors inFbf inare analytic
forΦ(h). Thus,Φ(h)is essentially self-adjoint onFbf in. In the sequel, we will
identify a∗(h), a(h) and Φ(h) with their closures. The Weyl operatorsW(h)
are given by W(h) = exp(iΦ(h)). They fulfill the CCR-relation for the Weyl operators, i.e.,
W(h)W(g) = exp(i/2Imhh|giHph)W(g+h),
which follows from explicit calculations onFbf in. The Weyl algebraW(f)over
a subspacefofHph is defined by
W(f) := cl LH{W(g)∈ B(Fb) : g∈f}. (4)
Here, cl denotes the closure with respect to the norm of B(Fb), and "LH"
denotes the linear hull. Letα : R
3 → [0,∞)be a locally bounded Borel function anddom(α) := {f ∈ Hph : αf ∈ Hph}. Note, that (αf)(k) is given by α(k)f(k, p) for photons.
If dom(α) is dense subspace of Hph, α defines a self-adjoint multiplication
operator on Hph. In this case, the second quantization dΓ(α) of αis defined
by
(dΓ(α)fn)(k1, . . . , kn) := (α(k1) +α(k2) +. . .+α(kn))fn(k1, . . . , kn)
2.2 Hilbert space and Hamiltonian for the particles
LetHel be a closed, separable subspace of L2(X, dµ), that is invariant under
complex conjugation. The HamiltonianHel for the particle is a self-adjoint
op-erator onHelbeing bounded from below. We setHel,+ := Hel−infσ(Hel) + 1.
Partly, we need the assumption
Hypothesis 1. Let β >0. There exists a small positive constantǫ >0, and
TrHel{e
−(β−ǫ)Hel
} < ∞.
The condition implies the existence of a Gibbs state
ωβel(A) = Z−1 Tr
Hel{e
−β HelA
}, A∈ B(Hel),
for Z = TrHel{e
−β Hel
}.
Remark2.1. LetHel =L2(R
n, dnx)andH
el = −∆x+V1+V2, whereV1 is
a−∆x-bounded potential with relative bounda < 1andV2is inL2loc(R
n, dnx).
Thus Hel is essentially self-adjoint onCc∞(R
n). Moreover, if additionally
Z
e−(β−ǫ)V2(x)dnx <
∞, (5)
then one can show, using the Golden-Thompson-inequality, that Hypothesis 1 is satisfied.
2.3 Hilbert space and Hamiltonian for the interacting system
The Hilbert space for the joint system isH := Hel⊗ Fb. The vectors inHare
sequencesf = (fn)n∈N0 of wave functions,fn ∈ Hel⊗ H (n)
ph, obeying
kn 7→ fn(x, kn)∈ H
(n)
ph forµ- almost every x
x 7→ fn(x, kn)∈ Hel for Lebesgue - almost every kn,
wherekn = (k1, . . . , kn). The complex conjugate vector isf := (fn)n∈N0. Let Gj :=
{Gjk}k∈R
3, Hj := {Hkj}k∈
R
3 and F := {Fk}k∈
R
3 be
fam-ilies of closed operators on Hel for j = 1, . . . , r. We assume, that
dom(F∗
k),dom(Fk) ⊃ dom(Hel,1/2+)and that
k 7→ Gjk, (Hkj), FkHel,−1+/2, (Fk)∗Hel,−1+/2∈ B(Hel)
are weakly (Lebesgue-)measurable. For φ∈dom(Hel,1/2+)we assume that k 7→ (Gjkφ)(x), (H
j
kφ)(x), (Fkφ)(x)∈ Hph, (6)
Moreover we assume forG~ = (G1, . . . , Gr), ~H := (H1, . . . , Hr)andF, that
anda(F)f0(x) = 0. The (generalized) annihilation operator is
(a(F)fn+1)(x, k1, . . . , kn) (9)
:= (n+ 1)1/2
Z
(Fkn+1∗ fn+1)(x, k1, . . . , kn, kn+1)dkn+1.
Moreover, the corresponding (generalized) field operator is Φ(F) := 2−1/2(a(F) + a∗(F)). Φ(F) is symmetric on dom(H1/2
el,+)⊗ F
f in
b . The
bounds follow directly from Equations (8) and (9).
ka(F)Hel,−1+/2fk2 ans for the non-interacting, resp. interacting model are
Definition 2.2. Ondom(Hel)⊗dom(dΓ(α))∩ Fbf in we define
H0 := Hel⊗1 +1⊗dΓ(α), H := H0 + W, (11)
where W := Φ(G) Φ(~ H~) + h.c.+ Φ(F) and Φ(G) Φ(~ H)~ :=
Pr
j=1Φ(Gj) Φ(Hj). The abbreviation "h.c." means the formal adjoint operator
We give examples for possible configurations: Letγ∈Rbe a small coupling parameter. ◮The Nelson Model:
Hel⊂L2(R 3N),H
el := −∆ + V, Hph = L2(R
3)andα(k) =
|k|. The form factor isFk = γPNν= 1 e−i kxν|k|−1/21[|k| 6 κ], xν ∈R
3andHj, Gj = 0.
◮The Standard Model of Nonrelativistic QED: Hel⊂L2(R
3N),H
el := −∆ +V, Hph = L2(R 3
× {±})andα(k) = |k|. The form factors are
Fk = 4γ3/2π−1/2
N
X
ν=1
(−i∇xν·ǫ(k, p))e−i γ 1/2kxν
(2|k|)−1/21[|k| 6 κ] + h.c.,
Gi, νk = Hki, ν = 2γ3/2π−1/2ǫi(k, p)e−i γ 1/2
kxν(2
|k|)−1/21[|k| 6 κ] fori = 1,2,3, ν = 1, . . . , N, xν ∈R
3andk= (k, p)∈ R
3× {±}. ǫ
i(k,±)∈
R
3are polarization vectors. ◮The Pauli-Fierz-Model: Hel ⊂L2(R
3N), H
el := −∆ + V, Hph = L2(R 3)or
Hph = L2(R 3
× {±}), and α(k) = |k|. The form factor is Fk = γPNν= 11[|k| 6 κ]k·xν and
Gjk=Hkj= 0
3 The Representation π
In order to describe the particle system at inverse temperatureβ we introduce the algebraic setting. For f = {f ∈ Hph : α−1/2f ∈ Hph} we define the
algebra of observables by
A = B(Hel)⊗ W(f).
For elementsA∈Awe defineτ˜0
t(A) := ei t H0A e−i t H0 and
˜
τtg(A) := ei t HAe−i t H. We first discuss the model without interaction.
3.1 The Representation πf
The time-evolution for the Weyl operators is given by ei tHˇW(f)e−i tHˇ = W(ei t αf). For this time-evolution an equilibrium stateωfβis defined by
ωfβ(W(f)) = hf|(1 + 2̺β)fiHph,
where ̺β(k) = exp(β α(k)) − 1
−1
. It describes an infinitely extended gas of bosons with momentum density ̺β at temperature β. Since ωfβ is a
creation and annihilation operators. One has
ωfβ(a(f)) = ωfβ(a∗(f)) = ωfβ(a(f)a(g)) = ωfβ(a∗(f)a∗(g)) = 0, ωβf(a∗(f)a(g)) = hg|̺βfiHph.
For polynomials of higher degree one can apply Wick’s theorem for quasi-free states, i.e.,
ωfβ(aσ2m(f2m)· · ·aσ1(f1)) =
X
P∈Z2
Y
{i,j}∈P i>j
ωfβ aσi(fi)aσj(fj), (12)
whereaσk = a∗ or aσk = afork= 1, . . . ,2m.
Z2are the pairings, that is P ∈ Z2,iffP = {Q1, . . . , Qm}, #Qi= 2andSmi=1Qi = {1, . . . , 2m}.
The Araki-Woods isomorphismπf : W(f) → B(Fb⊗ Fb)is defined by
πf[W(f)] := Wβ(f) := exp(iΦβ(f)),
Φβ(f) := Φ((1 + ̺β)1/2f)⊗1 +1⊗Φ(̺1β/2f).
The vectorΩβf := Ω⊗Ωfulfills
ωβf(W(f)) = hΩβf|πf[W(f)] Ωβfi. (13)
3.2 The representation πel
The particle system without interaction has the observablesB(Hel), the states
are defined by density operatorsρ, i.e., ρ∈ B(Hel), 0 6ρ, Tr{ρ} = 1. The
expectation ofA∈ B(Hel)inρat timet is
Tr{ρ ei t HelA e−i t Hel}.
Sinceρis a compact, self-adjoint operator, there is an ONB(φn)n of
eigenvec-tors, with corresponding (positive) eigenvalues(pn)n. Let
σ(x, y) =
∞
X
n=1 p1/2
n φn(x)φn(y)∈ Hel⊗ Hel. (14)
Forψ∈ Hel we defineσ ψ := Rσ(x, y)ψ(y)dµ(y). Obviously,σis an operator
of Hilbert-Schmidt class. Note,σ ψ := σ ψ has the integral kernelσ(x, y). It is a straightforward calculation to verify that
Tr{ρ ei t HelA e−i t Hel} = he−i tLelσ
|(A⊗1)e−i tLelσ
iHel⊗Hel,
whereLel = Hel⊗1−1⊗Hel. This suggests the definition of the representation
πel :
Now, we define the representation map for the joint system by π : A → B(K), π := πel⊗πf,
where K := Hel⊗ Hel⊗ Fb⊗ Fb. LetMβ := π[A]′′ be theenveloping W∗
-algebra, hereπ[A]′denotes the commutant ofπ[A], andπ[A]′′the bicommutant.
We set D := U1⊗U1⊗ C, whereC is a subspace of vectors in Fbf in⊗ F f in b ,
with compact support, andU1:=∪∞
n=1ran1[Hel6n]. OnDthe operatorL0, given by
L0 := Lel⊗1+1⊗ Lf, onK,
Lf := dΓ(α)⊗1 −1⊗dΓ(α), onFb⊗ Fb,
is essentially self-adjoint and we can define
τt0(X) := ei tL0X e−i tL0 ∈Mβ, X∈Mβ, t∈R, It is not hard to see, that
π[˜τt0(A)] = τt0(π[A]), A∈A, t∈R OnK a we introduce a conjugation by
J (φ1⊗φ2⊗ψ1⊗ψ2) = φ2⊗φ1⊗ψ2⊗ψ1.
It is easily seen, thatJ L0 = −L0J. In this context one hasM′β = JMβJ,
see for example [4]. In the case, where Hel fulfills Hypothesis 1, we define the
vector representative Ωβel ∈ Hel ⊗ Hel of the Gibbs state ωβel as in (14) for
ρ=e−βHel
Z−1.
Theorem 3.1. Assume Hypothesis 1 is fulfilled. Then, Ωβ0 := Ωβel⊗Ωβf is a
cyclic and separatingvector forMβ. e−β/2L0 is a modular operatorandJ is
the modular conjugationfor Ωβ0, that is XΩβ0 ∈dom(e−β/2L0),
JXΩβ0 = e−β/2L0X∗Ωβ
0 (15)
for all X∈Mβ andL0Ωβ0 = 0. Moreover, ωβ0(X) := hΩ
β
0|XΩ
β
0iK, X ∈Mβ
is a (τ0, β)-KMS-state for M
β, i.e., for all X, Y ∈ Mβ exists Fβ(X, Y,·),
analytic in the stripSβ = {z∈C : 0 < Imz < β}, continuous on the closure
and taking the boundary conditions
Fβ(X, Y, t) = ω0β(X τt0(Y))
Fβ(X, Y, t +i β) = ω0β(τt0(Y)X)
4 The Liouvillean LQ
In this and the next section we will introduce the Standard LiouvilleanLQfor
a dynamicsτ onMβ, describing the interaction between particles and bosons
at inverse temperature β. The label Q denotes the interaction part of the Liouvillean, it can be deduced from the interaction partW of the corresponding Hamiltonian by means of formal arguments, which we will not give here. In a first step we prove self-adjointness ofLQ and of other Liouvilleans. A main
difficulty stems from the fact, that LQ and the other Liouvilleans, mentioned
before, are not bounded from below. The proof of self-adjointness is given in Theorem 4.2, it uses Nelson’s commutator theorem and auxiliary operators which are constructed in Lemma 4.1. The proof, thatτt(X)∈MβforX∈Mβ,
is given in Lemma 5.2. Assuming Ωβ0 ∈ dom(e−β/2(L0+Q)) we can ensure existence of a (τ, β)-KMS state ωβ(X) = hΩβ|XΩβi · kΩβk−2 onM
β, where
Ωβ = e−β/2(L0+Q)Ωβ
0. Moreover, we can show that e−βLQ is the modular operator forΩβ and conjugation
J. This is done in Theorem 5.3.
Our proof of 5.3 is inspired by the proof given in [6]. The main difference is that we do not assume, thatQ is self-adjoint and thatΩβ0 ∈dom(e−βQ). For this reason we need to introduce an additional approximationQN ofQ, which
is self-adjoint and affiliated with Mβ, see Lemma 5.1.
The interaction on the level of Liouvilleans between particles and bosons is given byQ, where
Q := Φβ(G) Φ~ β(H~) + h.c.+ Φβ(F), Φβ(G) Φ~ β(H) :=~ r
X
j=1
Φβ(Gj) Φβ(Hj).
For each familyK={Kk}k of closed operators onHel withkKkw,1/2<∞we set
Φβ(K) := a∗((1 + ̺β)1/2K)⊗1+ 1⊗a∗(̺1β/2K∗)
+ h.c. . Here, Kk acts as Kk ⊗1 on Hel ⊗ Hel. A Liouvillean, that describes the
dynamics of the joint system of particles and bosons is the so-called Standard Liouvillean
LQφ := (L0 +Q − QJ)φ, φ∈ D, (16) which is distinguished byJ LQ=−LQJ. For an operatorA, acting onK, the
symbol AJ is an abbreviation for JAJ. An important observation is, that
[Q , QJ] = 0on
D. Next, we define four auxiliary operators onD L(1)
a := (Hel,+⊗1+1⊗Hel,+⊗1+1⊗ Lf,a+1 (17)
L(2)a := H Q el,++ (H
Q el,+)
J +c1
1⊗ Lf,a+c2
L(3)a := H Q
el,++ (Hel,+)J +c11⊗ Lf,a+c2
whereLf,a is an operator onFb⊗ FbandHel,Q+ acts onK. Furthermore, Lf,a = dΓ(1 +α)⊗1+1⊗dΓ(1 +α) +1,
Lel,a = Hel,+⊗1 +1⊗Hel,+ Hel,Q+ := Hel,+⊗1+Q.
Obviously,L(ai), i = 1,2,3,4 are symmetric operators onD.
Lemma 4.1. For sufficiently large values ofc1, c2 > 0 we have thatL(ai), i =
1,2,3,4are essentially self-adjoint and positive. Moreover, there is a constant
c3 > 0 such that
c−31kL(1)
a φk 6 kL(ai)φk6c3kL(1)a φk, φ∈dom(L(1)a ). (18)
Proof. Let a, a′
∈ {l, r} and Ki, i = 1,2 be families of bounded operators
with kKikw <∞. Let Φl(Ki) = Φ(Ki)⊗1 and Φr(Ki) := 1⊗Φ(Ki). We
have for φ∈ D
kΦa(ηK1) Φa′(η′K2)φk 6 constk Lf,aφk (19)
kΦa(ηF)φk 6 constk(Lel,a)1/2(Lf,a)1/2φk,
whereη, η′
∈ {(1+̺β)1/2, ̺β1/2}. Note, that the estimates hold true, ifΦa(ηKi)
or Φa(ηF) are replaced byΦa(ηKi)J or Φa(ηF)J. Thus, we obtain for
suffi-ciently largec1 ≫ 1, depending on the form-factors, that kQ φk + kQJφk 6 1/2 Lel,a+c1Lf,aφ
. (20)
By the Kato-Rellich-Theorem ( [17], Thm. X.12) we deduce that L(ai) is
self-adjoint on dom(Lel,a+c1Lf,a), bounded from below and that Lel,a+c1Lf,a
isL(ai)-bounded for everyc2 > 0andi = 2,3,4. In particular,Dis a core of
L(ai). The proof follows now fromkL(ai)φk 6 k(Lel,a+c1Lf,a)φk 6 c1kL(1)a φk
forφ∈ D.
Theorem 4.2. The operators
L0, LQ = L0+Q−QJ, L0+Q, L0−QJ, (21)
defined on D, are essentially self-adjoint. Every core of L(1)a is a core of the
operators in line (21).
Proof. We restrict ourselves to the case of LQ. We check the assumptions
of Nelson’s commutator theorem ([17], Thm. X.37). By Lemma 4.1 it suf-fices to show kLQφk 6 constkL(1)a φk and |h LQφ| L(2)a φi − h L(2)a φ| LQφi| 6
To verify the second inequality we observe
LQφ
L(2)
a φ
−L(2)
a φ
LQφ (22)
6c1QφLf,aφ−Lf,aφQφ
+c1QJφLf,aφ−Lf,aφ
QJφ
+Lfφ
Qφ− hQφLfφ+
Lfφ
QJφ−QJφLfφ,
where we used, that Hel,Q+,(Hel,Q+)J = 0. Let K
i ∈ {Gj, Hj} and η, η′ ∈
{̺1/2,(1 + ̺)1/2
}. We remark, that
[Φa(η K1) Φa′(η′K2),Lf,a] = iΦa(i(1 +α)η K1) Φa′v(η′K2) (23)
+ iΦa(η K1) Φa′(i(1 +α)η′K2)
[Φa(η F), Lf,a] = iΦa(i(1 +α)η F).
Hence, for φ∈dom(L(2)a ), we have by means of (10) that
φ|[Φa(ηK1) Φa′(η′K2), Lf,a]φ 6 constkL1/2
f,aφk2 (24)
φ|[Φa(ηF), Lf,a]φ 6 constkLf,a1/2φk k(Lel,a)1/2φk.
Thus, (24) is bounded by a constant times k(L(1)a )1/2φk2. The essential
self-adjointness of LQ follows now from estimates analog to (23) and (24), where
Lf,ais replaced byLf in (23) and in the left side of (24). ForL0+QandL0−QJ one has to consider the commutator withL(3)a andL(4)a , respectively.
Remark4.3. In the same way one can show, thatH is essentially self-adjoint on any core ofH1:=Hel +dΓ(1 + α), even ifH is not bounded from below.
5 Regularized Interaction and Standard Form ofMβ
In this subsection a regularized interactionQN is introduced:
QN :=
n
Φβ(G~N) Φβ(H~N) + h.c.
o
+ Φβ(FN). (25)
The regularized form factors G~N, ~HN, FN are obtained by multiplying the
finite rank projectionPN :=1[Hel 6N]from the left and the right. Moreover,
an additional ultraviolet cut-off1[α6N], considered as a spectral projection, is added. The regularized form factors are
~
GN(k) := 1[α 6 N]PNG(k)~ PN, H~N(k) := 1[α 6 N]PNH(k)~ PN,
Lemma5.1. i)QN is essentially self-adjoint onD ⊂dom(QN). QN is affiliated
with Mβ, i.e,. QN is closed and
X′QN ⊂QNX′, ∀X′∈M′β.
ii) L0+QN, L0 − JQNJ and L0 +QN − JQNJ converges in the strong
resolvent sense toL0+Q,L0− JQJ andL0+Q− JQJ, respectively.
Proof. LetQN be defined onD. With the same arguments as in the proof of
Theorem 4.2 we obtain kQNφk 6 CkLf,aφk,
QNφ
Lf,aφ
− Lf,aφ
QNφ 6 C
(Lf,a)1/2φ
2,
for φ∈ D and some constant C > 0, where we have used that kFNkw < ∞.
Thus, from Theorem 4.2 and Nelson’s commutator theorem we obtain thatD is a common core forQN, L0+QN, L0 −QJN, L0 + QN −QJN and for the
operators in line (21). A straightforward calculation yields lim
N→∞QNφ = Qφ, Nlim→∞JQNJφ = JQJφ ∀φ∈ D.
Thus statement ii) follows, since it suffices to check strong convergence on the common coreD, see [16, Theorem VIII.25 a)].
LetNf := dΓ(1)⊗1+1⊗dΓ(1)be the number-operator. Sincedom(Nf)⊃ D
andWβ(f)J : dom(Nf) → dom(Nf), see [4], we obtain
QN(A⊗1⊗ Wβ(f))Jφ = (A⊗1⊗ Wβ(f))JQNφ (26)
forA∈ B(Hel), f ∈fandφ∈ D. By closedness ofQN and density arguments
the equality holds for φ∈dom(QN)andX ∈Mβ instead ofA⊗1⊗ Wβ(f).
ThusQN is affiliated with Mβ and thereforeei tQN ∈Mβ fort∈R.
Lemma 5.2. We have forX ∈Mβ andt∈R
τt(X) =eit(L0+Q)X eit(L0+Q), τt0(X) =eit(L0−Q
J)
X eit(L0−QJ) (27)
Moreover, τt(X)∈Mβ for allX ∈Mβ andt∈R, such as EQ(t) := ei t(L0+Q)e−i tL0 =ei tLQe−i t(L0−Q
J)
∈Mβ.
Proof. First, we prove the statement for QN, since QN is affiliated with Mβ
and thereforeeitQN ∈M
β. We set
ˆ
τtN(X) =ei t(L0+QN)X e−i t(L0+QN), ˆτt(X) =ei t(L0+Q)X e−i t(L0+Q)
On account of Lemma 5.1 and Theorem 4.2 we can apply the Trotter product
Thanks to Lemma 5.1 we also have
τt(X) = w-limn→∞τtN(X) = w-limN→∞ˆτ
Theorem5.3. Assume Hypothesis 1 andΩβ0 ∈dom(e−β/2 (L0+Q)). LetΩβ :=
e−β/2 (L0+Q)Ωβ
0. Then
JΩβ = Ωβ, Ωβ = eβ/2 (L0−QJ)Ωβ
0, (30) LQΩβ = 0, JX∗Ωβ = e−β/2LQXΩβ, ∀X ∈Mβ
Furthermore, Ωβ is separating and cyclic for M
β, and Ωβ ∈ C. The stateωβ
is defined by
ωβ(X) :=kΩβk−2
hΩβ|XΩβi, X ∈Mβ
is a(τ, β)-KMS state on Mβ.
Proof. First, we define Ω(z) = e−z(L0+Q)Ωβ
0 for z∈C with06Rez6β/2. Since Ωβ0 ∈ dom(e−β/2 (L0+Q)), Ω(z) is analytic on Sβ/2 := {z ∈ C : 0 < Re(z) < α} and continuous on the closure ofSβ/2, see Lemma A.2 below.
◮Proof ofJΩ(β/2) = Ω(β/2): We pick φ ∈ Sn∈Nran1[|L0|
6 n]. Let f(z) := hφ| JΩ(z)i and g(z) := he−(β/2−z)L0φ|e−z(L0+Q)Ωβ
0i. Both f and g are analytic onSβ/2 and continuous on its closure. Thanks to Lemma 5.2 we have EQ(t) ∈ Mβ,
and hence
f(it) = hφ| JEQ(t) Ωβ0i = hφ|e−β/2L0EQ(t)∗Ωβ0i = g(i t), t∈R. By Lemma A.1, f andg are equal, in particular in z = β/2. Note that φis any element of a dense subspace.
◮Proof ofΩβ0 ∈dom(eβ/2 (L0−Q
J
))andΩ(β/2) = eβ/2 (L0−QJ)Ωβ
0: Let φ ∈ Sn∈Nran1[|L0 − Q
J| 6 n]. We set g(z) :=
hez(L0−QJ)φ|e−zL0Ωβ
0i. Since EQ(t)J = ei t(L0−Q
J)
e−i tL0, g
coin-cides forz = i twithf(z) := hφ| JΩ(z)i. Hence they are equal inz = β/2. The rest follows sinceeβ/2 (L0−QJ)is self-adjoint.
◮Proof ofLQΩ(β/2) = 0:
Choose φ ∈ Sn∈ N
ran1[|LQ| 6 n]. We define g(z) :=
he−zLQφ|ez(L0−QJ)Ωβ
0i and f(z) := hφ|Ω(z)i for z in the closure of Sβ/2. Again both functions are equal on the line z = i t, t ∈ R. Hence f and g are identical, and therefore Ω(β/2) ∈ dom(e−β/2LQ) and
e−β/2LQΩ(β/2) = Ω(β/2). We conclude that L
QΩ(β/2) = 0.
◮Proof ofJX∗Ω(β/2) = e−β/2LQXΩ(β/2), ∀X∈M β:
ForeA∈Manaβ we have, that J A∗Ω(
−it) = JA∗E
Q(t) Ωβ0 = e−β/2L0EQ(t)∗AΩβ0 = e−(β/2−i t)L0e−it(L0+Q)AΩβ
0
=e−(β/2−it)L0τ
Let φ ∈ Sn∈ N
ran1[|L0| 6 n]. We define f(z) = hφ| JA∗Ω(z)i and g(z) = he−(β/2−z)L0φ|τ
iz(A) Ω(z)i. Since f and g are analytic and equal
for z = it, we have JA∗Ω(β/2) = τ
i β/2(A) Ω(β/2). To finish the proof we
pick φ ∈ Sn∈ N
ran1[|LQ| 6 n], and set f(z) := hφ|τiz(A) Ω(β/2)i and
g(z) := he−zLQφ|AΩ(β/2)i. Forz = i twe see
g(it) = hφ|e−itLQA eitLQΩ(β/2)
i = hφ|τ−t(A) Ω(β/2)i = f(i t).
Hence AΩ(β/2)∈dom(e−β/2LQ)andJA∗Ω(β/2) = e−β/2LQAΩ(β/2).
SinceMana
β is dense in the strong topology, the equality holds for allX∈Mβ.
◮Proof, thatΩβ is separating forMβ:
LetA∈Manaβ . We chooseφ∈Sn∈Nran1[|(L0+Q)|
6 n]. First, we have JA∗Ω(β/2) = τiβ/2(A) Ω(β/2).
Letfφ(z) =hφ|τz(A)Ω(β/2)iandgφ(z) =hez(L0+Q)φ|Ae−(β/2+z)(L0+Q)Ωβ0i for−β/2 6 Rez 6 0. Both functions are continuous and analytic if−β/2 < Rez < 0. Furthermore, fφ(i t) = gφ(i t)for t ∈R. Hence fφ = gφ and for z = −β/2
hφ| JA∗Ω(β/2)i = he−β/2 (L0+Q)φ|AΩβ
0i.
This equation extends to all A ∈ Mβ, we obtain AΩβ0 ∈ dom(e−β/2 (L0+Q)), such as e−β/2(L0+Q)AΩβ
0 = JA∗Ω(β/2) for A ∈ Mβ. Assume
A∗Ω(β/2) = 0, then
e−β/2 (L0+Q)AΩβ
0 = 0and hence AΩ
β
0 = 0. Since Ω
β
0 is separating, it follows that A= 0and thereforeA∗ = 0.
◮Proof ofΩβ
∈ C, and thatΩβ is cyclic forM β:
To prove that φ ∈ C it is sufficient to check that hφ|AJAΩβ0i > 0 for all A∈Mβ. We have
hΩ(β/2)|AJAΩ0βi = h JA∗Ω(β/2)|AΩ
β
0i
= he−β/2(L0+Q)AΩβ
0|AΩ
β
0i > 0. The proof follows, since every separating element ofCis cyclic. ◮Proof, thatωβ is a (τ, β)-KMS state:
ForA, B∈Mβ and z∈Sβ we define
Fβ(A, B, z) = che−iz/2LQA∗Ωβ|eiz/2LQBΩβi,
wherec := kΩβ
k−2. First, we observe
Fβ(A, B, t) = che−it/2LQA∗Ωβ|eit/2LQBΩβi = chΩβ|Aτt(B)Ωβi
and
ωβ(τt(B)A) = chτt(B∗)Ωβ|AΩβi = ch JAΩβ| Jτt(B∗)Ωβi
= che−β/2LQA∗Ωβ
|e−β/2LQτ
t(B)Ωβi
= che−i(iβ+t)/2LQA∗Ωβ
|ei(iβ+t)/2LQBΩβ
i = Fβ(A, B, t+iβ).
The requirements on the analyticity ofFβ(A, B,·)follow from Lemma A.2.
6 Proof of Theorem 1.3
Forsn := (sn, . . . , s1)∈R
n we define
QN(sn) := QN(sn)· · ·QN(s1), QN(s) := e−sL0QNesL0, s∈R (31) At this point, we check thatQN(sn)Ω
β
0 is well defined, and that it is an analytic vector of L0, see Equation (25). The goal of Theorem 1.3 is to give explicit conditions onHel andW, which ensureΩβ0 ∈dom(e−β/2 (L0+Q)). Let
η1 :=
Z
kG(k)~ k2
B(Hel)+kH(k)~ k
2
B(Hel)
(2 + 4α(k)−1)dk (32)
η2 :=
Z
kF(k)Hel,−γ+k2B(Hel)+kF(k)
∗H−γ
el,+kB(Hel)
(2 + 4α(k)−1)dk
The idea of the proof is the following. First, we expande−β/2(L0+QN)eL0 in a
Dyson-series, i.e.,
e−β/2(L0+QN)eL0 (33)
=1+
∞
X
n=1 (−1)n
Z
∆n β/2
e−snL0Q
NesnL0· · ·e−s1L0QNes1L0dsn.
Under the assumptions of Theorem 1.3 we obtain an upper bound, uniform in N, for
hΩβ0|e−β(L0+QN)Ω
β
0i (34)
= 1 +
∞
X
n=1 (−1)n
Z
∆n β
hΩβ0|e−snL0QNesnL0· · ·e−s1L0QNes1L0Ωβ0idsn.
This is proven in Lemma 6.4 below, which is the most important part of this section. In Lemma 6.1 and Lemma 6.2 we deduce from the upper bound for (34) an upper bound for ke−(β/2)(L0+QN)Ωβ
Lemma 6.1. Assume
simplex of dimension nand sidelengthx.
Proof. Letφ∈ran1[|L0 +QN| 6 k]and0 6 x 6 β/2 be fixed. Anm-fold
application of the fundamental theorem of calculus yields
he−x(L0+QN)φ|exL0Ωβ
We turn now to the second expression on the right side of Equation (36), after a linear transformation depending onsm+1 we get
(−1)m+1 with at most2mbosons, we obtain the upper bound
Lemma 6.2. Let 0 < x 6 β/2. We have the identity
Proof. Recall Theorem 3.1 and Lemma 5.1. SinceJ is a conjugation we have hφ|ψi=h J ψ| Jφi, and for every operatorX, that is affiliated withMβ, we
The second statement of the Lemma follows by choosingn = m.
SinceL0 + QN tends toL0 +Qin the strong resolvent sense asN → ∞, we know from [16] thatlimN→ ∞ψN = f(L0 + Q)φ =: ψand
lim
N→∞e
−x(L0+QN)ψ
N = lim
N→∞g(L0+QN)φ = g(L0+Q)φ = e
−x(L0+Q)ψ.
Thus,
|he−x(L0+Q)ψ
|Ωβ0i| = lim
N→∞|he
−x(L0+QN)ψ
N|Ωβ0i| 6 sup
N∈N
ke−x(L0+QN)Ωβ
0k kψk,
Since{f(L0+Q)φ∈ K : φ∈ K, f ∈ C0∞(R)}is a core ofe
−x(L0+Q), we obtain
Ωβ0 ∈dom(e−x(L0+Q)).
Lemma 6.4. For someC >0 we have
Z
∆n β
Ωβ0|QN(sn)Ω β
0dsn
6const (n+ 1)2(1 +β)n8η
1+
(8Cη2)1/2 (n+ 1)(1−2γ)/2
n
,
whereη1 andη2 are defined in (32).
Proof of 6.4. First recall the definition of QN and QN(sn) in Equation (25)
and Equation (31), respectively. Let
Z
∆n β
Ωβ0|QN(sn) Ω β
0
dsn =:
Z
∆n 1
βnJn(β, s)dsn,
The functionsJn(β, s)clearly depends onN, but since we want to find an upper
bound independent of N, we drop this index. Let W1 = Φ(G) Φ(~ H~) + h.c., W2 := Φ(F)and W :=W1+W2. By definition of ω0β in (3.1), see also (13), we obtain
Jn(β, sn) = ω β
0
e−β snH0W eβsnH0· · · e−β s1H0W eβ s1H0 = (Z)−1 X
κ∈{1,2}n
ωβfTrHel
e−β Hel e−β snH0Wκ(n)eβ snH0
· · ·
· · · e−β s1H0Wκ(1)eβ s1H0
By definition ofωβf it suffices to consider expressions with an even number of
operators. In this case the expectations in ωβf can be expressed by an integral over R
2m× {±}2m with respect to ν, which is defined in Lemma A.4 below.
To give a precise formula we define
M(m1, m2) = {κ∈ {1,2}n : #κ−1({i}) = mi, i= 1,2}.
Thus we obtain Jn(β, sn) = (Z)−1
X
(n1, n2)∈N2
n1+ 2n2=n
X
κ∈M(n1,2n2 ) m:=n1+n2
Z
ν(dk2m⊗dτ2m) (40)
TrHel
e−(β−β(s1−s2m))HelI2
me−β(s2m−1−s2m)Hel· · ·e−β(s1−s2)HelI1 ,
Of courseIj depends onk2m×τ2m, namely forκ(j) = 1,2we have
Ij =
(
Ij(m, τ, m′, τ′), κ(j) = 1
Ij(m, τ), κ(j) = 2,
where(m, τ), (m′, τ′)
∈ {(kj, τj) : j = 1, . . . , m}. Forκ(j) = 1we have that
Ij(m,+, m′, −) = G~∗(m)H~(m′) + H~∗(m)G(m~ ′)
Ij(m,−, m′, +) = G(m)~ H~∗(m′) + H~(m)G~∗(m′)
Ij(m,+, m′, +) = G~∗(m)H~∗(m′) + H~∗(m)G~∗(m′)
Ij(m,−, m′, −) = G(m)~ H~(m′) + H~(m)G(m~ ′)
and forκ(j) = 2we have that
Ij(m,+) = F∗(m)
Ij(m,−) = F(m).
In the integral (40) we insert for(m, τ)and(m′, τ′)in the definition ofI
jfrom
left to rightk2m, τ2m, . . . , k1, τ1.
For fixed(k2m, τ2m)the integrand of (40) is a trace of a product of4m
oper-ators inHel. We will apply Hölder’s-inequality for the trace, i.e.,
|TrHel{A2mB2m· · ·A1B1}|6
2m
Y
j=1
kBjkB(Hel)·
2m
Y
j=1
TrHel{A pj i }p
−1 j .
In our casepi := (si−1−si)−1fori= 2, . . . ,2mandp1 := (1−s1+s2m)−1
and
(Aj, Bj) :=
(
e−β p−j1Hel, I
j(m, τ, m′, τ′), κ(j) = 1
e−β p−j1HelHγ el,+, H
−γ
el,+Ij(m, τ,), κ(j) = 2
We define
LetEgs := inf σ(Hel). The spectral theorem for self-adjoint operators implies
ke−ǫ HelHel,pi+γk
Inserting this estimates we get TrHel
Now, we recall the definition of ν. Roughly speaking, one picks a pair of variables(ki, kj)and integrates overδki,kjcoth(β/2α(ki))dkidkj. Subsequently
one picks the next pair and so on. At the end one sums up all 2(2mmm)!! pairings
and all4mcombinations ofτ
2m. Inserting Estimate (41) and that
By Lemma A.3 below and since(2m)!/(m!)264mwe have
Z
∆n β
Ωβ0|QN(sn) Ω β
0dsn
6const(1 +β)n X
(n1, n2)∈N
2 0 n1+ 2n2=n
n
n1
(8η
1)
n1(8C′η
2)
n2
(n+ 1)(1−2γ)n2−2
This completes the proof.
7 The Harmonic Oscillator
LetL2(X, dµ) =L2(
R)andHel =:Hosc:=−∆q+Θ
2q2be the one dimensional harmonic oscillator andHph=L2(R
3). We define
H = Hosc + Φ(F) + ˇH, Hˇ := dΓ(|k|), (44)
whereΦ(F) = q · Φ(f), withλ(|k|−1/2 +
|k|1/2)f ∈L2(
R 3).
Hosc is the harmonic oscillator, the form-factor F comes from the dipole
ap-proximation.
The Standard Liouvillean for this model is denoted by Losc. Now we prove
Theorem 1.4.
Proof. We define the creation and annihilation operators for the electron.
A∗ = Θ 1/2q
−iΘ−1/2p √
2 , A =
Θ1/2q +iΘ−1/2p √
2 , p = −i ∂x, (45) Φ(c) = c1q + c2p, forc = c1 +i c2∈C, ci∈R. (46) These operators fulfill the CCR-relations and the harmonic- oscillator is the number-operator up to constants.
[A, A∗] = 1, [A∗, A∗] = [A, A] = 0, Hosc = ΘA∗A + Θ/2, (47)
[Hosc, A] = −ΘA, [Hosc, A∗] = ΘA∗. (48)
The vectorΩ := Θ
π
1/4
e−Θq2/2
is called the vacuum vector. Note, that one can identifyFb[C]withL
2(
R), sinceLH{(A
∗)nΩ
|n∈N 0
}is dense inL2( R). It follows, thatωosc
β is quasi-free, as a state overW(C)and ωoscβ (W(c)) = (Z)−1TrHel{e
−βHelW(c)
} = exp −1/4 coth(βΘ/2)|c|2, (49) whereZ = TrHel{e
−βHel}is the partition function forH el.
that Inserting the identities of Equation (52) in Equation (51) and applying Wick’s theorem [5, p. 40] yields
h2n(β, λ) = (β λ)2n
The last equality in (53) holds, since the integrand is invariant with respect to a change of the axis of coordinates.
We interpret two pairingsP andP′∈ Z
2as an indirected graphG = G(P, P′), where M2n = {1, . . . ,2n} is the set of points. Any graph inGhas two kinds
of lines, namely lines in Losc(G), which belong to elements of P and lines in
Lf(G), which belong to elements ofP′.
LetG(A)be the set of undirected graphs with points inA⊂M2n, such that for
exact one line inLosc(G), which begins with "i". Gc(A)is the set of connected
graphs. We do not distinguish, if points are connected by lines inLf(G)or by
lines inLosc(G).
be the family of decompositions ofM2n inkdisjoint set. It follows
h2n(β, λ) =
From the first to the second line we summarize terms with graphs, having con-nected components containing the same set of points. From the second to the third line the order of the components is respected, hence the correction factor
1
Let now be Aa ⊂M2n with #Aa = 2ma > 2 fixed. In Ga are #Aa lines in
Losc(Ga), since such lines have no points in common, we have ma(2ma! 2ma)! choices.
Let now be the lines inLosc(Ga)fixed. We have now (2ma−2)(2ma−4)· · ·1
choices forma lines inLf(Ga), which yield a connected graph. Thus
X
For#Aa = 2exists only one connected graph. We obtain for h2n
h2n(β, λ) = (λ)2n
Since thePnm=1m1 can be considered as a lower Riemann sum for the integral
Rm+ 1
Conversely, Equation (58) and Lemma 7.2 imply
h2n(β, λ) > λ2n
Lemma 7.2. Following statements are true.
J(G, A, β) =J(#A, β), G∈ Gc(A)
J(#A, β)6(2k|k|−1/2f
k2(Θβ)−1)#A·(C β + 1) J(#A, β)>
Θ−1
Z
|f(k)|2
sinh(|k|β/2) sinh(Θβ/2)dk
#A/2 ,
where#A = 2m andC = (1/2)k|kk|1/2fk2fk2.
Proof of 7.2. A relabeling of the integration variables yields
J(G, A, β)6Kf
Z
[0,1]2m
Kosc(|t1−t2|, β)Kf(|t2 − t3|, β)· · ·
· · ·Kosc(|t2m−1 − t2m|, β)dt
for Kf := sups∈[0,1]Kf(s, β). We transform due to si := ti − ti+ 1, i 6
2m−1ands2m = t2m, hence−1 6 si 6 1, i = 1, . . . ,2m, since integrating
a positive function we obtain
J(G, A, β)6 Z 1
−1
Kosc(|s|, β)ds
m Z 1
−1
Kf(|s|, β)ds
m−1
· sup
s∈[0,1]
Kf(s, β).
We recall that
Z 1
−1
Kosc(|s|, β)ds = (2Θ)−1
Z 1
−1
cosh(βΘ|s| −Θβ/2)
sinh(Θβ/2) ds = 2(Θ 2β)−1
and
Z 1
−1
Kf(|s|, β)ds =
Z 1
−1
Z cosh(β
|s| |k| −β|k|/2)|f(k)|2 2 sinh(β|k|/2) dk ds = 2
Z
|f(k)|2 β|k| dk.
Usingcoth(x)61 + 1/xand using convexity ofcosh, we obtain
sup
s∈[0,1]
Kf(s, β)6(1/2)
Z
|f(k)|2dk + 1 β
Z
|f(k)|2 |k| dk.
Due to the fact, thatt 7→ Kf(t, β)andt 7→ Kosc(t, β)attain their minima at
Remark 7.3. In the literature there is one criterion for Ωβ0 ∈ dom(e−β/2 (L0+Q)), to our knowledge, that can be applied in this
situa-tion [6]. One has to show that ke−β/2QΩβ
0k < ∞. If we consider the case,
where the criterion holds for ±λ, then the expansion inλ converges,
ke−β/2QΩβ0k2 =
∞
X
n= 0 (λ β)2n
(2n)! ω
β el(q
2n)ωβ f(Φ(f)
2n)
=
∞
X
n= 0 (λ β)2n
(2n)!
(2n)!
n! 2n
2
Kosc(0, β)nKf(0, β)n
=
∞
X
n=0
(λ β)2nΘ−n
2n
n
2−2ncoth(Θβ/2)
Z
|f(k)|2coth(β
|k|/2)dk
n
>
∞
X
n= 0
(λ β)2n(4 Θ)−n Z |f(k)|2dkn.
Obviously, for any value ofλ 6= 0, there is aβ > 0, for whichke−β/2QΩβ
0k < ∞ is not fulfilled.
Acknowledgments:
This paper is part of the author’s PhD requirements. I am grateful to Volker Bach for many useful discussions and helpful advice. The main part of this work was done during the author’s stay at the Institut for Mathematics at the University of Mainz. The work has been partially supported by the DFG (SFB/TR 12).
A
Lemma A.1. Let f, g : {z ∈ C : 0 6 Re(z) 6 α} → C continuous and
analytic in the interior. Moreover, assume that f(t) = g(t) for t∈R. Then f = g.
Proof of A.1. Leth : {z∈C : |Im(z) < α} →Cdefined by
h(z) :=
(
f(z) −g(z), on{z∈C : 0 6 Im(z) < α} f(z)−g(z), on{z∈C : −α < Im(z) < 0}
(61)
Thanks to the Schwarz reflection principlehis analytic. Sinceh(t) = 0for all t∈R, we geth = 0. Hence f =gon{z∈C : 0 6 Re(z) < α}. Since both f andg are continuous, we infer thatf =g on the whole domain.
Lemma A.2. Let H be some self-adjoint operator in H, α > 0 and φ ∈ dom(eα H). Then φ ∈ dom(ez H) for z ∈ {z ∈ C : 0 6 Re(z) 6 α}. z 7→ez Hφ is continuous on {z ∈ C : 0 6 Re(z) 6 α} and analytic in the
Proof of A.2. Due to the spectral calculus we have
Z
e2Rez sd
hφ|Esφi6
Z
(1 +e2α s)dhφ|Esφi=:C 2 1 <∞.
Thus φ∈dom(ez H). Letψ
∈ Hand f(z) =hψ|ez Hφ
i. There is a sequence {ψn} withψn ∈Sm∈N
ran1[|H|6m] andlimn→∞ψn =ψ. We set fn(z) = hψn|ez Hφi. It is not hard to see thatfn is analytic, sinceψn is an analytic
vector for H, and that |fn(z)| 6 C1kψnk and limn→∞fn(z) = f(z). Thus
f is analytic and hence z 7→ ez Hφ is analytic. Thanks to the dominated
convergence theorem the right side of
keznHφ−ezHφk26Z (e2Rezns+e2Rezs
−ezns¯ +zs−ezs¯+zns)dhφ|Esφi (62)
tends to zero forlimn→∞zn =z. This implies the continuity ofz7→ez Hφ.
Lemma A.3. We have forn1+n2>1
Z
∆n 1
Cκ(s)dsn 6
constCn2
(n1+n2)! (n+ 1)(1−2γ)n2−2 (63)
Proof of A.3. We turn now to the integral
Z
∆n 1
Cκ(s)dsn =
Z
∆n 1
(1 −s1 + sn)−α1 nY−1
i=1
(si − si+ 1)−αidsn. (64)
We define fork = 1, . . . ,2n, a change of coordinates bysk = r1 −Pkj= 2rj,
the integral transforms to
Z
Sn
(1 −(r2 +· · ·+rn))−α1 n
Y
i= 2
ri−αidrn (65)
=
Z
Tn−1
(1 −(r2 +· · ·+rn))1−α1 n
Y
i=2
r−iαidrn−1
= Γ(1− α1)
−1Γ 1−γ2n2 Γ n1 + 2n2(1−γ) where S2n := {r∈
R
2n : 06 r
i 6 1, r2 +· · ·+ r2n 6 r1} andT2n−1 :=
{r∈R
2n−1 : 0 6 r
i 6 1, r2 +· · ·+ r2n 6 1}. From the first to the second
formula we integrate over dr1. The last equality follows from [11, Formula 4.635 (4)], hereΓdenotes the Gamma-function.
From Stirling’s formula we obtain
Sincen1 +n2 > 1get Γ(n1+n2+ 1)
Γ(n1+ 2(1−γ)n2) 6 (n+ 1)
2n1+ 2(1−γ)n2 e
−(1−2γ)n2
. (67)
Note thatΓ(n1+n2+ 1) = (n1+n2)!.
Lemma A.4. Let (1 + α(k)−1/2)f1, . . . ,(1 + α(k)−1/2)f2
m ∈ Hph and σ ∈
{+,−}2m. Let a+ = a∗ and a− = a
ωfβ aσ2m(e−σ2ms2mα(k)f2m)· · ·aσ1(e−σ1s1α(k)f1)
=
Z
f2σ2mm (k2m, τ2m)· · ·f1σ1(k1, τ1)ν(dk2m⊗dτ2m),
where ν(dk2m⊗dτ2m) is a measure on (R 3)2m
× {+,−}2m for phonons,
re-spectively on(R 3
× {±})2m
× {+,−}2m for photons, and
ν(dk2m⊗dτ2m)6
X
P∈Z2m
X
τ∈{+,−}2m
Y
{i > j}∈P
δki, kjcoth(β α(ki)/2)
dk2m.
(68)
for f+(k, τ) := f(k)1[τ = +] andf+(k, τ) := f(k)1[τ = −].
Proof of A.4. Sinceωβf is quasi-free, we obtain witha+ := a∗ anda− := a
ωβf aσ2m(e−σ2ms2mα(k)f2
m)· · ·aσ1(e−σ1s1α(k)f1)
= X
P∈Z2
Y
{i, j}∈P i > j
ωfβ aσi(e−σisiα(k)fi)aσj(e−σjsjα(k)fj),
see Equation (12). For the expectation of the so called two point functions we obtain:
ωfβ a+(esiα(k)fi)a+(esjα(k)fj) = 0 = ωβf a(e−siα(k)fi)a(e−sjα(k)fj),
such as
ωfβ a+(exsiα(k)fi)a−(e−x sjα(k)fj) =
Z
fi(k)fj(k) e
x(si−sj)α(k)
eβ α(k)−1 dk ωfβ a−(exsiα(k)fi)a+(e−xsjα(k)fj) =
Z
fj(k)fi(k)
e(β+xsj−xsi)α(k) eβα(k)−1 dk Hence it follows
ωfβ aσ2m(e−σ2ms2mα(k)f2m)· · ·aσ1(e−σ1s1α(k)f1)
=
Z
wheref+(k, τ) := f(k)
1[τ= +]and f−(k, τ) := f(k)
1[τ = −]. ν(d3(2m)k
⊗d2mτ)is a measure on(
R 3)2m
× {+,−}2m, which is defined by
X
P∈Z2m
X
τ∈{+,−}2m
Y
{i > j}∈P
δτ,−τδki, kj (69)
δτ,+
ex(si−sj)α(ki)
eβ α(ki) −1 +δτ,−
e(β−x(si−sj))α(ki) eβ α(ki) −1
dk2m.
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Martin Könenberg Fakultät für Mathematik
und Informatik FernUniversität Hagen Lützowstraße 125
D-58084 Hagen, Germany martin.koenenberg@fernuni
