An Infinite Level Atom coupled to a Heat Bath

Martin Koenenberg^{1}

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 temperature β.

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 (𝔐_{β}, τ) which describes a system of finitely many particles interacting with an infinitely extended bosonic reservoir or heat bath at inverse temperature β. Here, 𝔐_{β} denotes the W^{∗}- algebra of observables and τ is an automorphism-group on 𝔐_{β}, which is defined by

τ_{t}(X) := e^{itL_{Q}} X e^{−itL_{Q}}, X ∈ 𝔐_{β}, t ∈ ℝ.  (1)

In this context, t is the time parameter. L_{Q} is the Liouvillean of the dynamical system at inverse temperature β, Q describes the interaction between particles and heat bath. On the one hand the choice of L_{Q} is motivated by heuristic arguments, which allow to derive the Liouvillean L_{Q} from the Hamiltonian H of the joint system of particles and bosons at temperature zero. On the other hand we ensure that L_{Q} anti-commutes with a certain anti-linear conjugation J, 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:

1Supported by the DFG (SFB/TR 12)

Documenta Mathematica 16 (2011) 177–208

Theorem 1.1. L_{Q}, defined in (16), has a unique self-adjoint realization and τ_{t}(X) ∈ 𝔐_{β} for all t ∈ ℝ and all X ∈ 𝔐_{β}.

The proof follows from Theorem 4.2 and Lemma 5.2. The main difficulty in the proof is, that L_{Q} 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 confined in space. This is reflected in Hypothesis 1, where we assume that the particle Hamiltonian H_{el} possesses a Gibbs state. In the case where H_{el} is a Schrödinger-operator, we give in Remark 2.1 a sufficient condition on the external potential V to ensure the existence of a Gibbs state for H_{el}. Our second theorem is

Theorem 1.2. Assume Hypothesis 1 and that Ω_{0}^{β} ∈ dom(e^{−(β/2)(L_{0}+Q)}). Then there exists a (β, τ)-KMS state ω^{β} on 𝔐_{β}.

This theorem ensures the existence of an equilibrium state on 𝔐_{β} for the dynamical system (𝔐_{β}, τ). Its proof is part of Theorem 5.3 below. Here, L_{0} denotes the Liouvillean for the joint system of particles and bosons, where the interaction part is omitted. Ω_{0}^{β} is the vector representative of the (β, τ)-KMS state for the system without interaction. In a third theorem we study the condition Ω_{0}^{β} ∈ dom(e^{−(β/2)(L_{0}+Q)}):

Theorem 1.3. Assume Hypothesis 1 is fulfilled. Then there are two cases,

1. If 0 ⩽ γ < 1/2 and \underline{η}_{1} (1 + β) ≪ 1, then Ω_{0}^{β} ∈ dom(e^{−β/2 (L_{0}+Q)}).

2. If γ = 1/2 and (1 + β)(\underline{η}_{1} + \underline{η}_{2}) ≪ 1, then Ω_{0}^{β} ∈ dom(e^{−β/2 (L_{0}+Q)}).

Here, γ ∈ [0, 1/2) is a parameter of the model, see (32) and \underline{η}_{1}, \underline{η}_{2} are parameters, which describe the strength of the interaction, see (32). In a last theorem we consider the case where H_{el} = −Δ_{q} +Θ^{2}q^{2} and the interaction Hamiltonian is λ q Φ(f) at temperature zero for λ ≠ 0. Then,

Theorem 1.4. Ω_{0}^{β} is in dom(e^{−β/2 (L_{0} + Q)}) for all β ∈ (0, ∞), whenever

|2Θ^{−1} λ| ∥|k|^{−1/2} f∥_{H_{ph}} < 1.

Furthermore, we show that our strategy can not be improved to obtain a result, which ensures existence for all values of λ, see (60). In the last decade there appeared a large number of mathematical contributions to the theory of open quantum system. Here we only want to mention some of them [3, 6, 8, 9, 10, 13, 14, 15], which consider a related model, in which the particle Hamilton H_{el} is represented as a finite symmetric matrix and the interaction part of the Hamiltonian is linear in annihilation and creation operators. In this case one can prove existence of a β- KMS without any restriction to the strength of the coupling. (In this case we can apply Theorem 1.3 with γ = 0 and \underline{η}_{1} = 0). We can show existence of KMS-states for an infinite 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, that Ω_{0}^{β} ∈ dom(e^{−(β/2)Q}), which implies Ω_{0}^{β} ∈ dom(e^{−(β/2)(L_{0}+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 interaction λ q · Φ(f), whenever (1 + β)λ∥(1 + |k|^{−1/2})f∥ ≪ 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 F_{b}, where

F_{b} := F_{b}[H_{ph}] := ℂ ⊕ ⨁_{n=1}^{∞} H_{ph}^{(n)}, H_{ph}^{(n)} := ⨂_{sym}^{n} H_{ph}.

H_{ph} is either a closed subspace of L^{2}(ℝ^{3}) or L^{2}(ℝ^{3} ×{±}), being invariant under complex conjugation. If phonons are considered we choose H_{ph} = L^{2}(ℝ^{3}), if photons are considered we choose H_{ph} = L^{2}(ℝ^{3} × {±}). In the latter case "+" or "-" labels the polarization of the photon. However, we will write ⟨ f | g ⟩_{H_{ph}} := ∫ \overline{f(k)} g(k) dk for the scalar product in both cases. This is an abbreviation for ∑_{p = ±} ∫ \overline{f(k, p)} g(k, p) dk in the case of photons.

H_{ph}^{(n)} is the n-fold symmetric tensor product of H_{ph}, that is, it contains all square integrable functions f_{n} being invariant under permutations π of the variables, i.e., f_{n}(k_{1}, …, k_{n}) = f_{n}(k_{π(1)}, …, k_{π(n)}). For phonons we have k_{j} ∈ ℝ^{3} and k_{j} ∈ ℝ^{3} × {±} for photons. The wave functions in H_{ph}^{n} are states of n bosons. The vector Ω := (1, 0, …) ∈ F_{b} is called the vacuum. Furthermore we denote the subspace F_{b} of finite sequences with F_{b}^{fin}. On F_{b}^{fin} the creation and annihilation operators, a^{∗}(h) and a(h), are defined for h ∈ H_{ph} by

(a^{∗}(h) f_{n})(k_{1}, …, k_{n+1})  (2)

= (n + 1)^{−1/2} ∑_{i=1}^{n+1} h(k_{i}) f_{n}(k_{1}, …, k_{i−1}, k_{i+1}, …, k_{n+1}),

(a(h) f_{n+1})(k_{1}, …, k_{n})  (3)

= (n + 1)^{1/2} ∫ \overline{h(k_{n+1})} f_{n+1}(k_{1}, …, k_{n+1}) dk_{n+1},

and a^{∗}(h) Ω = h, a(h) Ω = 0. Since a^{∗}(h) ⊂ (a(h))^{∗} and a(h) ⊂ (a^{∗}(h))^{∗}, the operators a^{∗}(h) and a(h) are closable. Moreover, the canonical commutation relations (CCR) hold true, i.e.,

[a(h) , a(h̃)] = [a^{∗}(h) , a^{∗}(h̃)] = 0, [a(h) , a^{∗}(h̃)] = ⟨ h | h̃ ⟩_{H_{ph}}.

Furthermore we define field operator by

Φ(h) := 2^{−1/2} (a(h) + a^{∗}(h)), h ∈ H_{ph}.

It is a straightforward calculation to check that the vectors in F_{b}^{fin} are analytic for Φ(h). Thus, Φ(h) is essentially self-adjoint on F_{b}^{fin}. In the sequel, we will identify a^{∗}(h), a(h) and Φ(h) with their closures. The Weyl operators W (h) are given by W (h) = exp(i Φ(h)). They fulfill the CCR-relation for the Weyl operators, i.e.,

W (h) W (g) = exp(i/2 Im ⟨ h | g ⟩_{H_{ph}}) W (g + h),

which follows from explicit calculations on F_{b}^{fin}. The Weyl algebra W (𝔣) over a subspace 𝔣 of H_{ph} is defined by

W (𝔣) := cl LH{W (g) ∈ B(F_{b}) : g ∈ 𝔣}.  (4)

Here, cl denotes the closure with respect to the norm of B(F_{b}), and "LH" denotes the linear hull. Let α : ℝ^{3} → [0, ∞) be a locally bounded Borel function and dom(α) := {f ∈ H_{ph} : αf ∈ H_{ph}}. Note, that (αf)(k) is given by α(k) f(k, p) for photons. If dom(α) is dense subspace of H_{ph}, α defines a self-adjoint multiplication operator on H_{ph}. In this case, the second quantization dΓ(α) of α is defined by

(dΓ(α) f_{n})(k_{1}, …, k_{n}) := (α(k_{1}) + α(k_{2}) + … + α(k_{n})) f_{n}(k_{1}, …, k_{n})

and dΓ(α) Ω = 0 on its maximal domain.

2.2 Hilbert space and Hamiltonian for the particles

Let H_{el} be a closed, separable subspace of L^{2}(X, dμ), that is invariant under complex conjugation. The Hamiltonian H_{el} for the particle is a self-adjoint operator on H_{el} being bounded from below. We set H_{el,+} := H_{el} − inf σ(H_{el}) + 1. Partly, we need the assumption

Hypothesis 1. Let β > 0. There exists a small positive constant ϵ > 0, and

Tr_{H_{el}}{e^{−(β − ϵ) H_{el}}} < ∞.

The condition implies the existence of a Gibbs state

ω_{el}^{β}(A) = Z^{−1} Tr_{H_{el}}{e^{−β H_{el}}A}, A ∈ B(H_{el}),

for Z = Tr_{H_{el}}{e^{−β H_{el}}}.

Remark 2.1. Let H_{el} = L^{2}(ℝ^{n}, d^{n} x) and H_{el} = −Δ_{x} + V_{1} + V_{2}, where V_{1} is a −Δ_{x}-bounded potential with relative bound a < 1 and V_{2} is in L_{loc}^{2}(ℝ^{n}, d^{n}x). Thus H_{el} is essentially self-adjoint on C_{c}^{∞}(ℝ^{n}). Moreover, if additionally

∫ e^{−(β − ϵ) V_{2}(x)} d^{n} x < ∞,  (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 is H := H_{el} ⊗ F_{b}. The vectors in H are sequences f = (f_{n})_{n∈ℕ_{0}} of wave functions, f_{n} ∈ H_{el} ⊗ H_{ph}^{(n)}, obeying

\underline{k}_{n} ↦ f_{n}(x, \underline{k}_{n}) ∈ H_{ph}^{(n)} for μ- almost every x
x ↦ f_{n}(x, \underline{k}_{n}) ∈ H_{el} for Lebesgue - almost every \underline{k}_{n},

where \underline{k}_{n} = (k_{1}, …, k_{n}). The complex conjugate vector is \overline{f} := ( \overline{f}_{n})_{n∈ℕ_{0}} . Let G^{j} := {G_{k}^{j}}_{k∈ℝ^{3}} , H^{j} := {H_{k}^{j}}_{k∈ℝ^{3}} and F := {F_{k}}_{k∈ℝ^{3}} be families of closed operators on H_{el} for j = 1, …, r. We assume, that dom(F_{k}^{∗}), dom(F_{k}) ⊃ dom(H_{el,+}^{1/2}) and that

k ↦ G_{k}^{j}, (H_{k}^{j}), F_{k}H_{el,+}^{−1/2}, (F_{k})^{∗}H_{el,+}^{−1/2} ∈ B(H_{el})

are weakly (Lebesgue-)measurable. For ϕ ∈ dom(H_{el,+}^{1/2}) we assume that

k ↦ (G_{k}^{j} ϕ)(x), (H_{k}^{j} ϕ)(x), (F_{k} ϕ)(x) ∈ H_{ph},  (6)
k ↦ ((G_{k}^{j})^{∗} ϕ)(x), ((H_{k}^{j})^{∗} ϕ)(x), ((F_{k})^{∗} ϕ)(x) ∈ H_{ph}, for x ∈ X. (7)

Moreover we assume for G⃗ = (G^{1}, …, G^{r}), H⃗ := (H^{1}, …, H^{r}) and F, that

∥G⃗∥_{w} < ∞, ∥H⃗∥_{w} < ∞, ∥F ∥_{w,1/2} < ∞,

where

∥G_{j}∥_{w}^{2} := ∫ (α(k) + α(k)^{−1}) (∥(G_{k}^{j})^{∗}∥_{B(H_{el})}^{2} + ∥G_{k}^{j}∥_{B(H_{el})}^{2}) dk

∥G⃗∥_{w}^{2} := ∑_{j=1}^{r} ∥G_{j}∥_{w}^{2}, ∥F ∥_{w,1/2}^{2} := ∥F H_{el,+}^{−1/2}∥_{w}^{2} + ∥F^{∗}H_{el,+}^{−1/2}∥_{w}^{2}.

We define for f = (f_{n})_{n=0}^{∞} ∈ dom(H_{el,+}^{1/2}) ⊗ F_{b}^{fin} the (generalized) creation operator

(a^{∗}(F) f_{n})(x, k_{1}, …, k_{n+1})  (8)

:= (n + 1)^{−1/2} ∑_{i=1}^{n+1}(F_{k_{i}} f_{n})(x, k_{1}, …, k_{i−1}, k_{i+1}, …, k_{n+1})

and a(F) f_{0}(x) = 0. The (generalized) annihilation operator is

(a(F) f_{n+1})(x, k_{1}, …, k_{n})  (9)
:= (n + 1)^{1/2} ∫ (F_{k_{n+1}}^{∗} f_{n+1})(x, k_{1}, …, k_{n}, k_{n+1}) dk_{n+1}.

Moreover, the corresponding (generalized) field operator is Φ(F) := 2^{−1/2} (a(F) + a^{∗}(F)). Φ(F) is symmetric on dom(H_{el,+}^{1/2}) ⊗ F_{b}^{fin}. The bounds follow directly from Equations (8) and (9).

∥ a(F)H_{el,+}^{−1/2} f∥_{H}^{2} ⩽ ∫ |α(k)|^{−1}∥F_{k}^{∗} H_{el,+}^{−1/2}∥_{B(H_{el})}^{2} dk · ∥dΓ(|α|)^{1/2}f∥_{H}^{2} (10)
∥a^{∗}(F)H_{el,+}^{−1/2} f∥_{H}^{2} ⩽ ∫ |α(k)|^{−1}∥F_{k} H_{el,+}^{−1/2}∥_{B(H_{el})}^{2} dk · ∥dΓ(|α|)^{1/2} f∥_{H}^{2}
+ ∫ ∥F_{k} H_{el,+}^{−1/2}∥_{B(H_{el})}^{2} dk · ∥f∥_{H}^{2}.

For (G_{k})^{j}, (H_{k})^{j} ∈ B(H_{el}), the factor H_{el,+}^{−1/2} can be omitted. The Hamiltonians for the non-interacting, resp. interacting model are

Definition 2.2. On dom(H_{el}) ⊗ dom(dΓ(α)) ∩ F_{b}^{fin} we define

H_{0} := H_{el} ⊗ 1 + 1 ⊗ dΓ(α),  H := H_{0} + W,  (11)

where W := Φ(G⃗) Φ(H⃗) + h. c. + Φ(F) and Φ(G⃗) Φ(H⃗) := ∑_{j=1}^{r} Φ(G^{j}) Φ(H^{j}). The abbreviation "h.c." means the formal adjoint operator of Φ(G⃗) Φ(H⃗).

We give examples for possible configurations: Let γ ∈ ℝ be a small coupling parameter. ▶ The Nelson Model: H_{el} ⊂ L^{2}(ℝ^{3N}), H_{el} := −Δ + V, H_{ph} = L^{2}(ℝ^{3}) and α(k) = |k|. The form factor is F_{k} = γ ∑_{ν = 1}^{N} e^{−i kx_{ν}} |k|^{−1/2} 1[ |k| ⩽ κ], x_{ν} ∈ ℝ^{3} and H^{j}, G^{j} = 0. ▶ The Standard Model of Nonrelativistic QED: H_{el} ⊂ L^{2}(ℝ^{3N}), H_{el} := −Δ + V, H_{ph} = L^{2}(ℝ^{3} ×{±}) and α(k) = |k|. The form factors are

F_{k} = 4γ^{3/2} π^{−1/2} ∑_{ν=1}^{N}(−i∇_{x_{ν}} · ϵ(k, p))e^{−i γ^{1/2}kx_{ν}} (2|k|)^{−1/2} 1[ |k| ⩽ κ] + h. c.,

G_{k}^{i, ν} = H_{k}^{i, ν} = 2γ^{3/2} π^{−1/2} ϵ_{i}(k, p) e^{−i γ^{1/2} kx_{ν}} (2|k|)^{−1/2} 1[ |k| ⩽ κ]

for i = 1, 2, 3, ν = 1, …, N, x_{ν} ∈ ℝ^{3} and k = (k, p) ∈ ℝ^{3} × {±}. ϵ_{i}(k, ±) ∈ ℝ^{3} are polarization vectors. ▶ The Pauli-Fierz-Model: H_{el} ⊂ L^{2}(ℝ^{3N}), H_{el} := −Δ + V, H_{ph} = L^{2}(ℝ^{3}) or H_{ph} = L^{2}(ℝ^{3} × {±}), and α(k) = |k|. The form factor is F_{k} = γ ∑_{ν = 1}^{N} 1[ |k| ⩽ κ] k · x_{ν} and G_{k}^{j} = H_{k}^{j} = 0

3 The Representation π

In order to describe the particle system at inverse temperature β we introduce the algebraic setting. For 𝔣 = {f ∈ H_{ph} : α^{−1/2}f ∈ H_{ph}} we define the algebra of observables by

𝔄 = B(H_{el}) ⊗ W(𝔣).

For elements A ∈ 𝔄 we define τ̃_{t}^{0}(A) := e^{i t H_{0}} A e^{−i t H_{0}} and τ̃_{t}^{g}(A) := e^{i t H}Ae^{− i t H}. We first discuss the model without interaction.

3.1 The Representation π_{f}

The time-evolution for the Weyl operators is given by

e^{i t Ȟ} W(f) e^{−i t Ȟ} = W(e^{i t α} f).

For this time-evolution an equilibrium state ω_{f}^{β} is defined by

ω_{f}^{β}(W(f)) = ⟨ f | (1 + 2 ϱ_{β}) f ⟩_{H_{ph}},

where ϱ_{β}(k) = ( exp(β α(k)) − 1)^{−1}. It describes an infinitely extended gas of bosons with momentum density ϱ_{β} at temperature β. Since ω_{f}^{β} is a quasifree state on the Weyl algebra, the definition of ω_{f}^{β} extends to polynomials of creation and annihilation operators. One has

ω_{f}^{β}(a(f)) = ω_{f}^{β}(a^{∗}(f)) = ω_{f}^{β}(a(f) a(g)) = ω_{f}^{β}(a^{∗}(f) a^{∗}(g)) = 0,
ω_{f}^{β}(a^{∗}(f) a(g)) = ⟨ g | ϱ_{β} f ⟩_{H_{ph}}.

For polynomials of higher degree one can apply Wick’s theorem for quasi-free states, i.e.,

ω_{f}^{β}(a^{σ_{2m}}(f_{2m}) ⋯ a^{σ_{1}}(f_{1})) = ∑_{P ∈Z_{2}} ∏_{i>j^{{i,j}∈P}} ω_{f}^{β}(a^{σ_{i}}(f_{i}) a^{σ_{j}}(f_{j})),  (12)

where a^{σ_{k}} = a^{∗} or a^{σ_{k}} = a for k = 1, …, 2m. Z_{2} are the pairings, that is

P ∈ Z_{2}, iff P = {Q_{1}, …, Q_{m}}, #Q_{i} = 2 and ⋃_{i=1}^{m} Q_{i} = {1, …, 2m}.

The Araki-Woods isomorphism π_{f} : W(𝔣) → B(F_{b} ⊗ F_{b}) is defined by

π_{f}[W(f)] := W_{β}(f) := exp(i Φ_{β}(f)),
Φ_{β}(f) := Φ((1 + ϱ_{β})^{1/2} f) ⊗ 1 + 1 ⊗ Φ(ϱ_{β}^{1/2} \overline{f}).

The vector Ω_{f}^{β} := Ω ⊗ Ω fulfills

ω_{f}^{β}(W(f)) = ⟨ Ω_{f}^{β} | π_{f}[W(f)] Ω_{f}^{β} ⟩.  (13)

3.2 The representation π^{el}

The particle system without interaction has the observables B(H_{el}), the states are defined by density operators ρ, i.e., ρ ∈ B(H_{el}), 0 ⩽ ρ, Tr{ρ} = 1. The expectation of A ∈ B(H_{el}) in ρ at time t is

Tr{ ρ e^{i t H_{el}} A e^{−i t H_{el}}}.

Since ρ is a compact, self-adjoint operator, there is an ONB (ϕ_{n})_{n} of eigenvectors, with corresponding (positive) eigenvalues (p_{n})_{n}. Let

σ(x, y) = ∑_{n=1}^{∞} p_{n}^{1/2} ϕ_{n}(x) \overline{ϕ_{n}(y)} ∈ H_{el} ⊗ H_{el}.  (14)

For ψ ∈ H_{el} we define σ ψ := ∫ σ(x, y) ψ(y) dμ(y). Obviously, σ is an operator of Hilbert-Schmidt class. Note, \overline{σ} ψ := \overline{σ \overline{ψ}} has the integral kernel \overline{σ(x, y)}. It is a straightforward calculation to verify that

Tr{ρ e^{i t H_{el}} A e^{−i t H_{el}}} = ⟨ e^{−i t L_{el}} σ | (A ⊗ 1) e^{−i t L_{el}}σ ⟩_{H_{el}⊗H_{el}},

where L_{el} = H_{el}⊗1−1⊗\overline{H}_{el}. This suggests the definition of the representation

π^{el} : B(H_{el}) → B(H_{el} ⊗ H_{el}), A ↦ A ⊗ 1.

Now, we define the representation map for the joint system by

π : 𝔄 → B(K), π := π_{el} ⊗ π_{f},

where K := H_{el} ⊗ H_{el} ⊗ F_{b} ⊗ F_{b}. Let 𝔐_{β} := π[𝔄]′′ be the enveloping W^{∗}- algebra, here π[𝔄]′ denotes the commutant of π[𝔄], and π[𝔄]′′ the bicommutant. We set D := U_{1} ⊗ \overline{U_{1}} ⊗ C, where C is a subspace of vectors in F_{b}^{fin} ⊗ F_{b}^{fin}, with compact support, and U_{1} := ∪_{n=1}^{∞} ran 𝟙[H_{el} ⩽ n]. On D the operator L_{0}, given by

L_{0} := L_{el} ⊗ 1 + 1 ⊗ L_{f}, on K,
L_{f} := dΓ(α) ⊗ 1 − 1 ⊗ dΓ(α), on F_{b} ⊗ F_{b},

is essentially self-adjoint and we can define

τ_{t}^{0}(X) := e^{i t L_{0}} X e^{−i t L_{0}} ∈ 𝔐_{β}, X ∈ 𝔐_{β}, t ∈ ℝ,

It is not hard to see, that

π[τ̃_{t}^{0}(A)] = τ_{t}^{0}(π[A]), A ∈ 𝔄, t ∈ ℝ

On K a we introduce a conjugation by

J (ϕ_{1} ⊗ ϕ_{2} ⊗ ψ_{1} ⊗ ψ_{2}) = \overline{ϕ_{2}} ⊗ \overline{ϕ_{1}} ⊗ \overline{ψ_{2}} ⊗ \overline{ψ_{1}}.

It is easily seen, that J L_{0} = −L_{0} J. In this context one has 𝔐′_{β} = J 𝔐_{β} J, see for example [4]. In the case, where H_{el} fulfills Hypothesis 1, we define the vector representative Ω_{el}^{β} ∈ H_{el} ⊗ H_{el} of the Gibbs state ω_{el}^{β} as in (14) for ρ = e^{−βH_{el}} Z^{−1}.

Theorem 3.1. Assume Hypothesis 1 is fulfilled. Then, Ω_{0}^{β} := Ω_{el}^{β} ⊗ Ω_{f}^{β} is a cyclic and separating vector for 𝔐_{β}. e^{−β/2L_{0}} is a modular operator and J is the modular conjugation for Ω_{0}^{β}, that is

XΩ_{0}^{β} ∈ dom(e^{−β/2 L_{0}}), J X Ω_{0}^{β} = e^{−β/2 L_{0}} X^{∗} Ω_{0}^{β}  (15)

for all X ∈ 𝔐_{β} and L_{0} Ω_{0}^{β} = 0. Moreover,

ω_{0}^{β}(X) := ⟨ Ω_{0}^{β} | X Ω_{0}^{β} ⟩_{K}, X ∈ 𝔐_{β}

is a (τ^{0}, β)-KMS-state for 𝔐_{β}, i.e., for all X, Y ∈ 𝔐_{β} exists F_{β}(X, Y, ·), analytic in the strip S_{β} = {z ∈ ℂ : 0 < Im z < β}, continuous on the closure and taking the boundary conditions

F_{β}(X, Y, t) = ω_{0}^{β}(X τ_{t}^{0}(Y))
F_{β}(X, Y, t + i β) = ω_{0}^{β}(τ_{t}^{0}(Y) X)

For a proof see [14].

4 The Liouvillean L_{Q}

In this and the next section we will introduce the Standard Liouvillean L_{Q} for a dynamics τ on 𝔐_{β}, 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 part W of the corresponding Hamiltonian by means of formal arguments, which we will not give here. In a first step we prove self-adjointness of L_{Q} and of other Liouvilleans. A main difficulty stems from the fact, that L_{Q} 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, that τ_{t}(X) ∈ 𝔐_{β} for X ∈ 𝔐_{β}, is given in Lemma 5.2. Assuming Ω_{0}^{β} ∈ dom(e^{−β/2(L_{0}+Q)}) we can ensure existence of a (τ, β)-KMS state ω^{β}(X) = ⟨Ω^{β} |X Ω^{β}⟩ · ∥Ω^{β}∥^{−2} on 𝔐_{β}, where Ω^{β} = e^{−β/2(L_{0}+Q)}Ω_{0}^{β}. Moreover, we can show that e^{−βL_{Q}} is the modular operator for Ω^{β} 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, that Q is self-adjoint and that Ω_{0}^{β} ∈ dom(e^{−βQ}). For this reason we need to introduce an additional approximation Q_{N} of Q, which is self-adjoint and affiliated with 𝔐_{β}, see Lemma 5.1. The interaction on the level of Liouvilleans between particles and bosons is given by Q , where

Q := Φ_{β}(G⃗) Φ_{β}(H⃗) + h. c. + Φ_{β}(F), Φ_{β}(G⃗) Φ_{β}(H⃗) := ∑_{j=1}^{r} Φ_{β}(G^{j}) Φ_{β}(H^{j}).

For each family K = {K_{k}}_{k} of closed operators on H_{el} with ∥K∥_{w,1/2} < ∞ we set

Φ_{β}(K) := (a^{∗}((1 + ϱ_{β})^{1/2} K) ⊗ 1 + 1 ⊗ a^{∗}(ϱ_{β}^{1/2} K^{∗})) + h. c. .

Here, K_{k} acts as K_{k} ⊗ 1 on H_{el} ⊗ H_{el}. A Liouvillean, that describes the dynamics of the joint system of particles and bosons is the so-called Standard Liouvillean

L_{Q} ϕ := (L_{0} + Q − Q^{J}) ϕ,  ϕ ∈ D,  (16)

which is distinguished by J L_{Q} = −L_{Q} J. For an operator A, acting on K, the symbol A^{J} is an abbreviation for J A J. An important observation is, that [Q , Q^{J}] = 0 on D. Next, we define four auxiliary operators on D

L_{a}^{(1)} := (H_{el,+} ⊗ 1 + 1 ⊗ \overline{H}_{el,+}) ⊗ 1 + 1 ⊗ L_{f,a} + 1  (17)
L_{a}^{(2)} := H_{el,+}^{Q} + (H_{el,+}^{Q})^{J} + c_{1}1 ⊗ L_{f,a} + c_{2}
L_{a}^{(3)} := H_{el,+}^{Q} + (H_{el,+})^{J} + c_{1}1 ⊗ L_{f,a} + c_{2}
L_{a}^{(4)} := H_{el,+} ⊗ 1 + (H_{el,+}^{Q})^{J} + c_{1}1 ⊗ L_{f,a} + c_{2},

where L_{f,a} is an operator on F_{b} ⊗ F_{b} and H_{el,+}^{Q} acts on K. Furthermore,

L_{f,a} = dΓ(1 + α) ⊗ 1 + 1 ⊗ dΓ(1 + α) + 1,
L_{el,a} = H_{el,+} ⊗ 1 + 1 ⊗ \overline{H}_{el,+} H_{el,+}^{Q} := H_{el,+} ⊗ 1 + Q.

Obviously, L_{a}^{(i)}, i = 1, 2, 3, 4 are symmetric operators on D.

Lemma 4.1. For sufficiently large values of c_{1}, c_{2} ⩾ 0 we have that L_{a}^{(i)}, i = 1, 2, 3, 4 are essentially self-adjoint and positive. Moreover, there is a constant c_{3} > 0 such that

c_{3}^{−1} ∥L_{a}^{(1)} ϕ∥ ⩽ ∥L_{a}^{(i)} ϕ∥ ⩽ c_{3} ∥L_{a}^{(1)} ϕ∥,  ϕ ∈ dom(L_{a}^{(1)}).  (18)

Proof. Let a, a′ ∈ {l, r} and K_{i}, i = 1, 2 be families of bounded operators with ∥K_{i}∥_{w} < ∞. Let Φ_{l}(K_{i}) = Φ(K_{i}) ⊗ 1 and Φ_{r}(K_{i}) := 1 ⊗ Φ(K_{i}). We have for ϕ ∈ D

∥ Φ_{a}(ηK_{1}) Φ_{a′}(η′K_{2}) ϕ ∥ ⩽ const ∥ L_{f,a} ϕ ∥  (19)
∥ Φ_{a}(ηF) ϕ ∥ ⩽ const ∥(L_{el,a})^{1/2}(L_{f,a})^{1/2} ϕ ∥,

where η, η′ ∈ {(1+ϱ_{β})^{1/2}, ϱ_{β}^{1/2}}. Note, that the estimates hold true, if Φ_{a}(ηK_{i}) or Φ_{a}(ηF) are replaced by Φ_{a}(ηK_{i})^{J} or Φ_{a}(ηF)^{J}. Thus, we obtain for sufficiently large c_{1} ≫ 1, depending on the form-factors, that

∥Q ϕ∥ + ∥Q^{J} ϕ∥ ⩽ 1/2 ∥(L_{el,a} + c_{1} L_{f,a}) ϕ∥.  (20)

By the Kato-Rellich-Theorem ( [17], Thm. X.12) we deduce that L_{a}^{(i)} is selfadjoint on dom(L_{el,a} + c_{1} L_{f,a}), bounded from below and that L_{el,a} + c_{1} L_{f,a} is L_{a}^{(i)}-bounded for every c_{2} ⩾ 0 and i = 2, 3, 4. In particular, D is a core of L_{a}^{(i)}. The proof follows now from ∥L_{a}^{(i)} ϕ∥ ⩽ ∥(L_{el,a} +c_{1} L_{f,a}) ϕ∥ ⩽ c_{1} ∥L_{a}^{(1)} ϕ∥ for ϕ ∈ D.

Theorem 4.2. The operators

L_{0}, L_{Q} = L_{0} + Q − Q^{J} , L_{0} + Q, L_{0} − Q^{J} ,  (21)

defined on D, are essentially self-adjoint. Every core of L_{a}^{(1)} is a core of the operators in line (21).

Proof. We restrict ourselves to the case of L_{Q}. We check the assumptions of Nelson’s commutator theorem ([17], Thm. X.37). By Lemma 4.1 it suffices to show ∥L_{Q}ϕ∥ ⩽ const ∥L_{a}^{(1)}ϕ∥ and |⟨ L_{Q}ϕ | L_{a}^{(2)}ϕ⟩ − ⟨ L_{a}^{(2)}ϕ| L_{Q}ϕ⟩| ⩽ const ∥(L_{a}^{(1)})^{1/2}ϕ∥^{2} for ϕ ∈ D. The first inequality follows from Equation (20).

To verify the second inequality we observe

|⟨ L_{Q} ϕ | L_{a}^{(2)} ϕ ⟩ − ⟨ L_{a}^{(2)} ϕ | L_{Q} ϕ ⟩|  (22)
⩽ c_{1}|⟨ Qϕ | L_{f,a}ϕ⟩ − ⟨ L_{f,a}ϕ | Qϕ⟩|
+ c_{1}|⟨ Q^{J} ϕ | L_{f,a}ϕ ⟩ − ⟨ L_{f,a}ϕ | Q^{J} ϕ ⟩|
+ |⟨ L_{f}ϕ| Qϕ ⟩ − ⟨ Qϕ | L_{f}ϕ ⟩| + |⟨ L_{f}ϕ | Q^{J} ϕ ⟩ − ⟨ Q^{J} ϕ | L_{f}ϕ ⟩|,

where we used, that [H_{el,+}^{Q}, (H_{el,+}^{Q})^{J} ] = 0. Let K_{i} ∈ {G_{j}, H_{j}} and η, η′ ∈ {ϱ^{1/2}, (1 + ϱ)^{1/2}}. We remark, that

[Φ_{a}(η K_{1}) Φ_{a′}(η′ K_{2}) , L_{f,a}] = i Φ_{a}(i (1 + α) η K_{1}) Φ_{a′}v(η′ K_{2})  (23)
+ i Φ_{a}(η K_{1}) Φ_{a′}(i (1 + α) η′ K_{2})
[Φ_{a}(η F) , L_{f,a}] = i Φ_{a}(i (1 + α) η F).

Hence, for ϕ ∈ dom(L_{a}^{(2)}), we have by means of (10) that

|⟨ ϕ | [Φ_{a}(ηK_{1}) Φ_{a′}(η′K_{2}), L_{f,a}]ϕ ⟩| ⩽ const ∥L_{f,a}^{1/2}ϕ∥^{2}  (24)
|⟨ϕ | [Φ_{a}(ηF), L_{f,a}]ϕ ⟩| ⩽ const ∥L_{f,a}^{1/2}ϕ∥ ∥(L_{el,a})^{1/2}ϕ∥.

Thus, (24) is bounded by a constant times ∥(L_{a}^{(1)})^{1/2}ϕ∥^{2}. The essential selfadjointness of L_{Q} follows now from estimates analog to (23) and (24), where L_{f,a} is replaced by L_{f} in (23) and in the left side of (24). For L_{0}+Q and L_{0}−Q^{J} one has to consider the commutator with L_{a}^{(3)} and L_{a}^{(4)}, respectively.

Remark 4.3. In the same way one can show, that H is essentially self-adjoint on any core of H_{1} := H_{el} + dΓ(1 + α), even if H is not bounded from below.

5 Regularized Interaction and Standard Form of 𝔐_{β}

In this subsection a regularized interaction Q_{N} is introduced:

Q_{N} := {Φ_{β}(G⃗_{N}) Φ_{β}(H⃗_{N}) + h. c. } + Φ_{β}(F_{N}).  (25)

The regularized form factors G⃗_{N}, H⃗_{N}, F_{N} are obtained by multiplying the finite rank projection P_{N} := 1[H_{el} ⩽ N] from the left and the right. Moreover, an additional ultraviolet cut-off 1[α ⩽ N], considered as a spectral projection, is added. The regularized form factors are

G⃗_{N}(k) := 1[α ⩽ N] P_{N} G⃗(k) P_{N},  H⃗_{N}(k) := 1[α ⩽ N] P_{N} H⃗(k) P_{N},
F_{N}(k) := 1[α ⩽ N] P_{N} F (k) P_{N}.

Lemma 5.1. i) Q_{N} is essentially self-adjoint on D ⊂ dom(Q_{N}). Q_{N} is affiliated with 𝔐_{β}, i.e,. Q_{N} is closed and

X′ Q_{N} ⊂ Q_{N} X′, ∀ X′ ∈ 𝔐′_{β}.

ii) L_{0} + Q_{N}, L_{0} − J Q_{N}J and L_{0} + Q_{N} − J Q_{N}J converges in the strong resolvent sense to L_{0} + Q, L_{0} − J QJ and L_{0} + Q − J QJ, respectively.

Proof. Let Q_{N} be defined on D. With the same arguments as in the proof of Theorem 4.2 we obtain

∥Q_{N}ϕ∥ ⩽ C∥L_{f,a}ϕ∥, |⟨ Q_{N}ϕ | L_{f,a}ϕ⟩ − ⟨ L_{f,a}ϕ | Q_{N}ϕ⟩| ⩽ C∥(L_{f,a})^{1/2}ϕ∥^{2},

for ϕ ∈ D and some constant C > 0, where we have used that ∥F_{N}∥_{w} < ∞. Thus, from Theorem 4.2 and Nelson’s commutator theorem we obtain that D is a common core for Q_{N}, L_{0} + Q_{N}, L_{0} − Q_{N}^{J}, L_{0} + Q_{N} − Q_{N}^{J} and for the operators in line (21). A straightforward calculation yields

lim_{N→∞} Q_{N}ϕ = Qϕ, lim_{N→∞} J Q_{N}J ϕ = J QJ ϕ ∀ ϕ ∈ D.

Thus statement ii) follows, since it suffices to check strong convergence on the common core D, see [16, Theorem VIII.25 a)]. Let N_{f} := dΓ(1)⊗1+1⊗dΓ(1) be the number-operator. Since dom(N_{f}) ⊃ D and W_{β}(f)^{J} : dom(N_{f}) → dom(N_{f}), see [4], we obtain

Q_{N}(A ⊗ 1 ⊗ W_{β}(f))^{J} ϕ = (A ⊗ 1 ⊗ W_{β}(f))^{J} Q_{N}ϕ  (26)

for A ∈ B(H_{el}), f ∈ 𝔣 and ϕ ∈ D. By closedness of Q_{N} and density arguments the equality holds for ϕ ∈ dom(Q_{N}) and X ∈ 𝔐_{β} instead of A ⊗ 1 ⊗ W_{β}(f). Thus Q_{N} is affiliated with 𝔐_{β} and therefore e^{i tQ_{N}} ∈ 𝔐_{β} for t ∈ ℝ.

Lemma 5.2. We have for X ∈ 𝔐_{β} and t ∈ ℝ

τ_{t}(X) = e^{it(L_{0}+Q)} X e^{it(L_{0}+Q)}, τ_{t}^{0}(X) = e^{it(L_{0}−Q^{J})} X e^{it(L_{0}−Q^{J})} (27)

Moreover, τ_{t}(X) ∈ 𝔐_{β} for all X ∈ 𝔐_{β} and t ∈ ℝ, such as

E_{Q}(t) := e^{i t (L_{0} + Q)} e^{−i t L_{0}} = e^{i t L_{Q}} e^{−i t (L_{0}−Q^{J})} ∈ 𝔐_{β}.

Proof. First, we prove the statement for Q_{N}, since Q_{N} is affiliated with 𝔐_{β} and therefore e^{itQ_{N}} ∈ 𝔐_{β}. We set

τ̂_{t}^{N}(X) = e^{i t (L_{0} + Q_{N})} X e^{−i t (L_{0} + Q_{N})}, τ̂_{t}(X) = e^{i t (L_{0} + Q)} X e^{−i t (L_{0} + Q)} (28)

On account of Lemma 5.1 and Theorem 4.2 we can apply the Trotter product formula to obtain

τ̂_{t}^{N}(X) = w-lim_{n → ∞} (e^{i \frac{t}{n} L_{0}} e^{i \frac{t}{n} Q_{N}} )^{n}X(e^{−i \frac{t}{n} Q_{N}} e^{−i \frac{t}{n} L_{0}})^{n}

= w-lim_{n→∞} τ 0_{\frac{t}{n}} (e^{i \frac{t}{n} Q_{N}} ⋯ τ 0_{\frac{t}{n}} (e^{i \frac{t}{n} Q_{N}} X e^{−i \frac{t}{n} Q_{N}}) ⋯ e^{−i \frac{t}{n} Q_{N}} ).

Since e^{i \frac{t}{n} Q_{N}} , X ∈ 𝔐_{β} and since τ^{0} leaves 𝔐_{β} invariant, τ̂_{t}^{N}(X) is the weak limit of elements of 𝔐_{β}, and hence τ̂_{t}^{N}(X) ∈ 𝔐_{β}. Moreover,

τ̂_{t}(X) = w-lim_{N → ∞} τ̂_{t}^{N}(X) ∈ 𝔐_{β}.

For E_{N}(t) := e^{i t (L_{0} + Q_{N})}e^{−i t L_{0}} ∈ B(K) we obtain

e^{it(L_{0} + Q_{N})}e^{−i t L_{0}} = s-lim_{n→∞} (e^{i \frac{t}{n} L_{0}}e^{i \frac{t}{n} Q_{N}} )^{n} e^{−i t L_{0}}

= s-lim_{n→∞} τ 0_{\frac{t}{n}} (e^{i \frac{t}{n} Q_{N}})τ 0_{\frac{2t}{n}} (e^{i \frac{t}{n} Q_{N}}) ⋯ τ 0_{\frac{nt}{n}} (e^{i \frac{t}{n} Q_{N}}) ∈ 𝔐_{β}.

By virtue of Lemma 5.1 we get E_{Q}(t) := e^{i t (L_{0} + Q)} e^{−i t L_{0}} = w-lim_{N → ∞} E_{N}(t) ∈ 𝔐_{β}. Since J leaves D invariant and thanks to Lemma 5.1, we deduce, that D is a core of J Q_{N}J. Moreover, we have e^{−itQ_{N}^{J}} = J e^{itQ_{N}} J ∈ 𝔐′_{β}. Since we have shown, that τ̂^{N} leaves 𝔐_{β} invariant, we get

τ_{t}^{N}(X) = w-lim_{n→∞}(e^{i \frac{t}{n} (L_{0}+Q_{N})}e^{i \frac{t}{n} (−Q_{N}^{J})})^{n}X (e^{−i \frac{t}{n} (−Q_{N}^{J})} e^{−i \frac{t}{n} (L_{0}+Q_{N})})^{n}

= w-lim_{n→∞} τ̂ N_{\frac{t}{n}}(e^{−i \frac{t}{n} Q_{N}^{J}} ⋯ τ̂ N_{\frac{t}{n}}(e^{−i \frac{t}{n} Q_{N}^{J}} X e^{i \frac{t}{n} Q_{N}^{J}}) ⋯ e^{i \frac{t}{n} Q_{N}^{J}} )

= τ̂_{t}^{N}(X).

Thanks to Lemma 5.1 we also have

τ_{t}(X) = w-lim_{n→∞} τ_{t}^{N}(X) = w-lim_{ℕ→∞} τ̂_{t}^{N}(X) = τ̂_{t}(X).  (29)

The proof of τ_{t}^{0}(X) = e^{it(L_{0}−Q^{J})} X e^{it(L_{0}−Q^{J})} follows analogously. Using the Trotter product formula we obtain

e^{it(L_{0}+Q_{N})} e^{−itL_{0}} = s-lim_{n → ∞} (e^{i \frac{t}{n} L_{0}}e^{i \frac{t}{n} Q_{N}} )^{n} e^{−itL_{0}}
= s-lim_{n → ∞} τ 0_{\frac{t}{n}} (e^{i \frac{t}{n} Q_{N}})τ 0_{\frac{2t}{n}} (e^{i \frac{t}{n} Q_{N}}) ⋯ τ 0_{\frac{nt}{n}} (e^{i \frac{t}{n} Q_{N}})
= s-lim_{n → ∞} (e^{i \frac{t}{n} (L_{0}−Q_{N}^{J})}e^{i \frac{t}{n} Q_{N}} )^{n} e^{−it(L_{0}−Q_{N}^{J})}
= e^{it(L_{0}+Q_{N} −J Q_{N} J)} e^{−it(L_{0}−Q_{N}^{J})}.

By strong resolvent convergence we may deduce E(t) = e^{itL_{Q}} e^{−it(L_{0}−Q^{J})} .

Let C be the natural positive cone associated with J and Ω_{0}^{β} and let 𝔐_{β}^{ana} be the τ-analytic elements of 𝔐_{β}, (see [4]).

Theorem 5.3. Assume Hypothesis 1 and Ω_{0}^{β} ∈ dom(e^{−β/2 (L_{0} + Q)}). Let Ω^{β} := e^{−β/2 (L_{0} + Q)} Ω_{0}^{β}. Then

J Ω^{β} = Ω^{β},  Ω^{β} = e^{β/2 (L_{0} − Q^{J})} Ω_{0}^{β},  (30)
L_{Q} Ω^{β} = 0, J X^{∗} Ω^{β} = e^{−β/2 L_{Q}}X Ω^{β}, ∀ X ∈ 𝔐_{β}

Furthermore, Ω^{β} is separating and cyclic for 𝔐_{β}, and Ω^{β} ∈ C. The state ω^{β} is defined by

ω^{β}(X) := ∥Ω^{β}∥^{−2} ⟨Ω^{β} |X Ω^{β}⟩, X ∈ 𝔐_{β}

is a (τ, β)-KMS state on 𝔐_{β}.

Proof. First, we define Ω(z) = e^{−z (L_{0}+Q)} Ω_{0}^{β} for z ∈ ℂ with 0 ⩽ Re z ⩽ β/2. Since Ω_{0}^{β} ∈ dom(e^{−β/2 (L_{0}+Q)}), Ω(z) is analytic on S_{β/2} := {z ∈ ℂ : 0 < Re (z) < α} and continuous on the closure of S_{β/2}, see Lemma A.2 below.

▶ Proof of J Ω(β/2) = Ω(β/2): We pick ϕ ∈ ⋃_{n∈ℕ} ran 1[|L_{0}| ⩽ n]. Let f(z) := ⟨ ϕ | J Ω(\overline{z}) ⟩ and g(z) := ⟨ e^{−(β/2 − \overline{z}) L_{0}} ϕ | e^{−z (L_{0} + Q)} Ω_{0}^{β} ⟩. Both f and g are analytic on S_{β/2} and continuous on its closure. Thanks to Lemma 5.2 we have E_{Q}(t) ∈ 𝔐_{β}, and hence

f(it) = ⟨ ϕ | J E_{Q}(t) Ω_{0}^{β} ⟩ = ⟨ ϕ | e^{−β/2 L_{0}} E_{Q}(t)^{∗} Ω_{0}^{β} ⟩ = g(i t), t ∈ ℝ.

By Lemma A.1, f and g are equal, in particular in z = β/2. Note that ϕ is any element of a dense subspace.

▶ Proof of Ω_{0}^{β} ∈ dom(e^{β/2 (L_{0}−Q^{J})}) and Ω(β/2) = e^{β/2 (L_{0}−Q^{J})} Ω_{0}^{β}: Let ϕ ∈ ⋃_{n∈ℕ} ran 1[|L_{0} − Q^{J} | ⩽ n]. We set g(z) := ⟨ e^{\overline{z}(L_{0} −Q^{J})} ϕ | e^{−z L_{0}} Ω_{0}^{β} ⟩. Since E_{Q}(t)^{J} = e^{i t (L_{0}−Q^{J})} e^{−i t L_{0}}, g coincides for z = i t with f(z) := ⟨ ϕ | J Ω(\overline{z}) ⟩. Hence they are equal in z = β/2. The rest follows since e^{β/2 (L_{0}−Q^{J})} is self-adjoint.

▶ Proof of L_{Q} Ω(β/2) = 0: Choose ϕ ∈ ⋃_{n∈ℕ} ran 1[|L_{Q}| ⩽ n]. We define g(z) := ⟨ e^{−\overline{z}L_{Q}}ϕ | e^{z (L_{0}−Q^{J})} Ω_{0}^{β} ⟩ and f(z) := ⟨ ϕ | Ω(z) ⟩ for z in the closure of S_{β/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/2 L_{Q}}) and e^{−β/2L_{Q}} Ω(β/2) = Ω(β/2). We conclude that L_{Q} Ω(β/2) = 0.

▶ Proof of J X^{∗} Ω(β/2) = e^{−β/2L_{Q}} X Ω(β/2), ∀ X ∈ 𝔐_{β}: Fore A ∈ 𝔐_{β}^{ana} we have, that

J A^{∗} Ω(−it) = J A^{∗} E_{Q}(t) Ω_{0}^{β} = e^{−β/2L_{0}} E_{Q}(t)^{∗}AΩ_{0}^{β}
= e^{−(β/2 − i t) L_{0}} e^{−it(L_{0}+Q)}A Ω_{0}^{β}
= e^{−(β/2−it)L_{0}} τ_{−t}(A) e^{−it(L_{0}+Q)} Ω_{0}^{β}.

Let ϕ ∈ ⋃_{n∈ℕ} ran 1[|L_{0}| ⩽ n]. We define f(z) = ⟨ ϕ | J A^{∗} Ω(\overline{z}) ⟩ and g(z) = ⟨ e^{−(β/2−\overline{z})L_{0}} ϕ | τ_{iz}(A) Ω(z) ⟩. Since f and g are analytic and equal for z = it, we have J A^{∗}Ω(β/2) = τ_{i β/2}(A) Ω(β/2). To finish the proof we pick ϕ ∈ ⋃_{n∈ℕ} ran 1[|L_{Q}| ⩽ n], and set f(z) := ⟨ ϕ | τ_{iz}(A) Ω(β/2) ⟩ and g(z) := ⟨ e^{−\overline{z}L_{Q}}ϕ | AΩ(β/2) ⟩. For z = i t we see

g(it) = ⟨ ϕ | e^{−itL_{Q}} A e^{itL_{Q}}Ω(β/2) ⟩ = ⟨ ϕ | τ_{−t}(A) Ω(β/2) ⟩ = f(i t).

Hence A Ω(β/2) ∈ dom(e^{−β/2 L_{Q}}) and J A^{∗}Ω(β/2) = e^{−β/2L_{Q}}AΩ(β/2). Since 𝔐_{β}^{ana} is dense in the strong topology, the equality holds for all X ∈ 𝔐_{β}.

▶ Proof, that Ω^{β} is separating for 𝔐_{β}: Let A ∈ 𝔐_{β}^{ana}. We choose ϕ ∈ ⋃_{n∈ℕ} ran 1[|(L_{0} + Q)| ⩽ n]. First, we have

J A^{∗} Ω(β/2) = τ_{iβ/2}(A) Ω(β/2).

Let f_{ϕ}(z) = ⟨ϕ|τ_{z}(A)Ω(β/2)⟩ and g_{ϕ}(z) = ⟨e^{\overline{z}(L_{0}+Q)}ϕ | Ae^{−(β/2+z)(L_{0}+Q)}Ω_{0}^{β}⟩ for −β/2 ⩽ Re z ⩽ 0. Both functions are continuous and analytic if −β/2 < Re z < 0. Furthermore, f_{ϕ}(i t) = g_{ϕ}(i t) for t ∈ ℝ. Hence f_{ϕ} = g_{ϕ} and for z = −β/2

⟨ ϕ | J A^{∗}Ω(β/2) ⟩ = ⟨ e^{−β/2 (L_{0}+Q)}ϕ | AΩ_{0}^{β} ⟩.

This equation extends to all A ∈ 𝔐_{β}, we obtain A Ω_{0}^{β} ∈ dom(e^{−β/2 (L_{0}+Q)}), such as e^{−β/2(L_{0}+Q)} A Ω_{0}^{β} = J A^{∗}Ω(β/2) for A ∈ 𝔐_{β}. Assume A^{∗} Ω(β/2) = 0, then e^{−β/2 (L_{0}+Q)}AΩ_{0}^{β} = 0 and hence AΩ_{0}^{β} = 0. Since Ω_{0}^{β} is separating, it follows that A = 0 and therefore A^{∗} = 0.

▶ Proof of Ω^{β} ∈ C, and that Ω^{β} is cyclic for 𝔐_{β}: To prove that ϕ ∈ C it is sufficient to check that ⟨ ϕ | AJ AΩ_{0}^{β}⟩ ⩾ 0 for all A ∈ 𝔐_{β}. We have

⟨ Ω(β/2)| AJ A Ω_{0}^{β} ⟩ = \overline{⟨ J A^{∗}Ω(β/2) |AΩ_{0}^{β} ⟩}
= \overline{⟨ e^{−β/2(L_{0}+Q)} AΩ_{0}^{β} | AΩ_{0}^{β} ⟩} ⩾ 0.

The proof follows, since every separating element of C is cyclic. ▶ Proof, that ω^{β} is a (τ, β)-KMS state: For A, B ∈ 𝔐_{β} and z ∈ S_{β} we define

F_{β}(A, B, z) = c ⟨ e^{−i\overline{z}/2L_{Q}}A^{∗}Ω^{β} | e^{iz/2L_{Q}}BΩ^{β} ⟩,

where c := ∥Ω^{β}∥^{−2}. First, we observe

F_{β}(A, B, t) = c ⟨ e^{−it/2L_{Q}}A^{∗}Ω^{β} | e^{it/2L_{Q}}BΩ^{β} ⟩ = c ⟨ Ω^{β} |Aτ_{t}(B)Ω^{β} ⟩

= ω^{β}(A τ_{t}(B)) and

ω^{β}(τ_{t}(B)A) = c ⟨ τ_{t}(B^{∗})Ω^{β} | AΩ^{β} ⟩ = c ⟨ J AΩ^{β} | J τ_{t}(B^{∗})Ω^{β} ⟩
= c ⟨ e^{−β/2L_{Q}}A^{∗}Ω^{β} | e^{−β/2 L_{Q}}τ_{t}(B)Ω^{β} ⟩
= c ⟨ e^{−i\overline{(iβ+t)}/2L_{Q}}A^{∗}Ω^{β} | e^{i (iβ+t)/2L_{Q}}BΩ^{β} ⟩
= F_{β}(A, B, t + iβ).

The requirements on the analyticity of F_{β}(A, B, ·) follow from Lemma A.2.

6 Proof of Theorem 1.3

For \underline{s}_{n} := (s_{n}, …, s_{1}) ∈ ℝ^{n} we define

Q_{N}(\underline{s}_{n}) := Q_{N}(s_{n}) ⋯ Q_{N}(s_{1}),  Q_{N}(s) := e^{−sL_{0}}Q_{N}e^{s L_{0}}, s ∈ ℝ (31)

At this point, we check that Q_{N}(\underline{s}_{n})Ω_{0}^{β} is well defined, and that it is an analytic vector of L_{0}, see Equation (25). The goal of Theorem 1.3 is to give explicit conditions on H_{el} and W, which ensure Ω_{0}^{β} ∈ dom(e^{−β/2 (L_{0}+Q)} ). Let

\underline{η}_{1} := ∫ (∥G⃗(k)∥_{B(H_{el})}^{2} + ∥H⃗(k)∥_{B(H_{el})}^{2})(2 + 4α(k)^{−1}) dk  (32)
\underline{η}_{2} := ∫ (∥F (k) H_{el,+}^{−γ}∥_{B(H_{el})}^{2} + ∥F (k)^{∗} H_{el,+}^{−γ}∥_{B(H_{el})})(2 + 4α(k)^{−1}) dk

The idea of the proof is the following. First, we expand e^{−β/2(L_{0}+Q_{N})}e^{L_{0}} in a Dyson-series, i.e.,

e^{−β/2(L_{0}+Q_{N})}e^{L_{0}}  (33)

= 1 + ∑_{n=1}^{∞}(−1)^{n} ∫_{Δ_{β/2}^{n}} e^{−s_{n}L_{0}}Q_{N}e^{s_{n} L_{0}} ⋯ e^{−s_{1}L_{0}}Q_{N}e^{s_{1} L_{0}} d\underline{s}_{n}.

Under the assumptions of Theorem 1.3 we obtain an upper bound, uniform in N, for

⟨Ω_{0}^{β} | e^{−β(L_{0}+Q_{N})}Ω_{0}^{β}⟩  (34)

= 1 + ∑_{n=1}^{∞}(−1)^{n} ∫_{Δ_{β}^{n}}⟨Ω_{0}^{β} | e^{−s_{n}L_{0}}Q_{N}e^{s_{n} L_{0}} ⋯ e^{−s_{1}L_{0}}Q_{N}e^{s_{1} L_{0}}Ω_{0}^{β}⟩ d\underline{s}_{n}.

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 ∥e^{−(β/2)(L_{0}+Q_{N})}Ω_{0}^{β}∥, which is uniform in N. The proof of Theorem 1.3 follows now from Lemma 6.3, where we show that Ω_{0}^{β} ∈ dom(e^{−(β/2)(L_{0}+Q)}).

Lemma 6.1. Assume

lim sup_{n → ∞} sup_{0 ⩽ x ⩽ β/2} ∥ ∫_{Δ_{x}^{n}} Q_{N}(\underline{s}_{n})Ω_{0}^{β} d\underline{s}_{n}∥^{1/n} < 1.

for all N ∈ ℕ. Then Ω_{0}^{β} ∈ dom(e^{−x(L_{0}+Q_{N})}), 0 < x ⩽ β/2 and

e^{−x (L_{0}+Q_{N})}Ω_{0}^{β} = Ω_{0}^{β} + ∑_{n=1}^{∞}(−1)^{n} ∫_{Δ_{x}^{n}} Q_{N}(\underline{s}_{n})Ω_{0}^{β} d\underline{s}_{n}.  (35)

In this context Δ_{x}^{n} = {(s_{1}, …, s_{n}) ∈ ℝ^{n} : 0 ⩽ s_{n} ⩽ … ⩽ s_{1} ⩽ x} is a simplex of dimension n and sidelength x.

Proof. Let ϕ ∈ ran 1[|L_{0} + Q_{N}| ⩽ k] and 0 ⩽ x ⩽ β/2 be fixed. An m-fold application of the fundamental theorem of calculus yields

⟨ e^{−x (L_{0} + Q_{N})} ϕ | e^{x L_{0}} Ω_{0}^{β} ⟩ = ⟨ ϕ | Ω_{0}^{β} + ∑_{n=1}^{m}(−1)^{n} ∫_{Δ_{x}^{n}} Q_{N}(\underline{s}_{n}) Ω_{0}^{β} d\underline{s}_{n} ⟩

+(−1)^{m+1} ∫_{Δ_{x}^{m+1}} ⟨ e^{−s_{m+1} (L_{0} + Q_{N})}ϕ | e^{s_{m+1} L_{0}} Q_{N}(\underline{s}_{m+1}) Ω_{0}^{β} ⟩ d\underline{s}_{m+1}. (36)

Since L_{0} Ω_{0}^{β} = 0 we have for r(\underline{s}_{m+1}) := (s_{m} − s_{m+1}, …, s_{1} − s_{m+1}) that

e^{s_{m+1}L_{0}} Q_{N}(\underline{s}_{m+1})Ω_{0}^{β} = Q_{N} Q_{N}(r(\underline{s}_{m+1}))Ω_{0}^{β},

We turn now to the second expression on the right side of Equation (36), after a linear transformation depending on s_{m+1} we get

(−1)^{m+1} ∫_{0}^{x} ⟨ e^{−s_{m+1}(L_{0}+Q_{N})}ϕ | Q_{N} ∫_{Δ_{x−s_{m+1}}^{m}} Q_{N}(\underline{r}_{m})Ω_{0}^{β} d\underline{r}_{m} ⟩ ds_{m+1}.

Since ∥e^{−s_{m+1} (L_{0} + Q_{N})} ϕ∥ ⩽ e^{β/2 k} ∥ϕ∥, and using that Q_{N}(\underline{r}_{m}) Ω_{0}^{β} is a state with at most 2m bosons, we obtain the upper bound

const ∥ϕ∥ \sqrt{(2m) (2m + 1)} sup_{0 ⩽ x ⩽ β/2} ∥ ∫_{Δ_{x−s_{m+1}}^{m}} Q_{N}(\underline{r}_{m}) Ω_{0}^{β} d\underline{r}_{m}∥.

Hence, for m → ∞ we get

⟨ e^{−x(L_{0}+Q_{N})}ϕ | Ω_{0}^{β} ⟩ = ⟨ ϕ | Ω_{0}^{β} + ∑_{n=1}^{∞}(−1)^{n} ∫_{Δ_{x}^{n}} Q_{N}(\underline{s}_{n}) Ω_{0}^{β} d\underline{s}_{n} ⟩.

Since ⋃_{k=1}^{∞} ran 1[|L_{0} + Q_{N}| ⩽ k] is a core of e^{−x(L_{0} + Q_{N})}, the proof follows from the self-adjointness of e^{−x(L_{0} + Q_{N})}.

Lemma 6.2. Let 0 < x ⩽ β/2. We have the identity

∫_{Δ_{x/2}^{n}} ∫_{Δ_{x/2}^{m}} ⟨ Q_{N}(\underline{r}_{m})Ω_{0}^{β} | Q_{N}(\underline{s}_{n})Ω_{0}^{β} ⟩ d\underline{r}_{m} d\underline{s}_{n}  (37)

= ∫_{Δ_{β}^{n+m}} 1[z_{m} ⩾ β − x ⩾ x ⩾ z_{m+1}] ⟨ Ω_{0}^{β} | Q_{N}(\underline{z}_{n+m})Ω_{0}^{β} ⟩ d\underline{z}_{n+m}.

For m = n it follows

∥ ∫_{Δ_{x/2}^{n}} Q_{N}(\underline{s}_{n})Ω_{0}^{β} d\underline{s}_{n}∥^{2} ⩽ ∫_{Δ_{β}^{2n}} |⟨ Ω_{0}^{β} | Q_{N}(\underline{s}_{2n})Ω_{0}^{β} ⟩| d\underline{s}_{2n}.  (38)

Proof. Recall Theorem 3.1 and Lemma 5.1. Since J is a conjugation we have ⟨ ϕ | ψ ⟩ = ⟨ J ψ | J ϕ ⟩, and for every operator X, that is affiliated with 𝔐_{β}, we have J X Ω_{0}^{β} = e^{−β/2 L_{0}} X^{∗} Ω_{0}^{β}. Thus,

∫_{Δ_{x/2}^{n}} ∫_{Δ_{x/2}^{m}} ⟨ Q_{N}(\underline{r}_{m})Ω_{0}^{β} | Q_{N}(\underline{s}_{n}) Ω_{0}^{β} ⟩ d\underline{r}_{m} d\underline{s}_{n}  (39)

= ∫_{Δ_{x/2}^{n}} ∫_{Δ_{x/2}^{m}} ⟨ e^{−β/2L_{0}} Q_{N}(\underline{s}_{n})^{∗} Ω_{0}^{β} | e^{−β/2L_{0}} Q_{N}(\underline{r}_{m})^{∗} Ω_{0}^{β} ⟩ d\underline{r}_{m} d\underline{s}_{n}

Since L_{0} Ω_{0}^{β} = 0 we have

e^{−βL_{0}} Q_{N}(\underline{r}_{m})^{∗} Ω_{0}^{β} = Q_{N}(β − r_{1}) ⋯ Q_{N}(β − r_{m}) Ω_{0}^{β}.

Next, we introduce new variables for \underline{r}, namely y_{i} := β − r_{m−i+1}. Let D_{x/2}^{m} := \{\underline{y}_{m} ∈ ℝ^{m} : β − x ⩽ y_{m} ⩽ … ⩽ y_{1} ⩽ β\}. Thus the right side of Equation (39) equals

∫_{Δ_{x/2}^{n}} ∫_{D_{x/2}^{m}} ⟨ Ω_{0}^{β} | Q_{N}(\underline{s}_{n}) Q_{N}(\underline{y}_{m})^{∗} Ω_{0}^{β} ⟩ d\underline{s}_{n} d\underline{y}_{m}

= ∫_{Δ_{β}^{n+m}} 1[z_{m} ⩾ β − x ⩾ x ⩾ z_{m+1}] ⟨ Ω_{0}^{β} | Q_{N}(\underline{z}_{n+m}) Ω_{0}^{β} ⟩ d\underline{z}_{n+m}.

The second statement of the Lemma follows by choosing n = m.

Lemma 6.3. Assume sup_{N∈ℕ} ∥e^{−x(L_{0}+Q_{N})}Ω_{0}^{β}∥ < ∞ then Ω_{0}^{β} ∈ dom(e^{−x(L_{0}+Q)}) and

∥e^{−x(L_{0}+Q)}Ω_{0}^{β}∥ ⩽ sup_{N∈ℕ} ∥e^{−x(L_{0}+Q_{N})} Ω_{0}^{β}∥

Proof. For f ∈ C_{0}^{∞}(ℝ) and ϕ ∈ K we define ψ_{N} := f(L_{0} + Q_{N}) ϕ. Obviously, for g(r) = e^{−x r} f(r) ∈ C_{0}^{∞}(ℝ) we have e^{−x (L_{0} + Q_{N})} ψ_{N} = g(L_{0} + Q_{N}) ϕ.

Since L_{0} + Q_{N} tends to L_{0} + Q in the strong resolvent sense as N → ∞, we know from [16] that lim_{N → ∞} ψ_{N} = f(L_{0} + Q) ϕ =: ψ and

lim_{N→∞} e^{−x (L_{0} + Q_{N})} ψ_{N} = lim_{N→∞} g(L_{0} + Q_{N}) ϕ = g(L_{0} + Q) ϕ = e^{−x (L_{0} + Q)} ψ.

Thus,

|⟨ e^{−x(L_{0}+Q)} ψ | Ω_{0}^{β} ⟩| = lim_{N→∞} |⟨ e^{−x(L_{0}+Q_{N})}ψ_{N} | Ω_{0}^{β} ⟩|

⩽ sup_{N∈ℕ} ∥e^{−x (L_{0}+Q_{N})}Ω_{0}^{β}∥ ∥ψ∥,

Since {f(L_{0} +Q) ϕ ∈ K : ϕ ∈ K, f ∈ C_{0}^{∞}(ℝ)} is a core of e^{−x(L_{0}+Q)}, we obtain Ω_{0}^{β} ∈ dom(e^{−x(L_{0}+Q)}).

Lemma 6.4. For some C > 0 we have

∫Δ_{β}^{n} |⟨ Ω_{0}^{β} | Q_{N}(\underline{s}_{n})Ω_{0}^{β} ⟩| d\underline{s}_{n}

⩽ const (n + 1)^{2} (1 + β)^{n} (8\underline{η}_{1} + \frac{(8C\underline{η}_{2})^{1/2}}{(n + 1)^{(1−2γ)/2}} )^{n},

where \underline{η}_{1} and \underline{η}_{2} are defined in (32).

Proof of 6.4. First recall the definition of Q_{N} and Q_{N}(\underline{s}_{n}) in Equation (25) and Equation (31), respectively. Let

∫_{Δ_{β}^{n}} |⟨ Ω_{0}^{β} | Q_{N}(\underline{s}_{n}) Ω_{0}^{β} ⟩| d\underline{s}_{n} =: ∫_{Δ_{1}^{n}} β^{n} J_{n}(β, \underline{s}) d\underline{s}_{n},

The functions J_{n}(β, \underline{s}) clearly depends on N, but since we want to find an upper bound independent of N, we drop this index. Let W_{1} = Φ(G⃗) Φ(H⃗) + h. c., W_{2} := Φ(F) and W := W_{1} + W_{2}. By definition of ω_{0}^{β} in (3.1), see also (13), we obtain

J_{n}(β, \underline{s}_{n}) = ω_{0}^{β}((e^{−β s_{n} H_{0}} W e^{βs_{n}H_{0}}) ⋯ (e^{−β s_{1} H_{0}} W e^{β s_{1} H_{0}}))

= (Z)^{−1} ∑_{κ∈{1, 2}^{n}} ω_{f}^{β}( Tr_{H_{el}} {e^{−β H_{el}}(e^{−β s_{n} H_{0}} W_{κ(n)} e^{β s_{n} H_{0}}) ⋯

⋯ (e^{−β s_{1} H_{0}} W_{κ(1)} e^{β s_{1} H_{0}})})

By definition of ω_{f}^{β} it suffices to consider expressions with an even number of field operators. In the next step we sum over all expression, where n_{1} times W_{1} occurs and 2n_{2} times W_{2}. The sum of n_{1} and n_{2} is denoted by m. For fixed n_{1} and n_{2} the remaining expressions are all expectations in ω_{f}^{β} of 2m field operators. In this case the expectations in ω_{f}^{β} 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(m_{1}, m_{2}) = {κ ∈ {1, 2}^{n} : #κ^{−1}({i}) = m_{i}, i = 1, 2}.

Thus we obtain

J_{n}(β, \underline{s}_{n}) = (Z)^{−1} ∑_{n_{1} + 2n_{2} = n^{(n_{1}, n_{2})∈ℕ^{2}}}  ∑_{m:= n_{1} + n_{2}^{κ∈M(n_{1}, 2n_{2})}} ∫ ν(d\underline{k}_{2m} ⊗ d\underline{τ}_{2m})  (40)

Tr_{H_{el}} {e^{−(β − β(s_{1} − s_{2m}))H_{el}}I_{2m}e^{−β (s_{2m−1} − s_{2m}) H_{el}} ⋯ e^{−β (s_{1} − s_{2}) H_{el}} I_{1}} ,

Of course I_{j} depends on \underline{k}_{2m} × \underline{τ}_{2m}, namely for κ(j) = 1, 2 we have

I_{j} = \begin{cases}I_{j}(m, τ, m′, τ′), & κ(j) = 1 \\ I_{j}(m, τ), & κ(j) = 2, \end{cases}

where (m, τ), (m′, τ′) ∈ {(k_{j}, τ_{j}) : j = 1, …, m}. For κ(j) = 1 we have that

I_{j}(m, +, m′, −) = G⃗^{∗}(m) H⃗(m′) + H⃗^{∗}(m) G⃗(m′)
I_{j}(m, −, m′, +) = G⃗(m) H⃗^{∗}(m′) + H⃗(m) G⃗^{∗}(m′)
I_{j}(m, +, m′, +) = G⃗^{∗}(m) H⃗^{∗}(m′) + H⃗^{∗}(m) G⃗^{∗}(m′)
I_{j}(m, −, m′, −) = G⃗(m) H⃗(m′) + H⃗(m) G⃗(m′)

and for κ(j) = 2 we have that

I_{j}(m, +) = F^{∗}(m)
I_{j}(m, −) = F (m).

In the integral (40) we insert for (m, τ) and (m′, τ′) in the definition of I_{j} from left to right k_{2m}, τ_{2m}, …, k_{1}, τ_{1}. For fixed (\underline{k}_{2m}, \underline{τ}_{2m}) the integrand of (40) is a trace of a product of 4m operators in H_{el}. We will apply Hölder’s-inequality for the trace, i.e.,

| Tr_{H_{el}}{A_{2m} B_{2m} ⋯ A_{1} B_{1}}| ⩽ ∏_{j=1}^{2m} ∥B_{j}∥_{B(H_{el})} · ∏_{j=1}^{2m} Tr_{H_{el}}{A_{i}^{p_{j}}}^{p_{j}^{−1}}.

In our case p_{i} := (s_{i−1} − s_{i})^{−1} for i = 2, …, 2m and p_{1} := (1 − s_{1} + s_{2m})^{−1} and

(A_{j}, B_{j}) := \begin{cases}(e^{−β p_{j}^{−1} H_{el}}, I_{j}(m, τ, m′, τ′)), & κ(j) = 1 \\ (e^{−β p_{j}^{−1} H_{el}} H_{el,+}^{γ}, H_{el,+}^{−γ} I_{j}(m, τ,)), & κ(j) = 2 \end{cases} .

We define

η_{1}(k) = max {∥G⃗(k)∥_{B(H_{el})^{r}}, ∥H⃗(k)∥_{B(H_{el})^{r}}}
η_{2}(k) = max {∥F (k) H_{el,+}^{−γ}∥_{B(H_{el})}, ∥F^{∗}(k) H_{el,+}^{−γ}∥_{B(H_{el})}}.

By definition of B_{j} we have

∥B_{j}∥_{B(H_{el})} ⩽ \begin{cases}η_{1}(m)η_{1}(m′), & κ(j) = 1 \\ η_{2}(m), & κ(j) = 2 \end{cases} .  (41)

Furthermore,

Tr_{H_{el}}{A_{i}^{p_{j}}}^{p_{j}^{−1}} = Tr_{H_{el}} {e^{−βH_{el}} H_{el,+}^{p_{j}γ}}^{p_{j}^{−1}}
⩽ ∥e^{−ϵH_{el}} H_{el,+}^{p_{j}γ}∥_{H_{el}}^{p_{j}^{−1}} Tr_{H_{el}} {e^{−(β−ϵ) H_{el}}}^{p_{j}^{−1}}, k(j) = 2

Let E_{gs} := inf σ(H_{el}). The spectral theorem for self-adjoint operators implies

∥e^{−ϵ H_{el}} H_{el,+}^{p_{i} γ}∥_{H_{el}}^{p_{i}^{−1}} ⩽ sup_{r ⩾ E_{gs}} e^{−ϵ p_{i}^{−1} r}(r − E_{gs} + 1)^{γ} ⩽ ϵ^{−γ} p_{i}^{γ} e^{−ϵ p_{i}^{−1} (E_{gs} − 1)}.

Inserting this estimates we get

Tr_{H_{el}}{e^{−(β − β(s_{1} − s_{2m}))H_{el}}I_{2m}e^{−β (s_{2m−1} − s_{2m}) H_{el}} ⋯ e^{−β (s_{1} − s_{2}) H_{el}} I_{1}}

⩽ C_{κ}(\underline{s}_{n}) ∏_{j=1}^{2m} ∥B_{j}∥_{B(H_{el})}

where

C_{κ}(\underline{s}_{n}) := (1 − s_{1} + s_{n})^{−α_{1}} ∏_{i=1}^{n−1}(s_{i} − s_{i+1})^{−α_{i}}  (42)

and

α_{i} = \begin{cases}0, & κ(i) = 1 \\ 1/2, & κ(i) = 2 \end{cases}  (43)

Now, we recall the definition of ν. Roughly speaking, one picks a pair of variables (k_{i}, k_{j}) and integrates over δ_{k_{i},k_{j}} coth(β/2α(k_{i})) dk_{i}dk_{j}. Subsequently one picks the next pair and so on. At the end one sums up all \frac{(2m)!}{2^{m} m!} pairings and all 4^{m} combinations of \underline{τ}_{2m}. Inserting Estimate (41) and that

∫ η_{ν}(k)η_{ν′}(k) coth(β/2α(k)) dk ⩽ (1 + β^{−1})\underline{η}_{ν}^{1/2}\underline{η}_{ν′}^{1/2},

we obtain

|J_{n}(β, \underline{s})| ⩽ \frac{(1 + β^{−1})^{n}}{Z} ∑_{n_{1} + 2n_{2} = n^{(n_{1}, n_{2})∈ℕ^{2}_{0}}} ∑_{m:= n_{1} + n_{2}^{κ∈M(n_{1}, 2n_{2})}}( \underline{η}_{1})^{n_{1}} (C\underline{η}_{2})^{n_{2}} \frac{(2m)!2^{m}}{m!} C_{κ}(\underline{s})

By Lemma A.3 below and since (2m)!/(m!)^{2} ⩽ 4^{m} we have

∫Δ_{β}^{n} |⟨ Ω_{0}^{β} | Q_{N}(\underline{s}_{n}) Ω_{0}^{β} ⟩| d\underline{s}_{n}

⩽ const(1 + β)^{n} ∑_{n_{1} + 2n_{2} = n^{(n_{1}, n_{2})∈ℕ^{2}_{0}}} \binom{n}{n_{1}} \frac{(8\underline{η}_{1})^{n_{1}} (8C′\underline{η}_{2})^{n_{2}}}{(n + 1)^{(1−2γ) n_{2}−2}}

This completes the proof.

7 The Harmonic Oscillator

Let L^{2}(X, dμ) = L^{2}(ℝ) and H_{el} =: H_{osc} := −Δ_{q}+Θ^{2}q^{2} be the one dimensional harmonic oscillator and H_{ph} = L^{2}(ℝ^{3}). We define

H = H_{osc} + Φ(F) + Ȟ,  Ȟ := dΓ(|k|),  (44)

where Φ(F) = q · Φ(f), with λ (|k|^{−1/2} + |k|^{1/2}) f ∈ L^{2}(ℝ^{3}). H_{osc} is the harmonic oscillator, the form-factor F comes from the dipole approximation. The Standard Liouvillean for this model is denoted by L_{osc}. Now we prove Theorem 1.4.

Proof. We define the creation and annihilation operators for the electron.

A^{∗} = \frac{Θ^{1/2} q − i Θ^{−1/2} p}{\sqrt{2}}, A = \frac{Θ^{1/2} q + i Θ^{−1/2} p}{\sqrt{2}}, p = −i ∂_{x}, (45)
Φ(c) = c_{1} q + c_{2} p, for c = c_{1} + i c_{2} ∈ ℂ, c_{i} ∈ 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,  H_{osc} = ΘA^{∗} A + Θ/2, (47)
[H_{osc}, A] = −Θ A,  [H_{osc}, A^{∗}] = Θ A^{∗}.  (48)

The vector Ω := (\frac{Θ}{π})^{1/4} e^{−Θ q^{2} /2} is called the vacuum vector. Note, that one can identify F_{b}[ℂ] with L^{2}(ℝ), since LH{(A^{∗})^{n} Ω | n ∈ ℕ^{0}} is dense in L^{2}(ℝ). It follows, that ω_{β}^{osc} is quasi-free, as a state over W (ℂ) and

ω_{β}^{osc}(W (c)) = (Z)^{−1} Tr_{H_{el}}{e^{−βH_{el}} W (c)} = exp ( − 1/4 coth(β Θ/2) |c|^{2}),

(49)

where Z = Tr_{H_{el}}{e^{−β H_{el}}} is the partition function for H_{el}. First, we remark, that Equation (31) is defined for this model without regularization by P_{N} := 1[H_{el} ⩽ N]. Moreover we obtain from Lemma 6.2, that

∥ ∫_{Δ_{β/2}^{n}} Q(\underline{s}_{n})Ω_{0}^{β} d\underline{s}_{2n}∥^{2} ⩽ ∫_{Δ_{β}^{2n}} |⟨ Ω_{0}^{β} | Q(\underline{s}_{2n}) Ω_{0}^{β} ⟩| d\underline{s}_{2n} =: h_{2n}(β, λ).

(50)

To show that Ω^{β} ∈ dom(e^{−β/2 (L_{0} + Q)}) is suffices to prove, that ∑_{n = 0}^{∞} h_{2n}(β, λ)^{1/2} < ∞. We have

h_{2n}(β, λ) = \frac{(−β λ)^{2n}}{Z} ∫_{Δ_{1}^{2n}} ω_{β}^{osc}((e^{−β s_{2n} H_{el}} q e^{β s_{2n} H_{el}})  (51)
⋯ (e^{−β s_{1} H_{el}} q e^{β s_{1} H_{el}}))
·ω_{f}^{β}((e^{−β s_{2n} Ȟ} Φ(f) e^{β s_{2n} Ȟ}) ⋯ (e^{−β s_{1} Ȟ} Φ(f) e^{β s_{1} Ȟ})) d\underline{s}_{2n}.

Moreover, we have

e^{−β s_{i} H_{el}} q e^{β s_{i} H_{el}} = (2Θ)^{−1/2}(e^{−β Θ s_{i}} A^{∗} + e^{β Θ s_{i}} A )
e^{−β s_{i} Ȟ} Φ(f) e^{β s_{i} Ȟ} = 2^{−1/2}(a^{∗}(e^{−β s_{i} |k|} f) + a(e^{β s_{i} |k|} f)).  (52)

Inserting the identities of Equation (52) in Equation (51) and applying Wick’s theorem [5, p. 40] yields

h_{2n}(β, λ) = (β λ)^{2n} ∫_{Δ_{1}^{2n}} ∑_{P ∈Z_{2}} ∏_{{i, j}∈P} K_{osc}(|s_{i} − s_{j}|, β)

· ∑_{P′∈Z_{2}} ∏_{\{k, l\}∈P′} K_{f}(|s_{k} − s_{l}|, β) d\underline{s}_{2n}

= \frac{(β λ)^{2n}}{(2n)!} ∫_{[0, 1]^{2n}} ∑_{P, P′∈Z_{2}} ∏_{\{k, l\}∈P′^{\{i, j\}∈P}} K_{osc}(|s_{i} − s_{j}|, β) K_{f}(|s_{k} − s_{l}|, β) d\underline{s}_{2n}, (53)

where for k < l and i < j, such as

K_{f}(|s_{k} − s_{l}|, β) := ω_{f}^{β}((e^{−β s_{k} Ȟ} Φ(f) e^{β s_{k} Ȟ}) (e^{−β s_{l} Ȟ} Φ(f) e^{β s_{l} Ȟ}))
K_{osc}(|s_{i} − s_{j}|, β) := ω_{β}^{osc}(e^{−β s_{i} H_{el}} q e^{β s_{i} H_{el}} e^{−β s_{j} H_{el}} q e^{β s_{j} H_{el}}).

The last equality in (53) holds, since the integrand is invariant with respect to a change of the axis of coordinates. We interpret two pairings P and P′ ∈ Z_{2} as an indirected graph G = G(P, P′), where M_{2n} = {1, …, 2n} is the set of points. Any graph in G has two kinds of lines, namely lines in L_{osc}(G), which belong to elements of P and lines in L_{f}(G), which belong to elements of P′. Let G(A) be the set of undirected graphs with points in A ⊂ M_{2n}, such that for each point "i" in A, there is exact one line in L_{f}(G), which begins in "i", and exact one line in L_{osc}(G), which begins with "i". G_{c}(A) is the set of connected graphs. We do not distinguish, if points are connected by lines in L_{f}(G) or by lines in L_{osc}(G). Let

P_{k} := {P : P = {A_{1}, …, A_{k}}, ∅ ≠ A_{i} ⊂ M_{2n},

A_{i} ∩ A_{j} = ∅ for i ≠ j, ⋃_{i = 1}^{k} A_{i} = M_{2n}}

be the family of decompositions of M_{2n} in k disjoint set. It follows

h_{2n}(β, λ) = \frac{(β λ)^{2n}}{(2n)!} ∑_{G∈G(M_{2n})} ∫_{M_{2n}} ∏_{\{k, l\}∈L_{f} (G)^{\{i,j\}∈Losc(G)}} K_{osc}(|s_{i} − s_{j}|, β)

K_{f}(|s_{k} − s_{l}|, β) d\underline{s}_{n}

= \frac{(βλ)^{2n}}{(2n)!} ∑_{k = 1}^{2n} ∑_{\{A_{1},..., A_{k}\}∈P_{k}} ∑_{G_{a}∈G_{c}(A_{a})^{(G_{1},..., G_{k})}} ∏_{a = 1}^{k} J(G_{a}, A_{a}, β)

= \frac{(βλ)^{2n}}{(2n)!} ∑_{k = 1}^{2n} \frac{1}{k!} ∑_{\{A_{1},..., A_{k}\}∈P_{k}^{A_{1},..., A_{k}⊂M_{2n},}} ∑_{G_{a}∈G_{c}(A_{a})^{(G_{1},..., G_{k})}} ∏_{a = 1}^{k} J(G_{a}, A_{a}, β),(54)

where

J(G_{a}, A_{a}, β) := ∫_{A_{a}} ∏_{\{k,l\}∈L_{f} (G_{a})^{\{i, j\}∈Losc(Ga)}} K_{osc}(|s_{i} − s_{j}|, β)K_{f}(|s_{k} − s_{l}|, β) d\underline{s}. (55)

∫_{A_{a}} d\underline{s} means, ∫_{−1}^{1} ds_{j_{1}} ∫_{−1}^{1} ds_{j_{2}} … ∫_{−1}^{1} ds_{j_{m}}, where A_{a} = \{j_{1}, …, j_{m}\} and #A_{a} = m. From the first to the second line we summarize terms with graphs, having connected 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 \frac{1}{k!} is introduced. Due to Lemma 7.2 the integral depends only on the number of points in the connected graph, i. e. J(G, A, β) = J(#A, β). Moreover, Lemma 7.2 states that β^{#A} ·J(#A, β) ⩽ (2∥|k|^{−1/2} f∥_{2} (Θ β)^{−1})^{#A} (C β + 1). To ensure that G_{c}(A_{a}) is not empty, #A_{a} must be even. For (m_{1}, …, m_{k}) ∈ ℕ^{k} with m_{1} + ⋯ + m_{k} = n we obtain

∑_{\{A_{1},..., A_{k}\}∈P_{k}^{A_{1},..., A_{k}⊂M_{2n},#A_{i} = 2m_{i}}} 1 = \frac{(2n)!}{(2m_{1})! ⋯ (2m_{k})!}.  (56)

Documenta Mathematica 16 (2011) 177–208

Let now be A_{a} ⊂ M_{2n} with #A_{a} = 2m_{a} > 2 fixed. In G_{a} are #A_{a} lines in L_{osc}(G_{a}), since such lines have no points in common, we have \frac{(2m_{a})!}{m_{a}! 2^{ma}} choices. Let now be the lines in L_{osc}(G_{a}) fixed. We have now ((2m_{a} − 2)(2m_{a} − 4) ⋯ 1) choices for m_{a} lines in L_{f}(G_{a}), which yield a connected graph. Thus

∑_{G_{a}∈G_{c}(A_{a}),} 1 = \frac{(2m_{a})!}{m_{a}! 2^{m_{a}}} ((2m_{a} − 2) (2m_{a} − 4) ⋯ 1) = \frac{(2m_{a})!}{2m_{a}}.  (57)

For #A_{a} = 2 exists only one connected graph. We obtain for h_{2n}

h_{2n}(β, λ) = (λ)^{2n} ∑_{k = 1}^{2n} \frac{1}{k!} ∑_{m_{1} +...+ m_{k} = n^{(m_{1},..., m_{k})∈ℕ^{k}}} ∏_{a = 1}^{k} \frac{J(2m_{a}, β)(β^{2})^{m_{a}}}{2m_{a}}  (58)

⩽ (2Θ^{−1} ∥|k|^{−1/2} f∥ λ)^{2n} ∑_{k = 1}^{2n} \frac{1}{k!} ∑_{m_{1} +...+ m_{k} = n^{(m_{1},..., m_{k})∈ℕ^{k}}} ∏_{a = 1}^{k} \frac{(C β + 1)}{2m_{a}}

⩽ (2Θ^{−1} ∥|k|^{−1/2} f∥ λ)^{2n} ∑_{k = 1}^{2n} \frac{((C β + 1)/2 ∑_{m = 1}^{n} \frac{1}{m})^{k}}{k!} .

Since the ∑_{m=1}^{n} \frac{1}{m} can be considered as a lower Riemann sum for the integral ∫_{1}^{m + 1} r^{−1} dr, we have ∑_{m = 1}^{n} \frac{1}{m} ⩽ ln(n + 1). Thus,

h_{2n}(β, λ) ⩽ (2Θ^{−1} ∥|k|^{−1/2} f∥ λ)^{2n} ∑_{k=1}^{2n} \frac{((C β + 1)/2 ln(n + 1))^{k}}{k!} (59)
⩽ (2Θ^{−1} ∥|k|^{−1/2} f∥ λ)^{2n}(n + 1)^{(C β + 1)/2}.

Since 2|λ| ∥|k|^{1/2} f∥ < Θ the series ∑_{n=0}^{∞} h_{2n}(β, λ)^{1/2} converges absolutely for all β > 0. It follows, that

e^{−β/2 (L_{0} + Q)} Ω_{0}^{β} = Ω_{0}^{β} + ∑_{n = 1}^{∞} ∫_{Δ_{β/2}^{n}} Q(\underline{s}_{n}) Ω_{0}^{β} d\underline{s}_{n}

exists.

Conversely, Equation (58) and Lemma 7.2 imply

h_{2n}(β, λ) ⩾ λ^{2n} \frac{J(2n, β) β^{2n}}{2n} = \frac{(Θ^{−1} ∫ \frac{β^{2} λ^{2} |f(k)|^{2}}{sinh(|k| β/2) sinh(β Θ/2)} dk)^{n}}{2n} . (60)

Hence for every β > 0 exists a λ ∈ ℝ, such that h_{2n}(β, λ) ⩾ \frac{1}{2n}. Thus ∑_{n=1}^{∞} h_{2n}(β, λ)^{1/2} = ∞

Remark 7.1. We can therefore not extended Theorem 1.4 to an existence proof for all λ > 0.

Lemma 7.2. Following statements are true.

J(G, A, β) = J(#A, β), G ∈ G_{c}(A)
J(#A, β) ⩽ (2∥|k|^{−1/2} f∥_{2} (Θ β)^{−1})^{#A} · (C β + 1)
J(#A, β) ⩾ (Θ^{−1} ∫ \frac{|f(k)|^{2}}{sinh(|k| β/2) sinh(Θ β/2)} dk)^{#A/2},

where #A = 2m and C = (1/2) \frac{∥f∥^{2}}{∥|k|^{1/2} f∥^{2}} .

Proof of 7.2. A relabeling of the integration variables yields

J(G, A, β) ⩽ \overline{K}_{f} ∫_{[0,1]^{2m}} K_{osc}(|t_{1} − t_{2}|, β) K_{f}(|t_{2} − t_{3}|, β) ⋯
⋯ K_{osc}(|t_{2m−1} − t_{2m}|, β) d\underline{t}

for \overline{K}_{f} := sup_{s∈[0,1]} K_{f}(s, β). We transform due to s_{i} := t_{i} − t_{i + 1}, i ⩽ 2m − 1 and s_{2m} = t_{2m}, hence −1 ⩽ s_{i} ⩽ 1, i = 1, …, 2m, since integrating a positive function we obtain

J(G, A, β) ⩽ ( ∫_{−1}^{1} K_{osc}(|s|, β) ds)^{m}( ∫_{−1}^{1} K_{f}(|s|, β) ds)^{m − 1}

· sup_{s∈[0,1]} K_{f}(s, β).

We recall that

∫_{−1}^{1} K_{osc}(|s|, β) ds = (2Θ)^{−1} ∫_{−1}^{1} \frac{cosh(β Θ |s| − Θ β/2)}{sinh(Θ β/2)} ds = 2(Θ^{2} β)^{−1}

and

∫_{−1}^{1} K_{f}(|s|, β) ds = ∫_{−1}^{1} ∫ \frac{cosh(β |s| |k| − β|k|/2) |f(k)|^{2}}{2 sinh(β |k|/2)} dk ds

= 2 ∫ \frac{|f(k)|^{2}}{β |k|} dk.

Using coth(x) ⩽ 1 + 1/x and using convexity of cosh, we obtain

sup_{s∈[0, 1]} K_{f}(s, β) ⩽ (1/2) ∫ |f(k)|^{2} dk + \frac{1}{β} ∫ \frac{|f(k)|^{2}}{|k|} dk.

Due to the fact, that t ↦ K_{f}(t, β) and t ↦ K_{osc}(t, β) attain their minima at t = 1/2, we obtain the lower bound for J(#A, β).

Remark 7.3. In the literature there is one criterion for Ω_{0}^{β} ∈ dom(e^{−β/2 (L_{0} + Q)}), to our knowledge, that can be applied in this situation [6]. One has to show that ∥e^{−β/2 Q} Ω_{0}^{β}∥ < ∞. If we consider the case, where the criterion holds for ±λ, then the expansion in λ converges,

∥e^{−β/2 Q} Ω_{0}^{β}∥^{2} = ∑_{n = 0}^{∞} \frac{(λ β)^{2n}}{(2n)!} ω_{el}^{β}(q^{2n}) ω_{f}^{β}(Φ(f)^{2n})

= ∑_{n = 0}^{∞} \frac{(λ β)^{2n}}{(2n)!} (\frac{(2n)!}{n! 2^{n}})^{2} K_{osc}(0, β)^{n} K_{f}(0, β)^{n}

= ∑_{n=0}^{∞}(λ β)^{2n} Θ^{−n} \binom{2n}{n}2^{−2n}( coth(Θ β/2) ∫ |f(k)|^{2} coth(β |k|/2) dk)^{n}

⩾ ∑_{n = 0}^{∞}(λ β)^{2n}(4 Θ)^{−n} ( ∫ |f(k)|^{2} dk)^{n}.

Obviously, for any value of λ ≠ 0, there is a β > 0, for which ∥e^{−β/2 Q} Ω_{0}^{β}∥ < ∞ 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 ⩽ Re (z) ⩽ 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. Let h : {z ∈ ℂ : | Im (z) < α} → ℂ defined by

h(z) := \begin{cases}f( z) − g( z), & on \{z ∈ ℂ : 0 ⩽ Im (z) < α\} \\ \overline{f( \overline{z})} − \overline{g( \overline{z})}, & on \{z ∈ ℂ : −α < Im (z) < 0\} \end{cases}  (61)

Thanks to the Schwarz reflection principle h is analytic. Since h(t) = 0 for all t ∈ ℝ, we get h = 0. Hence f = g on {z ∈ ℂ : 0 ⩽ Re (z) < α}. Since both f and g are continuous, we infer that f = g on the whole domain.

Lemma A.2. Let H be some self-adjoint operator in H, α > 0 and ϕ ∈ dom(e^{α H}). Then ϕ ∈ dom(e^{z H}) for z ∈ {z ∈ ℂ : 0 ⩽ Re (z) ⩽ α}. z ↦ e^{z H}ϕ is continuous on {z ∈ ℂ : 0 ⩽ Re (z) ⩽ α} and analytic in the interior.

Proof of A.2. Due to the spectral calculus we have

∫ e^{2Re z s}d⟨ϕ | 𝔼_{s} ϕ⟩ ⩽ ∫ (1 + e^{2α s})d⟨ϕ | 𝔼_{s} ϕ⟩ =: C_{1}^{2} < ∞.

Thus ϕ ∈ dom(e^{z H}). Let ψ ∈ H and f(z) = ⟨ψ | e^{z H} ϕ⟩. There is a sequence {ψ_{n}} with ψ_{n} ∈ ⋃_{m∈ℕ} ran 𝟙[|H| ⩽ m] and lim_{n→∞} ψ_{n} = ψ. We set f_{n}(z) = ⟨ψ_{n} | e^{z H} ϕ⟩. It is not hard to see that f_{n} is analytic, since ψ_{n} is an analytic vector for H, and that |f_{n}(z)| ⩽ C_{1} ∥ψ_{n}∥ and lim_{n→∞} f_{n}(z) = f(z). Thus f is analytic and hence z ↦ e^{z H}ϕ is analytic. Thanks to the dominated convergence theorem the right side of

∥e^{z_{n}H}ϕ − e^{zH}ϕ∥^{2} ⩽ ∫ (e^{2Re z_{n}s} + e^{2Re zs} − e^{z̄_{n}s+zs} − e^{z̄s+z_{n}s})d⟨ϕ | 𝔼_{s} ϕ⟩ (62)

tends to zero for lim_{n→∞} z_{n} = z. This implies the continuity of z ↦ e^{z H}ϕ.

Lemma A.3. We have for n_{1} + n_{2} ⩾ 1

∫_{Δ_{1}^{n}} C_{κ}(\underline{s}) d\underline{s}_{n} ⩽ \frac{const C^{n_{2}}}{(n_{1} + n_{2})! (n + 1)^{(1−2γ) n_{2}−2}}  (63)

Proof of A.3. We turn now to the integral

∫_{Δ_{1}^{n}} C_{κ}(\underline{s}) d\underline{s}_{n} = ∫_{Δ_{1}^{n}}(1 − s_{1} + s_{n})^{−α_{1}} ∏_{i=1}^{n − 1}(s_{i} − s_{i + 1})^{−α_{i}} d\underline{s}_{n}.  (64)

We define for k = 1, …, 2n, a change of coordinates by s_{k} = r_{1} − ∑_{j = 2}^{k} r_{j}, the integral transforms to

∫S^{n}(1 − (r_{2} + ⋯ + r_{n}))^{−α_{1}} ∏_{i = 2}^{n} r_{i}^{−α_{i}} d\underline{r}_{n}  (65)

= ∫_{T^{n−1}}(1 − (r_{2} + ⋯ + r_{n}))^{1 − α_{1}} ∏_{i=2}^{n} r_{i}^{−α_{i}} d\underline{r}_{n−1}

= \frac{Γ(1 − α_{1})^{−1}Γ(1 − γ)^{2n_{2}}}{Γ(n_{1} + 2n_{2} (1 − γ))}

where S^{2n} := \{\underline{r} ∈ ℝ^{2n} : 0 ⩽ r_{i} ⩽ 1, r_{2} + ⋯ + r_{2n} ⩽ r_{1}\} and T^{2n−1} := \{\underline{r} ∈ ℝ^{2n−1} : 0 ⩽ r_{i} ⩽ 1, r_{2} + ⋯ + r_{2n} ⩽ 1\}. From the first to the second formula we integrate over dr_{1}. The last equality follows from [11, Formula 4.635 (4)], here Γ denotes the Gamma-function. From Stirling’s formula we obtain

(2π)^{1/2} x^{x − 1/2} e^{−x} ⩽ Γ(x) ⩽ (2π)^{1/2} x^{x − 1/2} e^{−x + 1}, x ⩾ 1.  (66)

Since n_{1} + n_{2} ⩾ 1 get

\frac{Γ(n_{1} + n_{2} + 1)}{Γ(n_{1} + 2(1 − γ) n_{2})} ⩽ (n + 1)^{2}(\frac{n_{1} + 2(1 − γ)n_{2}}{e} )^{−(1−2γ)n_{2}}. (67)

Note that Γ(n_{1} + n_{2} + 1) = (n_{1} + n_{2})!.

Lemma A.4. Let (1 + α(k)^{−1/2}) f_{1}, …, (1 + α(k)^{−1/2}) f_{2m} ∈ H_{ph} and σ ∈ {+, −}^{2m}. Let a^{+} = a^{∗} and a^{−} = a

ω_{f}^{β}(a^{σ_{2m}}(e^{−σ_{2m} s_{2m} α(k)} f_{2m}) ⋯ a^{σ_{1}}(e^{−σ_{1} s_{1} α(k)} f_{1}))
= ∫ f_{2m}^{σ_{2m}}(k_{2m}, τ_{2m}) ⋯ f_{1}^{σ_{1}}(k_{1}, τ_{1}) ν(d\underline{k}_{2m} ⊗ d\underline{τ}_{2m}),

where ν(d\underline{k}_{2m} ⊗ d\underline{τ}_{2m}) is a measure on (ℝ^{3})^{2m} × \{+, −\}^{2m} for phonons, respectively on (ℝ^{3} × {±})^{2m} × {+, −}^{2m} for photons, and

ν(d\underline{k}_{2m} ⊗ d\underline{τ}_{2m}) ⩽ ∑_{P ∈Z_{2m}} ∑_{\underline{τ}∈\{+, −\}^{2m}} ∏_{\{i > j\}∈P} (δ_{k_{i}, k_{j}} coth(β α(k_{i})/2)) d\underline{k}_{2m}.

(68)

for f^{+}(k, τ) := f(k) 1[τ = +] and f^{+}(k, τ) := \overline{f(k)} 1[τ = −].

Proof of A.4. Since ω_{f}^{β} is quasi-free, we obtain with a^{+} := a^{∗} and a^{−} := a

ω_{f}^{β}(a^{σ_{2m}}(e^{−σ_{2m} s_{2m} α(k)} f_{2m}) ⋯ a^{σ_{1}}(e^{−σ_{1} s_{1} α(k)} f_{1}))
= ∑_{P ∈Z_{2}} ∏_{i > j^{{i, j}∈P}} ω_{f}^{β}(a^{σ_{i}}(e^{−σ_{i} s_{i} α(k)} f_{i}) a^{σ_{j}}(e^{−σ_{j} s_{j} α(k)} f_{j})),

see Equation (12). For the expectation of the so called two point functions we obtain:

ω_{f}^{β}(a^{+}(e^{s_{i} α(k)} f_{i}) a^{+}(e^{s_{j} α(k)} f_{j})) = 0 = ω_{f}^{β}(a(e^{−s_{i} α(k)} f_{i}) a(e^{−s_{j} α(k)} f_{j})),

such as

ω_{f}^{β}(a^{+}(e^{xs_{i}α(k)}f_{i})a^{−}(e^{−x s_{j}α(k)}f_{j})) = ∫ f_{i}(k) \overline{f_{j}(k)} \frac{e^{x (s_{i} − s_{j})α(k)}}{e^{β α(k)} − 1} dk
ω_{f}^{β}(a^{−}(e^{xs_{i}α(k)}f_{i}) a^{+}(e^{−xs_{j}α(k)}f_{j})) = ∫ f_{j}(k) \overline{f_{i}(k)} \frac{e^{(β+xs_{j}−xs_{i})α(k)}}{e^{βα(k)} − 1} dk

Hence it follows

ω_{f}^{β}(a^{σ_{2m}}(e^{−σ_{2m} s_{2m} α(k)} f_{2m}) ⋯ a^{σ_{1}}(e^{−σ_{1} s_{1} α(k)} f_{1}))
= ∫ f_{2m}^{σ_{2m}}(k_{2m}, τ_{2m}) ⋯ f_{1}^{σ_{1}}(k_{1}, τ_{1}) ν(d\underline{k}_{2m} ⊗ d\underline{τ}_{2m}),

where f^{+}(k, τ) := f(k) 1[τ = +] and f^{−}(k, τ) := \overline{f(k)} 1[τ = −]. ν(d^{3(2m)}k ⊗ d^{2m}τ) is a measure on (ℝ^{3})^{2m} × {+, −}^{2m}, which is defined by

∑_{P ∈Z_{2m}} ∑_{\underline{τ}∈\{+, −\}^{2m}} ∏_{\{i > j\}∈P} δ_{τ, −τ} δ_{k_{i}, k_{j}}  (69)

(δ_{τ, +} \frac{e^{x (s_{i} − s_{j}) α(k_{i})}}{e^{β α(k_{i})} − 1} + δ_{τ, −} \frac{e^{(β − x (s_{i} − s_{j})) α(k_{i})}}{e^{β α(k_{i})} − 1} ) d\underline{k}_{2m}.

References

[1] H. Araki, E. Woods. Representations of the Canonical Commutation Relations describing a non-relativistic infinite free Bose Gas. J. Math. Phys., 4:637–662, 1963.

[2] V. Bach, J. Fröhlich, and I. M. Sigal. Quantum electrodynamics of confined non-relativistic particles. Adv. in Math. , 137:299–395, 1998.

[3] V. Bach, J. Fröhlich, I. M.Sigal. Return to Equilibrium. J. Math. Phys., 41:3985–4060, 2000.

[4] O. Bratteli, D. Robinson. Operator Algebras and Quantum Statistical Mechanics 1, Text and Monographs in Physics, Springer-Verlag, 1987.

[5] O. Bratteli, D. Robinson. Operator Algebras and Quantum Statistical Mechanics 2, Text and Monographs in Physics, Springer-Verlag, 1996.

[6] Derezinski J., Jaksic V., Pillet C.-A. Perturbation theory of W*-dynamics, Liouvilleans and KMS-states. Rev. Math. Phys, 15: 447-489, 2003.

[7] J. Derezinski, V. Jaksic. Spectral theory of Pauli-Fierz operators. J. Funct. Anal., 180 : 243–327, 2001

[8] J. Derezinski, V. Jaksic. Return to Equilibrium for Pauli-Fierz Operators. Ann. Henri Poincare, 4 : 739–793, 2003

[9] J. Fröhlich, M. Merkli. Another Return to Equilibrium Commun. Math. Phys., 251: 235–262, 2004.

[10] J. Fröhlich, M. Merkli, I. M.Sigal. Ionization of Atoms in a Thermal Field. Journal of Statistical Physics, 116: 311–359, 2004. DOI 10.1023/B:JOSS.0000037226.16493.5e

[11] I.S. Gradstein, I.M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, 1980, 4

[12] R. Haag, N. Hugenholz, M. Winnink. On the Equilibrium States in Quantum Statistical Mechanics. Commun. Math. Phys., 5: 215–236, 1967.

[13] V. Jakšić, C. A. Pillet. On a Model for Quantum Friction. II: Fermi’s Golden Rule and Dynamics at Positive Temperature. Commun. Math. Phys., 176:619–643, 1996

[14] V. Jakšić, C. A. Pillet. On a Model for Quantum Friction III: Ergodic Properties of the Spin-Boson System. Commun. Math. Phys., 178:627– 651, 1996

[15] M. Merkli. Positive Commutators in Non-Equilibrium Statistical Quantum Mechanics, Commun. Math. Phys., 223: 327–362, 2001.

[16] M. Reed,B. Simon. Methods of Modern Mathematical Physics: I. Functional Analysis Academic Press, 1980.

[17] M. Reed,B. Simon. Methods of Modern Mathematical Physics: II. Fourier Analysis and Self-Adjointness Academic Press, 1980.

Martin Könenberg

Fakultät für Mathematik und Informatik FernUniversität Hagen Lützowstraße 125 D-58084 Hagen, Germany martin.koenenberg@fernuni -hagen.de