Documenta Math. 177 An Infinite Level Atom coupled to a Heat Bath

We consider a $W^*$-dynamical system $(\Mg,\taug)$, 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 $\taug$ and existence of a $(\beta,\taug)$ -KMS state under very explicit conditions on the strength of the interaction and on the inverse temperature $\beta$.


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 itLQ X e −itLQ , X ∈ M β , t ∈ Ê. (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: Theorem 1.1.L Q , defined in (16), has a unique self-adjoint realization and τ t (X) ∈ M β for all t ∈ Ê 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 β 0 ∈ dom(e −(β/2)(L0+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 β 0 is the vector representative of the (β, τ )-KMS state for the system without interaction.In a third theorem we study the condition Ω β 0 ∈ dom(e −(β/2)(L0+Q) ): Theorem 1.3.Assume Hypothesis 1 is fulfilled.Then there are two cases, 1.If 0 γ < 1/2 and η 1 (1 + β) ≪ 1, then Ω β 0 ∈ dom(e −β/2 (L0+Q) ).
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 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 ).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

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

H (n)
ph 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., annihilation operators, a * (h) and a(h), are defined for h ∈ H ph by 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., Furthermore we define field operator by It is a straightforward calculation to check that the vectors in F b f in are analytic for Φ(h).Thus, Φ(h) is essentially self-adjoint on F b f in .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., which follows from explicit calculations on F b f in .The Weyl algebra W (f) over a subspace f of H ph is defined by Here, cl denotes the closure with respect to the norm of B(F b ), and "LH" denotes the linear hull.
The condition implies the existence of a Gibbs state then one can show, using the Golden-Thompson-inequality, that Hypothesis 1 is satisfied.

Hilbert space and Hamiltonian for the interacting system
The Hilbert space for the joint system is ph , obeying where k n = (k 1 , . . ., k n ).The complex conjugate vector is f := ( f n ) n∈AE0 .Let G j := {G j k } k∈Ê 3 , H j := {H j k } k∈Ê 3 and F := {F k } k∈Ê 3 be families of closed operators on H el for j = 1, . . ., r.
el,+ can be omitted.The Hamiltonians for the non-interacting, resp.interacting model are Definition 2.2.On dom(H el ) ⊗ dom(dΓ(α)) ∩ F f in b we define where The abbreviation "h.c." means the formal adjoint operator of Φ( G) Φ( H).
◮ The Nelson Model: ◮ The Standard Model of Nonrelativistic QED: The form factors are ◮ The Pauli-Fierz-Model: 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 For elements A ∈ A we define τ 0 t (A) := e i t H0 A e −i t H0 and τ g t (A) := e i t H Ae − i t H .We first discuss the model without interaction.

The Representation π f
The time-evolution for the Weyl operators is given by For this time-evolution an equilibrium state ω β f is defined by It describes an infinitely extended gas of bosons with momentum density ̺ β at temperature β.Since ω β f is a quasifree state on the Weyl algebra, the definition of ω β f extends to polynomials of creation and annihilation operators.One has For polynomials of higher degree one can apply Wick's theorem for quasi-free states, i.e., where a σ k = a * or a σ k = a for k = 1, . . ., 2m.Z 2 are the pairings, that is

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, Since ρ is a compact, self-adjoint operator, there is an ONB (φ n ) n of eigenvectors, with corresponding (positive) eigenvalues 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, σ ψ := σ ψ has the integral kernel σ(x, y).It is a straightforward calculation to verify that where L el = H el ⊗1−1⊗H el .This suggests the definition of the representation

Documenta Mathematica 16 (2011) 177-208
Now, we define the representation map for the joint system by where given by is essentially self-adjoint and we can define On K a we introduce a conjugation by 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 β el ∈ H el ⊗ H el of the Gibbs state ω β el as in ( 14) for ρ = e −βH el Z −1 .Theorem 3.1.Assume Hypothesis 1 is fulfilled.Then, Ω β 0 := Ω β el ⊗ Ω β f is a cyclic and separating vector for M β .e −β/2L0 is a modular operator and J is the modular conjugation for Ω β 0 , that is for all X ∈ M β and L 0 Ω β 0 = 0.Moreover, analytic in the strip S β = {z ∈ : 0 < Im z < β}, continuous on the closure and taking the boundary conditions For a proof see [14].
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 Moreover, we can show that e −βLQ 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 β 0 ∈ 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 For each family K = {K k } k of closed operators on H el with K w,1/2 < ∞ we set 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 which is distinguished by Next, we define four auxiliary operators on D L (1)   a where L f,a is an operator on F b ⊗ F b and H Q el,+ acts on K. Furthermore, a , 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 a , i = 1, 2, 3, 4 are essentially self-adjoint and positive.Moreover, there is a constant Proof.Let a, a ′ ∈ {l, r} and K i , i = 1, 2 be families of bounded operators with 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 By the Kato-Rellich-Theorem ( [17], Thm.X.12) we deduce that L (i) a -bounded for every c 2 0 and i = 2, 3, 4. In particular, D is a core of a φ for φ ∈ D.
Theorem 4.2.The operators 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 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 (1) a ) 1/2 φ 2 for φ ∈ D. The first inequality follows from Equation (20).

Documenta Mathematica 16 (2011) 177-208
To verify the second inequality we observe where we used, that a ), we have by means of (10) Thus, (24) is bounded by a constant times (L a ) 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 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.

Regularized Interaction and Standard Form of M β
In this subsection a regularized interaction Q N is introduced: The regularized form factors G N , H 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 Proof.Let Q N be defined on D. With the same arguments as in the proof of Theorem 4.2 we obtain 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 N and for the operators in line (21).A straightforward calculation yields 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 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 Thus Q N is affiliated with M β and therefore e i tQN ∈ M β for t ∈ Ê.
Lemma 5.2.We have for X ∈ M β and t ∈ Ê Moreover, τ t (X) ∈ M β for all X ∈ M β and t ∈ Ê, such as Proof.First, we prove the statement for Q N , since Q N is affiliated with M β and therefore e itQN ∈ M β .We set On account of Lemma 5.1 and Theorem 4.2 we can apply the Trotter product formula to obtain Since e i t n QN , X ∈ M β and since τ 0 leaves M β invariant, τ N t (X) is the weak limit of elements of M β , and hence τ N t (X) ∈ M β .Moreover, For E N (t) := e i t (L0 + QN ) e −i t L0 ∈ B(K) we obtain By virtue of Lemma 5.1 we get E Q (t) := e i t (L0 + Q) e −i t L0 = 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 J N = J e itQN J ∈ M ′ β .Since we have shown, that τ N leaves M β invariant, we get Thanks to Lemma 5.1 we also have The proof of τ 0 t (X) = e it(L0−Q J ) X e it(L0−Q J ) follows analogously.Using the Trotter product formula we obtain By strong resolvent convergence we may deduce E(t) = e itLQ e −it(L0−Q J ) .
Let C be the natural positive cone associated with J and Ω β 0 and let M ana β be the τ -analytic elements of M β , (see [4]).
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 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.
Again both functions are equal on the line z = i t, t ∈ Ê.
◮ 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 β 0 0 for all A ∈ M β .We have 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 where c := Ω β −2 .First, we observe and The requirements on the analyticity of F β (A, B, •) follow from Lemma A.2.
6 Proof of Theorem 1.3 For s n := (s n , . . . ,s 1 ) ∈ Ê n we define At this point, we check that Q N (s n )Ω β 0 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 β 0 ∈ dom(e −β/2 (L0+Q) ).Let The idea of the proof is the following.First, we expand e −β/2(L0+QN ) e L0 in a Dyson-series, i.e., e −β/2(L0+QN ) e L0 (33) Under the assumptions of Theorem 1.3 we obtain an upper bound, uniform in N , for 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)(L0+QN ) Ω β 0 , which is uniform in N .The proof of Theorem 1.3 follows now from Lemma 6.3, where we show that Ω β 0 ∈ dom(e −(β/2)(L0+Q) ).
for all N ∈ AE.Then Ω β 0 ∈ dom(e −x(L0+QN ) ), 0 < x β/2 and In this context x} is a simplex of dimension n and sidelength x.
k] and 0 x β/2 be fixed.An m-fold application of the fundamental theorem of calculus yields Since L 0 Ω β 0 = 0 we have for r(s m+1 ) := (s m − s m+1 , . . ., s 1 − s m+1 ) that 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 Since e −sm+1 (L0 + QN ) φ e β/2 k φ , and using that Q N (r m ) Ω β 0 is a state with at most 2m bosons, we obtain the upper bound const φ (2m) (2m + 1) sup Hence, for m → ∞ we get is a core of e −x(L0 + QN ) , the proof follows from the self-adjointness of e −x(L0 + QN ) .Lemma 6.2.Let 0 < x β/2.We have the identity For m = n it follows 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 β 0 = e −β/2 L0 X * Ω β 0 .Thus, Next, we introduce new variables for r, namely y i := β − r m−i+1 .Let D m x/2 := {y m ∈ Ê m : β − x y m . . .y 1 β}.Thus the right side of Equation (39) equals The second statement of the Lemma follows by choosing n = m.
Proof of 6.4.First recall the definition of Q N and Q N (s n ) in Equation (25) and Equation (31), respectively.Let The functions J n (β, 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( G) Φ( H) + h.c. , W 2 := Φ(F ) and W := W 1 + W 2 .By definition of ω β 0 in (3.1), see also (13), we obtain By definition of ω β f 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 β f of 2m field operators.In this case the expectations in ω β f can be expressed by an integral over Ê 2m × {±} 2m with respect to ν, which is defined in Lemma A.4 below.
Inserting this estimates we get where and Now, we recall the definition of ν.Roughly speaking, one picks a pair of variables (k i , k j ) and integrates over δ ki,kj 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 (2m)! 2 m m! pairings and all 4 m combinations of τ 2m .Inserting Estimate (41) and that By Lemma A.3 below and since (2m)!/(m!) 2 4 m we have This completes the proof.

The Harmonic Oscillator
Let L 2 (X, dµ) = L 2 (Ê) and H el =: H osc := −∆ q +Θ 2 q 2 be the one dimensional harmonic oscillator and H ph = L 2 (Ê 3 ).We define where 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.
These operators fulfill the CCR-relations and the harmonic-oscillator is the number-operator up to constants.
The vector Ω := Θ π 1/4 e −Θ q 2 /2 is called the vacuum vector.Note, that one It follows, that ω osc β is quasi-free, as a state over W ( ) and For #A a = 2 exists only one connected graph.We obtain for h 2n m can be considered as a lower Riemann sum for the integral m + 1 1 r −1 dr, we have n m = 1 1 m ln(n + 1).Thus, Conversely, Equation (58) and Lemma 7.  tends to zero for lim n→∞ z n = z.This implies the continuity of z → e z H φ.
For phonons we have k j ∈ Ê 3 and k j ∈ Ê 3 × {±} for photons.The wave functions in H n ph 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 f in .On F b f in the creation and Documenta Mathematica 16 (2011) 177-208
) 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 byP N := 1[H el N ].Moreover we obtain from Lemma 6.2, 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 (2ma)!ma! 2 ma choices.Let now be the lines in L osc (G a ) fixed.We have now (2m a