KMS States of Weakly Coupled Anharmonic Crystals and the Resolvent CCR Algebra

0.26 MB·

by  T. Kanda

Regular KMS States
of
Weakly Coupled
6
1
0 Anharmonic Crystals
2

y and
a
M the Resolvent CCR Algebra

6
2

]
h
T. Kanda
p
Taku Matsui
-
h
t Graduate School of Mathematics, Kyushu University,
a
744 Motoka, Nishi-ku, Fukuoka 819-0395,JAPAN
m
[email protected]
[ [email protected]

2
v
9
0
8
4
0
.
1
0
6
1 Abstract: Weconsiderequilibriumstatesofweaklycoupledanharmonicquan-
: tum oscillators on Z. We consider the Resolvent CCR Algebra introduced by
v
i D.Buchholtz and H.Grundling, and we show that the infinite volume limit of
X
equilibrium states satisfies the KMS (Kubo-Martin-Schwinger) condition with
r regularity(= locally normal to Fock representations). Uniqueness of the KMS
a
states is proven as well.
Keywords: resolventalgebra,Heisenbergtimeevolution,Lieb-Robinsonbound.
AMS subject classification: 82B10
1 Introduction
In this article, we consider KMS states of certain one-parameter group of au-
tomorphisms of C∗-algebra associated with canonical commutation relations
(CCR) , which ,physically, correspond to equilibrium states of weakly coupled
anharmonic quantum oscillators on Z.
Let (X,σ) be a 2n dimensional real symplectic vector space with a non-
degenerate symplectic form σ and let A (σ) be the CCR algebra associated
CCR
with (X,σ) which is a *-algebraof unbounded operators generated by formally
self-adjoint elements Ψ(f) (f ∈X). Ψ(f) is linear on f, satisfying CCR
[Ψ(f),Ψ(g)]=iσ(f,g) , Ψ(f)∗ =Ψ(f)
1
By the Heisenberg time evolutionof a quantum observableQ associatedwith a
Hamiltonian, we mean
α (Q)=eitHQe−itH
t
This expression is formal . Both the Hamiltonian and elements Q of A (σ)
CCR
are unbounded operators and we have to specify domains of operators. Unless
the Hamiltonian is bilinear in Boson operators Ψ(f), it is likely that α (Q) is
t
not in A (σ). A traditional way to handle A (σ) is to consider unitaries
CCR CCR
generated by Ψ(f), W(f) = eitΨ(f). The C∗-algebra generated by W(f) is
called the Weyl CCR algebra. The Weyl CCR algebra is a simple C∗-algebra
and was used in study of Bose-Einstein condensation of the ideal gas. (c.f.[6])
Specialists have realized that the Weyl CCR algebra is not a C∗-algebra
suitableforscatteringtheoryandtostatisticalmechanicsofinteractingBosons.
The fundamental problem is (non-)existence of one-parameter group of auto-
morphisms, which is physically equivalent to existence of the Heisenberg time
evolution as automorphisms of a C∗-algebra. To be precise, let us consider
the case of one degree of freedom. Let X = R2 with a symplectic basis q,p
with σ(q,q) = σ(p,p) = 0 , σ(q,p) = −σ(p,q) = 1. We consider the standard
representationonL2(R),henceq isthe multiplicationoperatorandpisthe dif-
ferential operator. Let v(q) be a potential where v is continuous, with compact
support. Set h = p2 and h = h +v(q). In [8], D.Buchholz and H.Grundling
0 2 0
pointed out that, for any bounded operator Q on L2(R),
eithQe−ith−eith0Qe−ith0
is a compact operator. (See also [11].) This implies that any C∗-algebra on
L2(R) invariant for both the free time evolution and the time evolution with a
potentialv withacompactsupportmustcontaincompactoperators. Thisrules
out the Wey CCR algebra.
Intheirresearchofmathematicalfoundationofthesupersymmetricquantum
fieldtheory,D.BuchholzandH.Grunrdlingintroducedanewapproachforstudy
of the CCR algebra, the Resolvent CCR algebra in [7] (See [8], [9] as well) The
resolvent CCR algebra is a unital C∗-algebra generated by the resolvent of the
field operators
1
R(λ,f)=
iλ+Ψ(f)
2
where λ is a non-zero real parameter. (See [8] for the precise definition of the
resolvent CCR algebra.)
The Resolvent CCR algebra has several advantages.
(i) Singular representations of the CCR algebra with infinite field strength can
be characterizedin terms of the kernel of the semi-resolvent operators.
(ii)TheFockrepresentationisfaithfulandthereisaone-to-onecorrespondence
between the regular representations of the Weyl CCR algebra and those of the
resolvent CCR algebra.
(iii) The resolvent CCR algebra for a finite quantum system contains compact
operators.
Later,wewillseetheHamiltoniansofweaklycoupledanharmonicoscillators
gives rise to a one-parameter group of automorphisms on the resolvent CCR
algebra and we consider KMS states.
To describe precise statements of our results, we introduce notations now.
For any positive integer L we set
Λ ={ j ∈ Z |−L<j ≤L }⊂Z
L
H =⊗ L2(R,dx )
ΛL k∈ΛL k
and let B be the unital abelian C∗-algebra on H generated by the unit and
n ΛL
multiplication operators associated with the functions of the form
f(s x +s x ···+s x )
−L+1 −L+1 −L+2 −L+2 L L
where f(x) is a continuous function (with a single real variable) vanishing at
infinity and s ,···s are real constants. Let R be the unital C∗-algebra
−L+1 L L
on H generated by the unit and operators
ΛL
g(s x +···+s x +t p +···+t p )
−L+1 −L+1 L L −L+1 −L+1 L L
whereg(x)isacontinuousfunction(withasinglevariable)vanishingatinfinity,
s ,t (k,l ∈Λ ) are real constants and p be the quantum mechanical momen-
k l L k
tumoperatorp =−i ∂ . DuetothetensorproductstructureofH weobtain
the natural incklusion ∂Rxk ⊂ R if L < M. The inductive limit CΛ∗L-algebra of
L M
∪ R is denoted by R. We call R and R the resolvent CCR algebra.
L ΛL L
Theorem 1.1 Let H be the Schro¨dinger operator defined by
L
L L−1
H = p2 +ω2x2 +V(x ) + ϕ(x −x ) (1.1)
L k k k k k+1
k=X−L+1(cid:8) (cid:9) k=X−L+1
where the potential V and ϕ are rapidly decreasing smooth functions.
αL defined on the Fock representation via the following equation
t
e αL(Q)=eitHLQe−itHL, Q∈R (1.2)
t L
gives rise to a one-paraemeter group of automorphisms on R .
L
3
Combined with Lieb-Robinson bound techniques on Fock spaces, (c.f. [12],
[14], [16], [17], [18] ,[21]) a C∗-dynamical systems can be introduced for weakly
coupled anharmonic oscillators on the infinite lattice Z.
Theorem 1.2 The infinite volume limit
α (Q)= lim αL(Q) (1.3)
t t
L→∞
e
exists in the norm topology of R. Let Hfree be the Hamiltoninan of decoupled
oscillators
∞
Hfree = p2 +ω2x2 +V(x ) (1.4)
k k k
k=X−∞(cid:8) (cid:9)
and set αfree(Q)=eitHfreeQe−itHfree.
t
Then, α ◦αfree(Q)andαfree◦α (Q)arecontinuousintforthenormtopology
t −t −t t
of R.
Independently,D.Buchholzobtainedtheinfinitevolumelimitofdynamicsfor
more general models by different methods in [10]. We believe that application
of Lieb-Robinson bound techniques in itself will be useful for more advanced
researchin future.
The main results of this paper is uniqueness of the regular KMS states of
weakly coupled quantum oscillators. Uniqueness of the KMS states for one-
dimensional quantum systems is a well-known fact, however, in our case, the
timeevolutionα isnotnormcontinuousint. α iscontinuousinweaktopology
t t
on the GNS representations which is locally quasi-equivalent to the standard
Fock representation. We will see that there exists a infinite volume limit ψ of
KMS states of finite systems such that the restriction ψ to each finite volume
is normal to the Fock representation and ψ(Qα (R)) is continuous in t. By
t
regularity of states we mean states locally normal to the standard Fock state.
(See Definition 5.1 and Theorem 5.8 below .)
Theorem 1.3 The regular KMS state associated with the Hamiltonian (1.1)
exists, and is unique.
StatisticalMechanicsofanharmoniccrystalswiththe Hamiltonian(1.5)de-
fined below has been extensively studied by several people. (c.f. [1], [15] and
the references therein)
H = {p2+V(x )}+ |x −x |2 (1.5)
j j i j
jX∈Zd j,i∈ZdX,||i−j||=1
where V is a polynomial giving rise to a double well potential. Note that, in
ouranharmoniccrystal,Boseparticlesarefixedonthelatticesitesandtheyare
distinguishable.
Results obtainedso fararebasedonperturbationtheory andfordeveloping
a general theory a missing point is a suitable C∗-algebra describing full quan-
tum observables. We believe that the resolvent CCR algebra introduced by
4
D.Buchholz and H.Grunrdling is the right staff for handling the full quantum
system including momentum operators. We hope the results of this article is
the first step of understanding equilibrium states of anharmonic crystals.
2 Resolvent CCR algebra
In this section, we introduce the notation and recall the results of the resolvent
CCR algebra in [8].
For a given subset Λ ⊂ Z, we denote c (Λ) by the space of all finitely
c
supported function f : Λ → C. We define the symplectic form σ on c (Λ) by
c
σ(f,g)=Imhf,gi for f,g ∈c (Λ), where h,i is the canonical inner product
ℓ2 c ℓ2
on ℓ2(Z). Then c (Λ) equipped with σ is also a symplectic space.
c
We consider the Hilbert space H associated with any finite subset Λ of Z
Λ
defined by
H = L2(R,dx ),
Λ k
kO∈Λ
where dx is the Lebesgue measure on R. To simplify the notation, for any
k
finite subsets Λ ⊂ Γ ⊂ Z, we identify the linear operator A on H with the
Λ
linear operator A⊗ on H , where is the identity operator on H .
Γ\Λ Γ Γ\Λ Γ\Λ
1 1
Thus,foranyfinite subsetΛ⊂Z, weidentify the multiplicationoperatorx on
k
L2(R,dx ), k ∈ Λ, and x ⊗ on H . Also, we identify the differential
k k 1Λ\{k} ΛL
operator p =−i ∂ and p ⊗ . We denote the trace on H by Tr .
k ∂xk k 1Λ\{k} L L
ForanysubsetΛofZ,wedenoteW(Λ)andR(Λ)bytheWeylCCRalgebra
and the resolvent CCR algebra over (c (Λ),σ), respectively. The definitions of
c
the Weyl CCR algebra and the resolventCCR algebra are as follows.
The Weyl CCR algebra is the C∗-algebra generated by W(f), f ∈ c (Λ),
c
satisfying
W(f)∗ = W(−f),
W(f)W(g) = e−iσ(f2,g)W(f +g)
for all f,g ∈c (Λ) (see e.g. [6, Theorem 5.2.8.].).
c
The resolvent CCR algebra R(Λ) is the universal C∗-algebra generated by
R(λ,f), λ∈R\{0}, f ∈c (Λ), satisfying
c
i
R(λ,0) = − , (2.1)
λ
R(λ,f)∗ = R(−λ,f), (2.2)
νR(νλ,νf) = R(λ,f), (2.3)
R(λ,f)−R(µ,f) = i(µ−λ)R(λ,f)R(µ,f), (2.4)
[R(λ,f),R(µ,g)] = iσ(f,g)R(λ,f)R(µ,g)2R(λ,f), (2.5)
R(λ,f)R(µ,g) = R(λ+µ,f +g){R(λ,f)+R(µ,g)
+iσ(f,g)R(λ,f)R(µ,g)} (λ6=−µ), (2.6)
5
where λ,µ,ν ∈ R\{0} and f,g ∈ c (Λ). (See [8].) For any subset Λ ⊂ Z,
c
the resolvent CCR algebra R(Λ) is the inductive limit of the net of all finite
dimensionalnon-degeneratesymplecticsubspacesofc (Λ)[8,Theorem4.9(ii)].
c
For any positive integer L, we set Λ = {j ∈ Z | −L < j ≤ L}. For
L
simplicity, we set R = R(Z) and RL = R(ΛL). Also, we set RLc = R(Z\ΛL)
and R = R(Λ \Λ ), for any positive integers L ≤ L′. For the Weyl
L′\L L′ L
CCR algebra, we also set W = W(Z), W = W(Λ ), W = W(Z\Λ ) and
L L Lc L
W =W(Λ \Λ ), for any positive integers L≤L′.
L′\L L′ L
Letπ betheSchr¨odingerrepresentationofR(Λ)onH . Dueto[8,Theorem
0 Λ
4.10], π is a faithful representationof R(Λ).
0
3 Lieb-Robinson bounds and limiting dynamics
for the resolvent CCR algebra
In this section, we prove the LiebLieb-Robinson bounds for weakly coupled an-
harmonicquantumoscillatorsontheresolventCCRalgebra. First,weintroduce
the notation.
For any L ∈ N and positive constant ω ≥ 0, let Hh be the self-adjoint
L
operator on H defined by
ΛL
Hh = (p2 +ω2x2).
L k k
kX∈ΛL
We define the automorphism αh,L on B(H ) by
t ΛL
αht,L(Q)=eeitHLhQe−itHLh, Q∈B(HΛL).
Since the automorpehism αh,L induce the symplectic transform on (c (Λ ),σ),
t c L
αh,L is an automorphism on π (R ). Let Φ be the map from any finite subset
t 0 L
Λ of Z to B(H ) defined bey
Λ
e
V(x ) (Λ={k})
k
Φ(Λ)= ϕ(xk−xk+1) (Λ={k,k+1}) , (3.1)
 0 (otherwise)

where V and ϕ are real valued Schwarz functions on R. The function V repre-
sent anharmonicity of the potential of the system and ϕ corresponds to the
nearest neighbor interaction of particles. For any finite subset Γ ⊂ Z, we
set Υ(Γ) = Φ(Λ). For simplicity, we set Υ = Υ(Λ ), L ∈ N and
Λ⊂Γ L L
ΥL′\L = Υ(ΛPL′\ΛL) whenever L ≤ L′. Let HL be the self-adjoint operator on
H defined by
ΛL
H =Hh+Υ = (p2 +ω2x2 +V(x ))+ ϕ(x −x ). (3.2)
L L L k k k k k+1
kX∈ΛL k,k+X1∈ΛL
6
3.1 Proof of Theorem 1.1
As the Fock representation is faithful , we consider the convergence in strong
and norm topologies of operator valued integral on the Fock space. Let U (t)
L
be the unitary operator on HΛL defined by UL(t)= eitHLe−itHLh. By using the
Dyson series expansion of U (t), we obtain
L
t t1 tn−1
U (t)= + in dt dt ··· dt αh,L(Υ )···αh,L(Υ ).
L 1 Z 1Z 2 Z n tn L t1 L
nX≥1 0 0 0
e e
(See e.g. [6, Theorem 3.1.33].) Generally speaking, the above integral makes
senseintheweaktopologyontheFockspace. However,inourcurrentsituation,
D.Buchholz and H.Grundling have shown that
t t
αh,L(V(x ), αh,L(ϕ(x −x )
Z tn k Z tn k k−1
0 0
e e
are norm continuous families of elements of the resolvent CCR algebra. (See
[8, Proposition6.1]and[8, ProofofProposition7.1])andas a consequence,the
Dyson series converge in the norm topology.
Thus, αL is a one-parameter group of automorphism on π (R ).
t 0 L
e
3.2 Lieb-Robinson bounds and limiting dynamics
Next, we consider the Lieb-Robinson bound of the automorphism α L. Before
t
weprovetheLieb-Robinsonboundofα L,weintroducethefollowingnotations.
t
(See also [19], [20] and [21].) e
We set e
Hfree = (p2 +ω2x2 +V(x ))
L k k k
kX∈ΛL
and
αftree,L(Q)=eitHLfreeQe−itHLfree, Q∈RL.
For any subset Γ⊂Z and any finite subset Λ⊂Γ, we set
S (Λ)={X ⊂Γ|X∩Λ6=∅,X ∩(Γ\Λ)6=∅}.
Γ
If Γ = Z, then we denote S(Λ) by SZ(Λ). Let ∂ΦΛ be the subset of Λ defined
by
∂ Λ={x∈Λ|for someX ∈S(Λ)with x∈X,Φ(X)6=0}.
Φ
For any finite subsets Γ ,Γ ⊂Z, we set
1 2
1 1
D(Γ ,Γ )=min , .
1 2
x∈X∂ΦΓ1yX∈Γ2 1+|x−y| xX∈Γ1y∈X∂ΦΓ2 1+|x−y|
 
7
For L ∈ N, we set C = 4 1 . We define the norm k·k of the map
x∈Z (1+|x|)2 int
Φ by P
1
kΦk = sup kΦ(Λ)k
int x,y∈Z(1+|x−y|)2 Λ:x,Xy∈Λ⊂Z
x6=y
|Λ|<∞
where |Λ| is the number of elements of Λ.
Lemma 3.1 Let Γ and Γ be finite disjoint subsets of Z. For any finite subset
1 2
Λ of Z, L ∈ N, with Γ ∪Γ ⊂ Λ and arbitrary Q ∈ π (R(Γ )) and R ∈
L 1 2 L 0 1
π (R(Γ )), it follows that
0 1
2kQkkRk
(cid:13)hαLt(αf−rete,L(Q)),Ri(cid:13)≤ C (cid:16)e2kΦkintC|t|−1(cid:17)D(Γ1,Γ2). (3.3)
(cid:13) (cid:13)
holds(cid:13)foreaney t∈R. (cid:13)
Proof. Let ϕ(Λ )= ϕ(x −x ) and for any finite subset Λ⊂ Z
L k,k+1∈ΛL k+1 k
let Q(Λ) be the normPdense subset of π0(R(Λ)) defined by
Q(Λ)=span{π (R(λ ,f )···R(λ ,f ))|λ ∈R\{0},f ∈c (Λ), i=1,··· ,n, n∈N}.
0 1 1 n n i i c
Put
F(t)= αL αfree,L(Q) ,R , (3.4)
t −t
h (cid:16) (cid:17) i
for Q∈Q(Γ1) and R∈Q(Γ2). Sience Re(λ,f) preserves the domain of multipli-
cationanddifferentialoperatorsby[8,Theorem4.2(i)]andtheSchwarzfunction
in H is analytic elements for the operator (p2 +ω2x2 +V(x )), the
ΛL k∈ΛL k k k
function F is strongly differentiable. Thus, thePderivation of F is
d
F(t)=i αL(ϕ(Λ )),F(t) −i αL αfree,L(Q) , αL(ϕ(Λ )),R . (3.5)
dt t L t −t t L
(cid:2) (cid:3) h (cid:16) (cid:17) (cid:2) (cid:3)i
Since αL is stroengly continuous on He , tehe solution F(et) of the equation (3.5)
t ΛL
satisfies the following estimate by [21, Lemma 2.2]:
e
|t|
kF(t)k≤kF(0)k+ ds αL αfree,L(Q) , αL(ϕ(Λ )),R .
Z0 (cid:13)(cid:13)h s (cid:16) −s (cid:17) (cid:2) s L (cid:3)i(cid:13)(cid:13)
SinceforanysubsetΛ⊂Z,Q(Λ(cid:13))iesaneormdensesubesetinπ (R(Λ))(cid:13),theabove
0
inequality holds for any elements of Q∈π (R(Γ )) and R∈π (R(Γ )).
0 1 0 2
By the proof of [21, Theorem 3.1], we get
2kQkkRk
(cid:13)hαLt(αf−rete,L(Q)),Ri(cid:13)≤ C (cid:16)e2kΦkintC|t|−1(cid:17)D(X,Y). (3.6)
(cid:13) (cid:13)
for any(cid:13)Qe∈eπ (R(Γ )) and(cid:13)R∈π (R(Γ )).
0 1 0 2
We define the automorphisms αL and αh,L (t∈R) on R by
t t L
αL(Q)=π−1(eitHL)Qπ−1(e−itHL), (3.7)
t 0 0
αfree,L(Q)=π−1(eitHLfree)Qπ−1(e−itHLfree) (3.8)
t 0 0
for Q∈R . Note that the Schr¨odingerrepresentationπ is faithful representa-
L 0
tion. Thus, we get the following.
8
Corollary 3.2 Let Γ and Γ be finite disjoint subsets of Z. For any finite Λ
1 2 L
with Γ ∪Γ ⊂Λ and arbitrary Q∈R(Γ ) and R∈R(Γ ), it follows that
1 2 L 1 1
2kQkkRk
(cid:13)hαLt(αf−rete,L(Q)),Ri(cid:13)≤ C (cid:16)e2kΦkintC|t|−1(cid:17)D(Γ1,Γ2). (3.9)
(cid:13) (cid:13)
holds(cid:13)for all t∈R. (cid:13)
Theorem 3.3 For any t∈R, L∈N and Q∈R , the norm limit
L
lim αN(Q)=α (Q) (3.10)
t t
N→∞
exists and the convergence is uniform for t in compact sets.
Proof. The assertionfollows fromthe proofof[21, Theorem4.1]andthe above
corollary.
Finally, we give the proof of Theorem 1.2.
3.3 Proof of Theorem 1.2.
The existence of the infinite volume limit is proven in Theorem 3.3. Thus, we
prove the continuity of α ◦αfree in t ∈ R. We may assume that t ∈ [0,T]. By
t −t
the Dyson series expansions of eitHLe−itHLfree, we obtain
t tn−1
(cid:13)(cid:13)(cid:13)eitHLe−itHLfree −1(cid:13)(cid:13)(cid:13) ≤= e(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2nXL≥k1ϕikn∞ZT0−dt11,···Z0 dtnαtn(ϕ(ΛL))···αt1(ϕ(Λ(3L.)1)1(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13))
where ϕ(Λ ) = ϕ(x −x ). Note that kΦk ≤ 1kϕk , where
L k,k+1∈ΛL k+1 k int 4 ∞
kϕk is the suprPemum norm of ϕ. By using the estimate (83) of the proof of
∞
[21, Theorem 4.1], for any L∈N, L≤N ≤N′ and Q∈R , we have
L
αN′ ◦αfree,N′(Q)−αN ◦αfree,N(Q)
t t t t
(cid:13) (cid:13)
(cid:13)1 (cid:13) 1
≤ (cid:13)T(1+e21Ckϕk∞T)kQk (cid:13) .
2 (1+|k−l|)2
kX∈ΛLl∈ΛXN′\ΛN
Thus, when N =L and N′ →∞, we obtain
αt◦αftree(Q)−αLt ◦αftree,L(Q) ≤T(1+e21Ckϕk∞T)kQkLC. (3.12)
(cid:13) (cid:13)
(cid:13) (cid:13)
By (3.(cid:13)11) and (3.12), it follows that (cid:13)
α ◦αfree(Q)−Q ≤ α ◦αfree(Q)−αL◦αfree,L(Q)
t t t t t t
(cid:13)(cid:13) (cid:13)(cid:13) +(cid:13)(cid:13)(cid:13)2 eitHLe−itHLfree − kQk (cid:13)(cid:13)(cid:13)
(cid:13) 1(cid:13)
≤ T(1(cid:13)(cid:13)+e21Ckϕk∞T)kQk(cid:13)(cid:13)LC+2(e2TLkϕk∞ −1)kQk.
Thus, we are done.
9
4 Regular states of the resolvent CCR algebra
In this section, we consider regular states on R , L ∈ N, or R. Recall that a
L
state ψ on R or R is regular, if and only if ker(π (R(λ,f))) = {0} for any
L ψ
λ ∈ R\{0} and f ∈ c (Λ ) or f ∈ c (Z), respectively, where π is the GNS
c L c ψ
representation associated with ψ. ( c.f. [8, Definition 4.3] ) In another word ,
π (R(λ,f)) is the resolventofa closedoperatorif ψ is regular. Note thatthere
ψ
isaone-to-onecorrespondencebetweenaregularstateofRandthatofW. (See
[8, Corollary 4.4.] .) and by abuse of notations, we employ the same notation ,
ψ or ϕ etc. for the regular states of R and W.
The following claims are straightforwardimplicationof the Stone-vonNeu-
mann uniqueness theorem. (See e.g. [6, Corollary 5.2.15].)
Lemma 4.1 Let ψ be a regular state of R . Then. ψ is normal with respect to
L
the Fock representation.
Corollary 4.2 Let ψ be a regular state on R . Then there exists a positive
L
trace class operator ρ on H such that Tr (ρ) = 1 and ψ(Q) = Tr (ρπ (Q)),
ΛL L L 0
where π is the Schro¨dinger representation of R and Tr is the trace on H .
0 L L ΛL
Lemma 4.3 Let ψ be a regular state on R , L ∈ N. Let (H ,π ,ξ ) be the
L ψ ψ ψ
GNS representation of ψ. Put K(Λ )=π−1(K(H )), where K(H) is the set of
L 0 ΛL
all compact operator on a Hilbert space H. Then, π (K(Λ )) is weekly dense in
ψ L
π (R )′′.
ψ L
Proposition 4.4 Let ψ be a regular state on R , L ∈ N. Then, ψ(αL(Q)R)
L t
is continuous on t∈R for any Q,R∈R where αL is defined in (3.7).
L t
Proof. In B(HΛL), we consider the Dyson series of UL(t) = eitHLe−itHLh and
U (t)− has the following estimate:
L
1
t tn−1
kU (t)− k = (cid:13) in dt ··· dt αh,L(Υ )···αh,L(Υ )(cid:13)
L 1 (cid:13) Z 1 Z n tn L t1 L (cid:13)
(cid:13)nX≥1 0 0 (cid:13)
(cid:13) e e (cid:13)
≤ (cid:13)(cid:13)(e|t|kΥLk−1). (cid:13)(cid:13)
Note that αh,L preserve the set of all of compact operators K(H ) and for
t ΛL
Q∈K(H ), αh,L(Q) is norm continuous. Since for Q∈K(H ), we obtain
ΛLe t ΛL
eitHLQee−itHL −Q = UL(t)eitHLhQe−itHLhUL(t)−1−Q
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) ≤ 2(cid:13)(cid:13)(e|t|kΥLk−1)kQk+ αh,L(Q)−(cid:13)(cid:13)Q . (4.1)
t
(cid:13) (cid:13)
(cid:13) (cid:13)
Since the Schr¨odingerrepresentationπ is faithful, for an(cid:13)yeQ∈K(H (cid:13)), αL(Q)
0 ΛL t
is norm continuous for t∈R. By Corollary 4.2 and Lemma 4.3, ψ(RαL(Q)) is
t
continuous for R,Q∈R . e
L
10