*-Derivations and QM∞

*-Derivations and QM

systems:

preliminary results

Fabio Bagarello

Fukuoka, 2003

Physical motivation

Let S be a physical system. Its dynamical behavior is known (in principle) when the hamiltonian operator of S, H, is known:

Schr¨odinger representation:

idΨ(~r, t)

dt = HΨ(~r, t);

Heisenberg representation:

dA(t)

dt = i[H, A(t)].

Here Ψ is the wave function of S, while A is any observable of the system. The measured quantities are

< A >_Ψ (t) =< Ψ(~r, t), A(0)Ψ(~r, t) >=< Ψ(~r, 0), A(t)Ψ(~r, 0) >

For many physical systems, however, no H operator can be de- fined. We only have a regularized operator, H_L, corresponding, for instance, to a finite volume version of S. Therefore we can only know an approximated version of the dynamics of S, solving

idΨ_L(~r, t)

dt = H_LΨ_L(~r, t) or dA_L(t)

dt = i[H_L, A_L(t)].

Examples:

A class of examples is the mean field models. A discrete (spin) example is the Heisemberg model

H_V^(a) = 1

|V | X

i,j∈V 3

X

α,β=1

A^(α,β)σ_i^ασ_j^β, where i, j are lattice indexes,

σ³ =





1 0 0 −1



, σ¹ =



 0 1 1 0



, σ² =





0 −i i 0



 are the Pauli matrices and A^(α,β) is a 3 × 3 matrix.

Another spin model with long-range interaction is described by H_V^(b) = X

i,j∈V 3

X

α=1

J_i,j(σ_i^α − σ^α)(σ_j^α − σ^α),

where J_i,j is the long-range potential and σ^α is proportional to the 2 × 2 identity matrix.

In 1992 and 1993 [C. Trapani, F.B.], more spin models have been introduced with the properties that

kH_Vk → ∞ and kH_Vk

|V | → ∞,

when V → ∞: these are the AMFSM. While for H_V^(a) and H_V^(b) [G.

Morchio, F.B.] von Neumann algebras produce a good framework, for AMFSM these are not enough!

Then the following questions arise: is it possible to remove the cutoff ? For which systems? And how? Is this limit related to the original physical system?

Historical overview

The old algebraic formulation of the dynamical description of quantum systems is due to Haag and Kastler: its main ingredient is a C*-algebra of the quasi-local observables, A:

volume V −→ A_V,

Von Neumann algebra of the observables localized in V ; A_o = [

V

A_V ⇒ A = A_o^{k k}, where k k is the C*-norm induced by A_V.

However several models do not fit into this algebraic set-up:

already for long-ranged spin systems (H_V^(b)), or mean field spin systems (H_V^(a)), Robinson’s constraint on the potential is not sat- isfied and, as a consequence, the dynamics cannot be defined as a norm limit of the infrared cutoffed dynamics, since the time evolu- tion of a strictly local variable may involve sequences of completely delocalized operators. Even worse is the situation for continuous systems. To deal with these more realistic models, two possible ways have been developed.

On one hand, one can select a certain family of relevant states where the dynamics can be defined (Dubin and Sewell, Thirring, Strocchi and Morchio, B., etc...).

On the other hand, one can enlarge the algebraic set-up:

O*-algebras (C.Trapani, G. Epifanio,..); quasi *-algebras (G.

Lassner,...) ; partial *-algebras (J.P. Antoine, W. Karwowski,...);

CQ*-algebras (C.Trapani, F.B.,...).

The problem of performing rigorously the thermodynamical limit of some local observables was the physical reason motivating the introduction of quasi *-algebras (Lassner’s treatment of the BCS model of superconductivity):

STEP 1 (shared with Thirring and Wehrl)

The physical system S is considered inside a box of finite volume V . Under this condition (fixed cutoff), we write the hamiltonian of the finite system S_V, H_V, and its associated regularized Heisenberg evolution,

α^t_V(X) = e^iHVtXe^−iHVt



=⇒ dα_V^t (X)

dt = i[H_V, α^t_V(X)]

 , where X is a local observable of the system. Here

H_V = 2g

|V | X

i,j∈V

σ_i⁻σ_j⁺ + X

i∈V

σ_i³ = 2g|V |S_V⁻S_V⁺ + |V |S_V³, where σ_i^α is the α-component of the 2 × 2 Pauli matrices localized in the lattice site i ∈ V (σ_j⁺ = σ_j^x + iσ_j^y, σ_j⁻ = (σ_j⁺)^†), g and  are constants, and S_V^α = _{|V |}¹ P

i∈V σ_i^α.

STEP 2

For finite V , α^t_V(X) = f (t, X, S_V^α) belongs to the standard C*-algebra of the spin observables. To compute lim_{|V |→∞}α^t_V(X), Lassner first introduces the physical topology, τ , different from the usual topologies on C*-algebras. This was necessary since lim_{|V |,∞}α^t_V(X) does not exist, for generic X in the uniform, strong or weak topologies.

STEP 3

Lassner proved, using explicit estimates, that even if H_V does not converge (in any topology!), τ − lim_{|V |→∞}α_V^t (X) exists, for any local observable X, and it belongs to the τ -completion of the C*-spin algebra. This is because τ − lim_{|V |→∞}S_V^α exists!

This is the first physical application of a topological quasi *- algebra!

Remark both Lassner and Thirring and Wehrl introduced an effective hamiltonian H_{ef f} which essentially shares with H_V the

property to reproduce the same equation of motion when |V | →

∞:

idα_V^t (A)

dt = [H_V, α^t_V(A)] ^{|V |→∞}−→ idβ^t(A)

dt = [H_{ef f}, β^t(A)], which has the following solution: β^t(A) = e^{iHef ft}Ae^{−iHef ft}.

From the physical point of view replacing H_V with H_{ef f} amounts to replace a two spin interaction with an interaction of a single spin with an external magnetic field whose value is fixed by the representation of the algebra.

Our goal is to extend Lassner’s procedure to get the rigorous definition of the algebraic dynamics, i.e. the time evolution of observables and/or states, for a general physical system.

For that we follow [Sewell2002]: the description of a physical sys- tem S implies the knowledge of the set of the observables, the states and the dynamics. The physical motivations above suggest, as an algebraic structure, a quasi *-algebra (A, A₀). The set of states over (A, A₀), σ, is described for instance in [TrapaniReview1995], while the dynamics is usually a group (or a semigroup) of automor- phisms of the algebra, α^t. Therefore we put S = {(A, A₀), σ, α^t}.

The system S must be now regularized: we introduce some cutoff L, (e.g. a volume or an occupation number cutoff), so that S is replaced by a sequence of systems S_L, one for each value of L:

S = {(A, A₀), σ, α^t} −→ S_L = {A_L ⊂ (A, A₀), σ, α^t_L}

We call H_L, the bounded and self-adjoint operator which generates α_L^t .

1) H_L converges to an operator H

This is apparently the simplest situation. The convergence of the regularized dynamics α^t_L(A) = e^iHLtAe^−iHLt to the solution α^t(A) of the (formal) Heisenberg equation

idα^t(A)

dt = [H, α^t(A)].

has been analyzed in F. Bagarello, C. Trapani, Algebraic dynam- ics in O*-algebras: a perturbative approach, J. Math. Phys., 43, 3280-3292 (2002).

2) H_L does not converge (in any reasonable topology)

This is the most common situation, e.g. mean field and long- range interaction models. We have two possibilities:

(2a) consider the limit of α^t_L(A) = e^iHLtAe^−iHLt for some clev- erly chosen topology, see BCS or the AMFS models; it may exist even if H_L does not converge!

(2b) consider the derivations

δ_L(A) = i[H_L, A]

that give, at infinitesimal level, the dynamics of the system, [Sakai].

First question: do these derivations converge (in some sense)?

Which properties this limit δ enjoys? Is δ a derivation? and, in this case, is δ spatial (i.e., ∃H : δ(a) = i[H, A])?

Second question:whenever δ turns out to be a derivation, can we integrate it? In other words, since δ_L can be integrated, α_L^t (A) = e^iHLtAe^−iHLt, what can be said about the limit of α^t_L?

Mathematical preliminaries

Let A be a vector space and A_o a *-algebra contained in A. We say that A is a quasi *-algebra over A_o if

(i) the right and left multiplications of an element of A by an element of A_o are always defined and linear;

(ii) an involution * (which extends the involution of A_o) is defined in A with the property (AB)^∗ = B^∗A^∗ whenever the multiplication is defined.

A quasi *-algebra (A, A_o) is said to have a unit I if there exists an element I ∈ A_o such that AI = IA = A, ∀ A ∈ A. Finally, the quasi *-algebra (A, A_o) is said to be topological if A carries a locally convex topology ξ such that (a) the involution is continuous and the multiplications are separately continuous; and (b) A_o is dense in A[ξ].

Example:

Let H be an Hilbert space and N an unbounded, self adjoint op-

erator defined on a dense domain D(N ) ⊂ H.

We call D(N^k) the domain of the operator N^k, k ∈ N, and D := D^∞(N ) = ∩_k≥0D(N^k), which is dense in H.

topology t : φ ∈ D → kφk_n := kNⁿφk, n ∈ N0,

where k k is the norm of H. L⁺(D) is the *-algebra of all the closable operators defined on D which, together with their adjoints, map D into itself. Moreover, calling D⁰ the conjugate dual space of D, endowed with the strong dual topology t⁰, we define the set L(D, D⁰) of all the continuous linear maps from D[t] into D⁰[t⁰].

The topologies on L⁺(D) and L(D, D⁰) are introduced by means of the set C of all positive, bounded and continuous functions f (x) on R⁺ satisfying the condition sup_x≥0f (x)x^k < ∞, ∀k ∈ N. The seminorms of the topology τ on L⁺(D) are

X ∈ L⁺(D) → kXk^f,k := maxkf (N )XN^kk, kN^kXf (N )k , where k ≥ 0 and f belongs to C. Lassner proved that L⁺(D)[τ ] is a complete locally convex topological *-algebra, with involution

X^† := X^∗|_D.

The seminorms of the uniform topology τ_L on L(D, D⁰) are defined by

X ∈ L(D, D⁰) → kXk^f := kf (N )Xf (N )k. (1) where, again, f belongs to C. L(D, D⁰), with the topology τ_L, is a topological quasi *-algebra over L⁺(D).

PHYSICAL SYSTEM

⇓

OPERATOR N

⇓

D, L

(D), L(D, D

) and t, τ, τ

Examples:

In [BT-JSP1992], where H_V = _{|V |}¹γ

P

i,j∈V σ_i³σ_j³, γ ∈]0, 1], we have taken N = I+¹₂ P

i∈Z[Ii − (~σ_i · ~n_i)], where {~n_i} is a relevant sequence of 3-vectors.

In [BT-NCB1993], where H_V = _{|V |}¹γ

P

i,j∈V

P3

α=1σ_i^ασ_j^α, γ ∈ ]¹₂, 1], N has the same formal expression as above.

In [B-JMP1998], where H = a^†a, [a, a^†] = I, we have taken N = H.

The same choice was made in [B-NCB2002], where H = a^†a but aa^† − f (q)a^†a = I. Here f (q) is a function, with domain D(f ) ⊂ R, such that f (q) ≥ −1, ∀q ∈ D(f ); ∃ D⁺ ⊂ D(f ), of non-zero measure, such that f (q) > 1, ∀q ∈ D₊; ∃ q_f, q_b ∈ D(f ), such that lim_q→qf f (q) = −1, lim_q→qb f (q) = 1.

The existence of an effective hamiltonian can be formalized in the framework of quasi *-algebras introducing first the notion of a weak *-representation π:

given a quasi *-algebra (A, A_o), a linear map π : A → L(D, D⁰) is a weak *-representation of (A, A_o) if

(i) π(A^∗) = π(A)^†, ∀A ∈ A;

(ii) π restricted to A_o is a *-representation of A_o;

(iii) π(AB) = π(A)π(B), whenever A ∈ A_o or B ∈ A_o.

Then we say that the modeladmits an effective hamiltonian in the weak *-representation π if there exists a self-adjoint operator H_π^{ef f} in H_π(⊃ D) with the property

π(δ(A)) = iH_π^{ef f}, π(A) , ∀A ∈ A₀. This equation is understood in the following weak sense:

< π(δ(A))φ, ψ >= i{ < π(A)φ, H_π^{ef f}ψ > −

− < H_π^{ef f}φ, π(A^∗)ψ > }, ∀φ, ψ ∈ D(H_π^{ef f}), ∀A ∈ A₀. Remark:– This definition says, in practice, that the derivation

on π(A₀)

δ_π(π(A)) = π(δ(A))

is spatial and the implementing operator is self-adjoint. As is known, both these conditions require quite strong assumptions to be fulfilled.

We have consider some consequences of this definition in [BT- JMP1996]: for instance, the existence of the dynamics was proved under the following continuity assumption:

kπ(δ(A))k_N

π,N_π ≤ C_πkπ(A)k_N

π,N_π, C_π being a positive constant, kY k_N

π,N_π = ke^−NπY e^−Nπk, which must hold for all A in a certain dense *-subalgebra D of A, or even if

k[H_π^{ef f}, π(X)]_kk_Nπ,Nπ ≤ M_X^kL_X,π

is verified, where X ∈ A₀ and M_X, L_X,π are two positive constants which may depend on X.

Examples:

All the examples listed above satisfy these criteria, as well as the van der Waals spin model discussed by van Hemmen in Fortsch.

Phys, 1978.

But: when does a physical model admits an effective hamiltonian?

Dynamics at the infinitesimal level

For details we refer to:

F. B., A. Inoue, C Trapani, Derivations of quasi ∗-algebras, submitted to Int. Jour. Math. and Math. Sci.

Let (A, A₀) be a quasi *-algebra.

Definition 1 A *-derivation of A₀ is a map δ : A₀ → A with the following properties:

(i) δ(x^∗) = δ(x)^∗, ∀x ∈ A₀;

(ii) δ(αx + βy) = αδ(x) + βδ(y), ∀x, y ∈ A₀, ∀α, β ∈ C;

(iii) δ(xy) = xδ(y) + δ(x)y, ∀x, y ∈ A₀.

We need a slightly different kind of representation of a quasi

*-algebra, now:

Definition 2 Let (A, A₀) be a quasi *-algebra, D_π a dense do- main in a certain Hilbert space H_π, and π a linear map from A into L^†(D_π, H_π) such that:

(i) π(a^∗) = π(a)^†, ∀a ∈ A;

(ii) if a ∈ A, x ∈ A₀, then π(a)^ π(x) is well defined and π(ax) = π(a)^π(x).

We say that such a map π is a *-representation of A. More- over, if

(iii) π(A₀) ⊂ L^†(D_π),

then π is a *-representation of the quasi *-algebra (A, A₀).

Let π be a *-representation of A. The strong topology τ_s on π(A) is the locally convex topology defined by the following family of seminorms: {p_ξ(.); ξ ∈ D_π}, where p_ξ(π(a)) ≡ kπ(a)ξk, where a ∈ A, ξ ∈ D_π.

Let (A, A₀) be a quasi *-algebra and δ be a *-derivation of A₀. Let π be a *-representation of (A, A₀).

We will always assume that whenever for x ∈ A₀ π(x) = 0, then π(δ(x)) = 0.

Under this assumption, the linear map

δ_π(π(x)) = π(δ(x)), x ∈ A₀, (2) is well-defined on π(A₀) with values in π(A) and it is a *-derivation of π(A₀). We call δ_π the *-derivation induced by π.

Given such a representation π and its dense domain D_π, we consider the usual graph topology t_† generated by the seminorms

ξ ∈ D_π → kAξk, A ∈ L^†(D_π). (3) Calling D_π⁰ the conjugate dual of D_π we get the usual rigged Hilbert space D_π[t_†] ⊂ H_π ⊂ D⁰_π[t⁰_†], where t⁰_† denotes the strong dual topology of D_π⁰. As usual we introduce with L(D_π, D_π⁰ ) and L^†(D_π). In this case, L^†(D_π) ⊂ L(D_π, D⁰_π). We know that each operator A ∈ L^†(D_π) can be extended to all of D_π⁰ in the following way:

< ˆAξ⁰, η >=< ξ⁰, A^†η >, ∀ξ⁰ ∈ D_π⁰, η ∈ D_π.

Therefore the multiplication of X ∈ L(D_π, D⁰_π) and A ∈ L^†(D_π) can always be defined:

(X ◦ A)ξ = X(Aξ), and (A ◦ X)ξ = ˆA(Xξ), ∀ξ ∈ D_π. With these definitions it is known that (L(D_π, D_π⁰ ), L^†(D_π)) is a quasi *-algebra. π is a weak *-representation on (L(D_π, D⁰_π), L^†(D_π)).

We have the following

Theorem 3 Let (A, A₀) be a topological quasi *-algebra with identity I and δ be a *-derivation of A₀.

Then the following statements are equivalent:

(i) There exists a (τ −τ_s)-continuous, ultra-cyclic *-representation π of A, with ultra-cyclic vector ξ₀, such that the *-derivation

δ_π induced by π is spatial, i.e.

there exists H = H^† ∈ L(D_π, D_π⁰ ) such that Hξ₀ ∈ H_π and

δ_π(π(x)) = i{H ◦ π(x) − π(x) ◦ H}, ∀x ∈ A₀. (4)

(ii) There exists a positive linear functional f on A₀ such that:

f (x^∗x) ≤ p(x)², ∀x ∈ A₀, (5) for some continuous seminorm p of τ and, calling ˜f the con- tinuous extension of f to A, the following inequality holds:

| ˜f (δ(x))| ≤ C(p

f (x^∗x) + p

f (xx^∗)), ∀x ∈ A₀, (6) for some positive constant C.

(iii) There exists a positive sesquilinear form ϕ on A × A such that:

ϕ is invariant, i.e.

ϕ(ax, y) = ϕ(x, a^∗y), for all a ∈ A and x, y ∈ A₀; (7) ϕ is τ -continuous, i.e.

|ϕ(a, b)| ≤ p(a)p(b), for all a, b ∈ A, (8) for some continuous seminorm p of τ ; and ϕ satisfies the fol- lowing inequality:

|ϕ(δ(x), I)| ≤ C(p

ϕ(x, x) + p

ϕ(x^∗, x^∗)), ∀x ∈ A₀, (9) for some positive constant C.

Remarks:–(1) This result extends the analogous result for C*- algebras, [Bratteli and Robinson]; (2) if we add to a spatial *- derivation δ₀ a perturbation δ_p such that δ = δ₀ + δ_p is again a

*-derivation, it is reasonable to analyze under which conditions δ is still spatial. The answer is easily found under the following very general (and natural) assumptions: | ˜f (δ_p(x))| ≤ | ˜f (δ₀(x))|, for all x ∈ A₀, which is exactly what we expect since δ_p is small compared to δ₀. If we call H, H₀ and H_p the operators which implement

δ, δ₀ and δ_p, we can also prove that i[H, A]ψ = i[H₀ + H_p, A]ψ, for all A ∈ L^†(D_π) and ψ ∈ D_π.

In order to apply our results to QM_∞ we extend the above Theo- rem, assuming that there exists a (τ −τ_s)-continuous *-representation π in the Hilbert space H_π, which is ultra-cyclic with ultra-cyclic vector ξ₀, and a family of *-derivations (in the sense of Definition 1) {δ_n : n ∈ N} of a locally convex quasi *-algebra (A, A⁰) with identity. We define a related family of *-derivations δπ⁽ⁿ⁾ induced by π defined on π(A₀) and with values in π(A):

δ_π⁽ⁿ⁾(π(x)) = π(δ_n(x)), x ∈ A₀. (10) Proposition 4 Suppose that:

(i) {δ_n(x)} is τ -Cauchy for all x ∈ A₀;

(ii) For each n ∈ N, δ^π(n) is spatial, that is, there exits an

operator H_n such that

H_n = H_n^† ∈ L(D_π, D⁰_π),

H_nξ₀ ∈ H_π and δ_π⁽ⁿ⁾(π(x)) = i{H_n◦ π(x) − π(x) ◦ H_n}, ∀x ∈ A₀;

(iii)

sup

n

kH_nξ₀k =: L < ∞.

Then we have:

(a) ∃ δ(x) = τ − lim δ_n(x), for all x ∈ A₀, which is a *- derivation of A₀;

(b) δ_π, the *-derivation induced by π, is well-defined and spa- tial;

(c) if H is the self-adjoint operator which implements δ_π, if

< (H_n − H)ξ₀, ξ >→ 0 for all ξ ∈ D_π then H_n converges weakly to H.

Examples: (1) A radiation model In this example H = Pn

i=1a^†_ia_i. Here a_i and a^†_i satisfy the following CCR [a_i, a^†_j] = Iδ_i,j. Let Q_L be the projection operator on the subspace of H with at most L bosons. This operator can be written considering the spectral decomposition of H_(i) = a^†_ia_i = P∞

l=0lE_l⁽ⁱ⁾. We have Q_L = Pn i=1

PL

l=0E_l⁽ⁱ⁾. Let us now define a bounded operator H_L in H by H_L = Q_LHQ_L. It is easy to check that, for any vector Φ_M with M bosons (i.e., an eigenstate of the number operator N = H = Pn

i=1a^†_ia_i with eigenvalue M ), the condition sup_LkH_LΦ_Mk < ∞ is satisfied. In particular, for instance, sup_LkH_LΦ₀k = 0. It may be worth remarking that all the vectors Φ_M are cyclic. Calling δ_L the derivation implemented by H_L and δ the one implemented by H, it is clear that all the assumptions of the previous Proposition are satisfied, so that, in particular, the weak convergence of H_L to H follows. This is in agreement with [B-JMP1998].

(2) A mean-field spin model

In this case there exists no hamiltonian for the whole physical system but only for a finite volume subsystem: H_V = _{|V |}¹ P

i,j∈V σ₃ⁱσ₃^j. It is convenient to introduce the mean magnetization operator σ₃^V = _{|V |}¹ P

i∈V σ₃ⁱ. Let us indicate with ↑_i and ↓_i the eigenstates of σ₃ⁱ with eigenvalues +1 and −1. We define Φ_↑ = ⊗_i∈V ↑_i. It is clear that σ^V₃ Φ_↑ = Φ_↑, which implies that H_VΦ_↑ = |V |Φ_↑, which in turns implies that sup_V kH_VΦ_↑k = ∞. Therefore the cyclic vector Φ_↑ does not satisfy the main assumption of Proposition 4.

However, it is possible to consider a different cyclic vector Φ₀ = ....⊗ ↑_j−1 ⊗ ↓_j ⊗ ↑_j+1 ⊗ ↓_j+2 ⊗...,

(or local modifications of it) which is again an eigenstate of σ₃^V. Its eigenvalue depends on the volume V . However, it is clear that kσ^V₃ Φ₀k = _{|V |}¹ kΦ₀k_V, where _V can take only values 0, 1. Anal- ogously we have kH_VΦ₀k = _{|V |}¹ kΦ₀k²_V → 0, so that this vec- tor satisfies the assumptions of Proposition 4, and the derivation

δ_V(.) = i[H_V, .] converges to a derivation δ which is spatial and implemented by an operator H, which is the weak limit of H_V.

It is self-evident the special role played here by the cyclic vector.

Some new results: work in progress

As we have discussed in the Introduction, given S we have to replace it with a family of regularized systems S_L = {A_L ⊂ (A, A₀), σ, α^t_L}. We suppose that the dynamics α^t_L is generated by a *-derivation δ_L. The procedure of the previous section suggests to introduce the following

Definition 5 A family {S_L} is said to be c-representable if, among all the *-representations of (A, A₀), there exists at least one, π, such that:

(i) π is (τ − τ_s)-continuous;

(ii) π is ultra-cyclic with ultra-cyclic vector ξ₀;

(iii) if π is such that π(x) = 0, then π(δ_L(x)) = 0, ∀L.

Any such representation π is said to be a c-representation.

Proposition 6 Let {S_L} be a c-representable family and π a c-representation. Let H_L = H_L^∗ ∈ A_L be the operator which im-

plements δ_L: δ_L(X) = i[H_L, X], for all X ∈ A₀. Suppose that (1) δ_L(X) is τ -Cauchy for all X ∈ A₀; (2) sup_Lkπ(H_L)ξ₀k <

∞. Then

(a) δ(X) = τ − lim_Lδ_L(X) exists in A and is a *-derivation of A₀;

(b) δ_π, the *-derivation induced by π, is well defined and spa- tial.

Remark:– To prove the statement it is enough to check that δ_L^(π)(X) = π(δ_L(X)) satisfies the assumptions of Proposition 4.

Proposition 6 produces a sufficient condition for a model to admit an effective hamiltonian, in the sense of [BT1996].

Preliminary results on α

Let us now assume that the operator H_L introduced in the pre- vious section can be written in terms of some (intensive) operators S_L^α, α = 1, 2, .., N , which are assumed to be self-adjoint , and:

S^α = τ − lim

L S_L^α, [S^α, x] = 0, ∀x ∈ A₀.

This happens for mean field and AMF spin models, for instance.

Definition 7 We say that {S_L^α} is an uniformly τ -continuous sequence if, for all seminorms p of τ and all α there exist another seminorm q of τ and a positive constant c_p,q,α such that

p(S_L^αa) ≤ c_p,q,αq(a), ∀a ∈ A. (11)

(⇒ p(aS_L^α) ≤ c_p,q,αq(a), ∀a ∈ A and the same inequalities can be extended to S^α)

It is now straightforward to prove τ − lim_L(S_L^α)^k = (S^α)^k for α = 1, 2, .., N and k = 1, 2, ..., which implies the following result Proposition 8 Suppose that

(1) ∀x ∈ A₀ [H_L, x] depends on L only through S_L^α;

(2) S_L^α −→ S^{τ α} and {S_L^α} is an uniformly τ -continuous se- quence.

Then the following limits exist τ − lim

L i^k[H_L, x]_k = τ − lim

L δ_L^k(x), ∀x ∈ A₀, and define an element of A which we call δ^(k)(x).

Again:– the assumptions are satisfied, e.g., by mean field and AMF spin models.

From now on we suppose that e^iH(π)t ∈ L^†(D_π). Following [BT1996] we introduce the following

Definition 9 We say that x ∈ A₀ is a generalized analytic element of δ if

(i) P∞ k=0

(it)^k

k! π(δ^(k)(x)) is τ_s-convergent;

(ii) P∞ k=0

(it)^k

k! π(δ^(k)(x)) = e^iH(π)tπ(x)e^−iH(π)t. In this case we say that x ∈ G.

We can prove now the following

Proposition 10 Suppose that, whenever π(x_γ) −→ π(x) then^τs x_γ → x. Then, ∀x ∈ G the series^τ P∞

k=0 (it)^k

k! δ^(k)(x) converges in the τ -topology to an element of A which we call α^t(x). More- over, for all seminorms p of τ there exist two positive constants c₁ and c₂ such that

1 c₁ inf

ϕ∈Dπ

p_ϕ(π(x)) ≤ p(α^t(x)) ≤ c₂ sup

ϕ∈Dπ

p_ϕ(π(x)), ∀t ∈ R. (12) Finally, α^t can be extended to the τ -closure G of G.

S

↓ S_L

↓ H_L

τ

6−→

↓

δ_L(·) = i[H_L, ·]

τ ↓ δ(·) π .

δ_π(·) −→ H^(π) −→ e^iH(π)t · e^−iH(π)t .

α^t

Open problems:

• Is the set G rich enough?

• Under which conditions H^(π) belongs to L^†(D_π)?

• Under which conditions e^iH(π)t belongs to L^†(D_π)?

• What can be said about the role of the representations?

• We need more results about α^t

• ...