On the time evolution of QM∞

On the time evolution of QM

systems:

an algebraic approach

Fabio Bagarello

Wascom, 2003

Introduction

The old algebraic formulation of the dynamical description of quantum systems is due to Haag and Kastler: its main ingredi- ent 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 AV.

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

already for long-ranged spin systems Robinson’s constraint on the potential is not satisfied and, as a consequence, the dynam- ics cannot be defined as a norm limit of the infrared cutoffed dynamics, since the time evolution 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 ofrelevant 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 motivat- ing 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 reg- ularized Heisenberg evolution,

α^t_V(X) = e^iHVtXe^−iHVt,

where X is a local observable of the system. Here HV = 2g

|V | X

i,j∈V

σ_i⁻σ_j⁺+ X

i∈V

σ_i³ = 2g|V |S_V⁻S_V⁺+ X

i∈V

σ_i³, where σ_i^αis the α-component of the 2×2 Pauli matrices localized

in the lattice site i ∈ V , g and  are constants, and S_V^α =

1

|V |

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 Hef f which essentially shares with HV

the property to reproduce the same equation of motion when

|V | → ∞:

idα^t_V(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}.

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. We always will assume to know a regularized hamiltonian HL, (L is a system-depending cutoff), which is bounded and self-adjoint in the Hilbert space H of the physical system. Then...

1) HL converges to an operator H

This is apparently the simplest situation. The convergence of the regularized dynamics α_L^t(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) HL does not converge (in any reasonable topology)

This is the most common situation, e.g. mean field 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 model; it may exist even if H_V 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^t_L(A) = e^iHLtAe^−iHLt, what can be said about the limit of α^t_L?

Part 1: 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 Ao 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 Ao) is de- fined in A with the property (AB)^∗ = B^∗A^∗ whenever the mul- tiplication 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 con- tinuous and the multiplications are separately continuous; and (b) A_o is dense in A[ξ], (Lassner, Trapani,...)

Example:

Let H be an Hilbert space and N an unbounded, self adjoint operator 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 ad- joints, 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

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

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.

Definition 1 We say that the model admits an effective hamil- tonian in the weak *-representation π if there exists a self-adjoint operator H_π^{ef f} in Hπ(⊃ 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 deriva- tion 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 F.Bagarello, C.Trapani, The Heisenberg Dynamics of Spin Sis- tems: a Quasi*-Algebras Approach, J. Math. Phys., 37, 4219- 4234, (1996)

For instance:

1. existence of the dynamics α^t on a set of analytic elements;

2. the role of the representation π;

3. α^t as a group of automorphisms of a certain algebra.

....But: when does the model admits an effective hamil-

tonian?

Analysis of the dynamics at the infinitesimal level

For details we refer to:

F. B., A. Inoue, C Trapani, Derivations of quasi ∗-algebras, submitted to Journal de Math´ematiques Pures et Appliqu´ees.

Let (A, A₀) be a quasi *-algebra.

Definition 2 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 3 Let (A, A0) be a quasi *-algebra, Dπ 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 τs on π(A) is the locally convex topology defined by the following fam- ily of seminorms: {p_ξ(.); ξ ∈ D_π}, where p_ξ(π(a)) ≡ kπ(a)ξk, where a ∈ A, ξ ∈ Dπ.

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

δπ(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). (3)

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

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

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.

We can now prove the following

Theorem 4 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

δπ(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 contin- uous 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 δp 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+ Hp, A]ψ, for all A ∈ L^†(Dπ) and ψ ∈ Dπ.

In order to apply our results to QM_∞ we extend the above

Theorem, assuming that there exists a (τ − τs)-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 2) {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 5 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 oper- ator H_n such that

H_n = H_n^† ∈ L(D_π, D_π⁰),

Hnξ0 ∈ H_π and δπ⁽ⁿ⁾(π(x)) = i{Hn◦ 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) δπ, the *-derivation induced by π, is well-defined and spa- tial;

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

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

Example 1: A radiation model

In this example the representation π is just the identity map.

Let us consider a model of n free bosons, [B. 1998], whose dy- namics is given by the hamiltonian, H = Pn

i=1a^†_ia_i. Here a_i and a^†_i are respectively the annihilation and creation operators for the i-th mode. They satisfy the following CCR

[a_i, a^†_j] = Iδ_i,j. (11) 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^†_iai with eigenvalue M ), the condi- tion sup_LkH_LΦ_Mk < ∞ is satisfied. In particular, for instance, sup_LkH_LΦ₀k = 0. It may be worth remarking that all the vec- tors ΦM are cyclic. Calling δL 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 HL to H follows. This is in agreement with [B. 1998].

Example 2: A mean-field spin model

The situation described here is quite different from the one in the previous example. First of all, [B. 1998], 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, where i and j are the indices of the lattice site, σ₃ⁱ is the third component of the Pauli matrices, V is the volume cut-off and |V | is the number of the lattice sites in V . 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 σ₃^VΦ_↑ = Φ_↑, which implies that H_VΦ_↑ = |V |Φ_↑, which in turns implies that sup_V kH_VΦ_↑k =

∞. This means that the cyclic vector Φ_↑ does not satisfy the main assumption of Proposition 5, and for this reason nothing can be said about the convergence of H_V. However, it is possible to consider a different cyclic vector

Φ₀ = ....⊗ ↑_j−1 ⊗ ↓_j ⊗ ↑_j+1 ⊗ ↓_j+2 ⊗...,

which is again an eigenstate of σ^V₃ . Its eigenvalue depends on the volume V . However, it is clear that kσ₃^VΦ₀k = _{|V |}¹ kΦ₀k_V, where

_V can take only values 0, 1. Analogously we have kH_VΦ₀k =

1

|V |kΦ₀k²_V → 0, so that this vector satisfies the assumptions of Proposition 5, 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 vector Φ₀. It is also worth remarking that the same conclusions could also be found replacing Φ₀ with any vector which can be obtained as a locally perturbing Φ₀ itself.

Building up the spin-algebra

The model is defined on a lattice. To the lattice site p we attach the Hilbert space C² and the algebra of the 2×2 matrices generated by the identity matrix and the Pauli matrices σ_i^α. The Hilbert space of the infinite lattice is Hspin = ⊗_p∈ZC²p, and the algebra is the standard quasi local C*-spin algebra A_s.

Let now n = (n₁, n₂, n₃) be a unit vector in R³, and put (σ · n) = n1σ¹ + n2σ²+ n3σ³. The spectrum of σ · n, is {1, −1}.

We call |ni its unit eigenvector in C² associated with 1.

Let (n, n¹, n²) be an orthonormal basis of R³. We put n^± =

1

2(n¹ ± in²) and, for m = 0, 1, |m, ni = (σ · n⁻)^m|ni. We have (σ · n)|m, ni = (−1)^m|m, ni. Repeating this procedure for each lattice site we get, starting with a sequence of vectors in R³, {n} := {n_p}_{p∈Z}, a unit vector in Hspin defined as |{n}i =

⊗_p|n_pi. Furthermore, acting with the operators (σ_p· n⁻_p)^mp, p ∈ Z, on each |n^pi, we get a new vector of H_spin, |{m}, {n}i :=

⊗_p|m_p, npi. The set {|{m}, {n}i, m_p = 0, 1, X

p

mp < ∞}

forms an orthonormal basis in an Hilbert space {n}-depending, which we call H_{n}. On this space we define the unbounded self- adjoint (number) operator M_{n} by its action on the vectors of

the basis

M_{n}|{m}, {n}i = I + X

p

m_p

!

|{m}, {n}i.

Of course M_{n} depends on {n}, that is on the Hilbert space where the operator acts.

The sequence {n} cannot be chosen arbitrarily. We need to consider sequences such that n¹_p = n²_p = 0 for almost all p ∈ Z and lim_{|V |→∞ |V |}¹γ

P

p∈V n³_p exists in R.

The algebraic setting is fixed by the operator M_{n}: first we put D_{n} = T

kD(M_{n}^k ). Then we introduce the algebra L^†(D_{n}), endowed with the quasi-uniform topology τ_{n}. Fi- nally we introduce a *-representation of the C*-spin algebra A_s on L^†(D_{n}) and a topology τ₀ on A_s as follows:

π_{n}(σ_p^α) | {m}, {n}i = σ^α_p | m_p, n_pi⊗( ^Q

p⁰6=p

⊗ | m_p⁰, n_p⁰i) (α = 1, 2, 3), and

kAk^f,k_{n} := maxn

kM_{n}^k π_{n}(A)f (M_{n})k, kf (M_{n})π_{n}(A)M_{n}^k ko , where f belongs to C and k ≥ 0.

With these definitions, calling A the τ₀-completion of A_s, it is possible to prove that:

• (A[τ₀], As) is a topological quasi *-algebra;

• all the powers of the almost magnetization S₃^V are τ₀-converging in A;

• the finite volume dynamics α^t_V τ0-converges to a one-parameter group of automorphisms α^t of A (kα_V^t (A) − α^t_W(A)k^f,k_{n} → 0 for V, W → ∞)

• α^t solves the τ0-limit of the finite volume Heisenberg equa- tion of motion.