Algebras of unbounded operators and physical applications: a survey

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Motivations

Ordinary quantum . . .

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A physical application

The time evolution α^t

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Algebras of unbounded operators and physical applications: a survey

Fabio Bagarello

Bialowieza – July 2006

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I. Plan of the talk

1. few words of motivation

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I. Plan of the talk

2. few words on non relativistic ordinary quantum mechanics and quantum mechanics for systems with infinitely degrees of freedom

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I. Plan of the talk

3. a list of (physically relevant) results and open points...

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I. Plan of the talk

4. algebras of unbounded operators and their uses in QM∞

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I. Plan of the talk

4. algebras of unbounded operators and their uses in QM∞

5. future projects

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II. Motivations

Main goal: description of systems with a very large (10²⁴) number of degrees of freedom.

We cannot deal with these kind of systems considering separately their constituents, otherwise we cannot explain their collective effects. We also cannot try to solve 10²⁴ coupled Schrödinger equations and, after that, read the solution!

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II. Motivations

Main goal: description of systems with a very large (10²⁴) number of degrees of freedom.

We cannot deal with these kind of systems considering separately their constituents, otherwise we cannot explain their collective effects. We also cannot try to solve 10²⁴ coupled Schrödinger equations and, after that, read the solution!

Better to consider systems with infinite degrees of freedom: this allows a simpler analysis of, e.g., phase transitions.

Price to pay: the mathematical apparatus is rather sophisticated.

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III. Ordinary quantum mechanics [NR]

Possible descriptions:

Hilbert space description

Observable A of the physical system ←→ self-adjoint operator ˆA in some Hilbert space H;

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III. Ordinary quantum mechanics [NR]

pure states of the physical system ←→ normalized vectors of H;

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III. Ordinary quantum mechanics [NR]

expectation values of A ←→ < ψ, ˆAψ >= ρψ( ˆA) = tr (PψA);ˆ

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III. Ordinary quantum mechanics [NR]

mixed states: ˆρ = P

j w_j ρ_ψ_n, with P

j w_j = 1;

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III. Ordinary quantum mechanics [NR]

j w_j = 1;

dynamics (Schrödinger representation): U_t := e^{i Ht/~}

then ˆρ → ˆρt = U_t^∗ρUˆ t, H being the hamiltonian.

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III. Ordinary quantum mechanics [NR]

j w_j = 1;

dynamics (Heisenberg representation): Aˆ → ˆAt = U_tAUˆ ^∗: ^d Aˆ_t = ⁱ[H, ˆA_t].

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III. Ordinary quantum mechanics [NR]

j w_j = 1;

dynamics (Heisenberg representation): Aˆ → ˆAt = U_tAUˆ _t^∗: _{d t}^d Aˆ_t = ⁱ

~[H, ˆA_t].

They have the same physical content: ˆρ( ˆAt) = ˆρt( ˆA).

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Algebraic description

The observables are elements of the C*-algebra A(=

B(H)):

A, B ∈ A ⇒ AB, A + B, αA ∈ A, there exists a norm k.k : A → R⁺ such that:

kA + Bk ≤ kAk + kBk; kλAk = |λ|kAk;

kA^∗Ak = kAk²; kABk ≤ kAkkBk, and A is norm complete.

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Algebraic description

The observables are elements of the C*-algebra A(=

B(H)):

A, B ∈ A ⇒ AB, A + B, αA ∈ A, there exists a norm k.k : A → R⁺ such that:

kA + Bk ≤ kAk + kBk; kλAk = |λ|kAk;

kA^∗Ak = kAk²; kABk ≤ kAkkBk, and A is norm complete.

The states are linear, positive and normalized functional on A, ρ( ˆA)(= tr ( ˆρA), where ˆρ is a trace-class operator):

ρ(α₁A + α₂B) = α₁ρ(A) + α₂ρ(B);

ρ(A^∗A) ≥ 0; ρ(11) = 1. (⇒ |ρ(A)| ≤ kAk)

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Dynamics in HR (for conservative systems):

A 3 A → α^t(A) = UtAU_t^∗ ∈ A, ∀t.

α^t is a 1-parameter group of *-automorphisms of A:

α^t(λA) = λα^t(A),

α^t(A + B) = α^t(A) + α^t(B), α^t(AB) = α^t(A) α^t(B),

kα^t(A)k = kAk, α^t+s = α^t α^s.

Remark:– in SR the time evolution is ˆρ → ˆρt = α^{t ∗}ρ.ˆ

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von Neumann theorem (1931):

for finite quantum mechanical systems there exists only one irreducible representation (but for unitary equivalence):

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von Neumann theorem (1931):

let [Q, P ] = i ~I they can be represented on H = L²(R) as follows: ˆqf (q) = qf (q), ˆpf (q) = −i ~f⁰(q),

∀f ∈ S(R).

If ˆq⁰, ˆp⁰ is a different irreducible representation of Q, P on H⁰, then there exists an unitary map V : H → H⁰ such that ˆq⁰ = V ˆqV ^∗, ˆp⁰ = V ˆpV ^∗.

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von Neumann theorem (1931):

let [Q, P ] = i ~I they can be represented on H = L²(R) as follows: ˆqf (q) = qf (q), ˆpf (q) = −i ~f⁰(q),

∀f ∈ S(R).

If ˆq⁰, ˆp⁰ is a different irreducible representation of Q, P on H⁰, then there exists an unitary map V : H → H⁰ such that ˆq⁰ = V ˆqV ^∗, ˆp⁰ = V ˆpV ^∗.

There is no difference between the Hilbert

space and the algebraic descriptions of QM

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IV. QM

_∞

[T = 0]

Radical difference:

We may have inequivalent

representations of the same physical system.

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IV. QM

_∞

[T = 0]

Radical difference:

We may have inequivalent representations of the same physical system.

Example: infinite spin chain (Ising model): we have two inequivalent representations (T = 0):

ψ₀⁽⁺¹⁾ = . . . ⊗ ↑ ⊗ ↑ ⊗ ↑ ⊗ ↑ . . . and ψ₀⁽⁻¹⁾ = . . . ⊗ ↓ ⊗ ↓ ⊗ ↓ ⊗ ↓ . . .

where ψ^±1₀ are two possible ground states of the energy. They are labeled by different values of the mag- netization, m '< ψ₀^(±1),_{|V |}¹ P

j∈V π^(±1)(σ³_j) ψ₀^(±1) >

→ ±1, and, therefore, they are unitarely inequivalent.

This is a (first) example ofspontaneously broken symmetry: we have a symmetry of the (formal) hamiltonian H = −JP

j σ³_jσ_{j +1}³ (σ_j³ → −σ_j³) which is not a symmetry of the ground state.

Remark: local actions on ψ₀⁽⁺¹⁾ do not change the results.

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Haag and Kastler construction (∼1964)

Let Σ our physical system, V ⊂ R^d a finite d-dimensional region, H_V the related Hilbert space (whose construction depends on Σ), A_V = B(H_V) the associated C*- algebra and H_V the self-adjoint energy operator for ΣV, the restriction of Σ in V .

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Haag and Kastler construction (∼1964)

Let Σ our physical system, V ⊂ R^d a finite d-dimensional region, H_V the related Hilbert space (whose construction depends on Σ), A_V = B(H_V) the associated C*- algebra and H_V the self-adjoint energy operator for ΣV, the restriction of Σ in V .

{AV} satisfies the following properties:

• isotony: if V₁ ⊂ V₂ then A_V₁ ⊂ A_V₂. Moreover k.k2 ^V¹= k.k1 (⇒ AV1, AV2 ⊂ AV1∪V2);

• if V₁ ∩ V₂ = ∅ then [AV1, AV2] = 0;

Then we define A = A0

k.k, where A0 = ∪VAV.

A is the quasi-local C*-algebra of the bounded observables.

On A we introduce thespatial translations{γx}, which is a group of *-automorphisms of A: γxA_V = A_{V +x}, γ_x₁γ_x₂ = γ_x₁_+x₂.

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The states of Σ are positive, normalized linear func- tionals on A which, when restricted to V , reduces to the states over the finite system A_V: they corresponds to a family of density matrices ρV: ˆρ(A) = trV(ρVA) for each A ∈ A_V (Here tr_V is the trace in H_V) satisfy- ing the consistency condition tr_V(ρ_VA) = tr_V⁰(ρ_V⁰A)

∀A ∈ A_V, V ⊂ V⁰.

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∀A ∈ A_V, V ⊂ V⁰.

Ruelle, Dell’Antonio, Doplicher theorem (1966): these physical states have zero probability to describe an infinite number of particles in a finite region. (→locally finite states)

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∀A ∈ A_V, V ⊂ V⁰.

Ruelle, Dell’Antonio, Doplicher theorem (1966): these physical states have zero probability to describe an infinite number of particles in a finite region. (→locally finite states)

Pure state: ρ is pure if it is not a convex combination of other states, i.e. if there is no ρ₁, ρ2 and λ ∈]0, 1[

such that ρ = λρ + (1 − λ)ρ .

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Example 1: discrete system

Let X be an infinite lattice, 0 ∈ X, and H₀ a finite dimensional Hilbert space (e.g. H0 = C² for Pauli matrices). Let H_x a copy of H₀ localized in x ∈ X and H_V = ⊗x∈VH_x, A_V = B(HV).

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If V ⊂ V ⁰ then HV⁰ = HV ⊗ H_V⁰_\V and, ∀A ∈ AV, A⊗ IV⁰\V ∈ A_V⁰.

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The map γk(a_x⁽¹⁾₁ a_x⁽²⁾₂ . . . a⁽ⁿ⁾_x_n ) = a_x⁽¹⁾

1+k a⁽²⁾_x

2+k . . . a_x⁽ⁿ⁾

n+k is an automorphism for each k: it represents the spatial translations.

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The map γk(a_x⁽¹⁾₁ a_x⁽²⁾₂ . . . a⁽ⁿ⁾_x_n ) = a_x⁽¹⁾

1+k a⁽²⁾_x

2+k . . . a_x⁽ⁿ⁾

n+k is an automorphism for each k: it represents the spatial translations.

The local energy is given by considering the interactions of all the particles inside V ,

HV = X

r

X

x1,...,xr∈V

Vr(x1, x2, . . . , xr) where V is the r-body interaction.

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Example 2: continuous system

Starting point: Fermi or Bose commutation rules:

[Ψ[f ], Ψ^†[g]]± =< g, f >,

[Ψ[f ], Ψ[g]]_± = [Ψ^†[f ], Ψ^†[g]]_± = 0, ∀f , g ∈ L²(R).

Let Φ0 be thevacuum of the theory, i.e. a vector such that Ψ[f ]Φ₀ = 0, ∀f ∈ L²(R). HV is the norm clo- sure of Ψ^†[f1] . . . Ψ^†[fn]Φ0, where each f_j is supported in V . Observe that dim(H_V) = ∞!!

A_V = {X ∈ B(H_V) : [X, N_V] = 0}, where N_V = R

V Ψ^†(x )Ψ(x ) d x is the number operator.

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[Ψ[f ], Ψ^†[g]]± =< g, f >,

[Ψ[f ], Ψ[g]]_± = [Ψ^†[f ], Ψ^†[g]]_± = 0, ∀f , g ∈ L²(R).

The local hamiltonian, for 2-body interactions, is:

H_V = ~² 2m

Z

V

d x |∇Ψ(x )|²+

1 Z Z

0 † † 0 0 0

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[Ψ[f ], Ψ^†[g]]± =< g, f >,

[Ψ[f ], Ψ[g]]_± = [Ψ^†[f ], Ψ^†[g]]_± = 0, ∀f , g ∈ L²(R).

The local hamiltonian, for 2-body interactions, is:

H_V = ~² 2m

Z

V

d x |∇Ψ(x )|²+ +1

2 Z

V

d x Z

V

d x⁰Ψ^†(x )Ψ^†(x⁰)V (x , x⁰)Ψ(x⁰)Ψ(x ) Remark:– even in H_V unbounded operators appear!!

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Time evolution of Σ

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Time evolution of Σ

This is obtained from the dynamics of Σ_V in HR as follows:

[step ]1]: A 3 A → α^t_V(A) := e^{i H}^V^t/~Ae^{−i H}^V^t/~. [Step ]2]: α^t(A) = τ − lim_V α^t_V(A),

where τ is a reasonable topology of A. Possible topologies are:

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Time evolution of Σ

for short range interactions and discrete systems τ is usually the uniform topology [Haag, Hugenholtz, Winnink];

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Time evolution of Σ

for short range interactions and discrete systems τ is usually the uniform topology [Haag, Hugenholtz, Winnink];

for long range interactions α^t_V is not k.k−converging:

τ is the strong topology (restricted to a relevant family of states) [Sewell, Thirring, Werhl, Strocchi, Mor- chio, B.,. . .]: ρ is chosen in such a way that

ρ(α^t_V(A)) → ρ(α^t(A)) =: ρ_t(A),

(which gives also ρ_t). The existence of (sufficiently many) such ρ’s has to be checked in each model.

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An automorphism of A, γ, is a symmetry of Σ if α^t(γ(A)) = γ(α^t(A)) and is local if γ : AV → AV

and γ(H_V) = H_V.

γ is a symmetry of the state ρ if ρ_γ(A) := ρ(γ(A)) = ρ(A), ∀A ∈ A.

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and γ(H_V) = H_V.

A representation of a *-algebra is a map π : A → B(H), for a certain H, which preserves the algebraic structure of A:

π(A + B) = π(A) + π(B), π(λA) = λ π(A), π(AB) = π(A)π(B), π(A^∗) = π(A)^∗.

It follows that π(A) is a *-algebra as well.

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and γ(H_V) = H_V.

A representation of a *-algebra is a map π : A → B(H), for a certain H, which preserves the algebraic structure of A:

π(A + B) = π(A) + π(B), π(λA) = λ π(A), π(AB) = π(A)π(B), π(A^∗) = π(A)^∗.

It follows that π(A) is a *-algebra as well.

Important: any state ρ over the abstract C*-algebra Aproduces a unique (but for equivalence)GNS(Gelfand- Naimark-Segal) representation (H_ρ, π_ρ, Ω_ρ), in such a way that, ∀A ∈ A,

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Notice that:

1. GNS representations generated by different states need not be unitarely equivalent (e.g. Ising model)!

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Notice that:

2. Each (GNS) representation corresponds to a phase of the physical system. In particular, GNS representations generated by pure states correspond to pure phases [Ruelle].

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Notice that:

2. Each (GNS) representation corresponds to a phase of the physical system. In particular, GNS representations generated by pure states correspond to pure phases [Ruelle].

3. States which are only locally different are macro- scopically indistinguishable: all the macroscopic observables have the same expectation values (e.g.

Ising model again). [Hepp, Sewell]. Then they produce unitarely equivalent GNS representations.

Physical interpretation: equal values of the macroscopic observables (the order parameters) label unitarely equivalent GNS representations, which are interpreted as the same phase of the matter.

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In other words: two different phases of the matter correspond to GNS representations in which some macroscopic observable assumes different values.

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In other words: two different phases of the matter correspond to GNS representations in which some macroscopic observable assumes different values.

4. Under certain assumption on Σ, the dynamics in each representation πρ is hamiltonian: there exists a s.a. operator ˆHρ such that, ∀A ∈ A,

d

d t α^t_ρ(πρ(A)) = i [ ˆHρ, α^t_ρ(πρ(A))].

(This is not obvious for Σ at a pure algebraic level!) [Emch, Knops, Sewell]. ˆHρ is what is of- ten called in literature the effective hamiltonian.

Physical interpretation: different phases may have different dynamical behaviors.

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Motivations

QM∞[T = 0]

QM∞[T > 0]

A list of problems

Algebras of . . .

Work in progress

JJ II

J I

Page17

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V. QM

_∞

[T > 0]

We will give here only few considerations on equilibrium states and phase structure.

Case 1: finite system

(49)

Motivations

QM∞[T = 0]

QM∞[T > 0]

A list of problems

Algebras of . . .

Work in progress

JJ II

J I

Page17

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V. QM

_∞

[T > 0]

We will give here only few considerations on equilibrium states and phase structure.

Case 1: finite system

The following are equivalent:

ρ is a Gibbs state corresponding to the trace class operator ˆρ = _tr^e^−βHV

V(i d em), β⁻¹ = kT , ⇐⇒

it minimizes the free energy functional

FˆV(ρ) = trV(ρHV + β⁻¹ρ log(ρ)) ⇐⇒

it is a KMS (Kubo-Martin-Schwinger) state, i.e.

ρ(AtB) = ρ(BA_{t+i ~β}).

Therefore: for each temperature there exists an unique equilibrium state ⇒ an unique GNS representation ⇒ a single phase of Σ.

(50)

Motivations

QM∞[T = 0]

QM∞[T > 0]

A list of problems

Algebras of . . .

Work in progress

JJ II

J I

Page18

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Case 2: infinite system

Assumptions on H_V (relevant to prove the existence of thermodynamical limit of several physical quantities):

1. H_V₁_∪V₂− HV1− HV2 is a surface effect (short range forces);

2. there exists c > 0 such that kH_Vk ≤ c|V |.

Under these assumptions we have α^t(A) = k k − lim

V% α^t_V(A)