Quantum researches at the Engineering faculty

(1)

Recent research lines Algebras and QM∞

SUSY and Landau . . .

Home Page

JJ II

J I

Page1

Go Back

Full Screen

Close

Quantum researches at the Engineering faculty

Fabio Bagarello...

Palermo – June 2006

(2)

Home Page

JJ II

J I

Page2

Go Back

Full Screen

Close

Quit

...with the guilty illuminating support of A.M. Greco, who has always shown his interest and his enthusiasm for my quantum

world!!

(3)

Home Page

JJ II

J I

Page3

Go Back

Full Screen

Close

I. Recent research lines

1. Algebras and QM_∞

C.Trapani, T.Triolo (Palermo), A. Inoue (Fukuoka), M. Fragouloupolou (Atene), J.P. Antoine (Louvain- La-Neuve)

(4)

Home Page

JJ II

J I

Page3

Go Back

Full Screen

Close

Quit

I. Recent research lines

2. Supersymmetric QM and Landau levels S.T. Ali (Montreal)

(5)

Home Page

JJ II

J I

Page3

Go Back

Full Screen

Close

I. Recent research lines

3. Wavelets, coherent states, strange geometriesand applications to Quantum Hall effect

S.T. Ali, V. Hudon, (Montreal), J.P. Antoine (Louvain-La-Neuve), T.Triolo (Palermo)

(6)

Home Page

JJ II

J I

Page3

Go Back

Full Screen

Close

Quit

I. Recent research lines

4. Quantum open systems

G.L.Sewell (London), Y.G. Lu (Bari)

(7)

Home Page

JJ II

J I

Page3

Go Back

Full Screen

Close

I. Recent research lines

4. Quantum open systems

G.L.Sewell (London), Y.G. Lu (Bari)

(8)

Home Page

JJ II

J I

Page4

Go Back

Full Screen

Close

Quit

II. Algebras and QM

_∞

Case 1: Ordinary quantum mechanics [NR]

Possible descriptions:

(9)

Home Page

JJ II

J I

Page4

Go Back

Full Screen

Close

II. Algebras and QM

_∞

Hilbert space description

(10)

Home Page

JJ II

J I

Page4

Go Back

Full Screen

Close

Quit

II. Algebras and QM

_∞

Algebraic description

(11)

Home Page

JJ II

J I

Page4

Go Back

Full Screen

Close

II. Algebras and QM

_∞

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

B(H)): they are all bounded!!!

(12)

Home Page

JJ II

J I

Page4

Go Back

Full Screen

Close

Quit

II. Algebras and QM

_∞

But the von Neumann uniqueness theorem (1931) es- sentially states that

(13)

Home Page

JJ II

J I

Page4

Go Back

Full Screen

Close

II. Algebras and QM

_∞

But the von Neumann uniqueness theorem (1931) es- sentially states that

there is no difference between the Hilbert space and the algebraic descriptions of QM

(14)

Home Page

JJ II

J I

Page5

Go Back

Full Screen

Close

Quit

II.1. QM_∞ [T = 0]

Radical difference: We may have inequivalent representations of the same physical system.

(15)

Home Page

JJ II

J I

Page5

Go Back

Full Screen

Close

II.1. QM_∞ [T = 0]

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 .

(16)

Home Page

JJ II

J I

Page5

Go Back

Full Screen

Close

Quit

II.1. QM_∞ [T = 0]

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(HV) the associated C*- algebra and H_V the self-adjoint energy operator for Σ_V, the restriction of Σ in V .

{A_V} 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 [A_V₁, A_V₂] = 0;

Then we define A = A0

k.k, where A0 = ∪_VA_V.

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

(17)

Home Page

JJ II

J I

Page6

Go Back

Full Screen

Close

On A we introduce thespatial translations{γ_x}, which is a group of *-automorphisms of A: γ_xAV = AV +x, γx1γx2 = γx1+x2.

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 AV: they corresponds to a family of density matrices ρ_V: ˆρ(A) = tr_V(ρ_VA) for each A ∈ A_V (Here tr_V is the trace in H_V) satisfy- ing the consistency condition trV(ρVA) = trV⁰(ρV⁰A)

∀A ∈ A_V, V ⊂ V⁰.

(18)

Home Page

JJ II

J I

Page6

Go Back

Full Screen

Close

Quit

On A we introduce thespatial translations{γ_x}, which is a group of *-automorphisms of A: γ_xAV = AV +x, γx1γx2 = γx1+x2.

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 AV: they corresponds to a family of density matrices ρ_V: ˆρ(A) = tr_V(ρ_VA) for each A ∈ A_V (Here tr_V is the trace in H_V) satisfy- ing the consistency condition trV(ρVA) = trV⁰(ρV⁰A)

∀A ∈ A_V, V ⊂ V⁰.

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

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

(19)

Home Page

JJ II

J I

Page7

Go Back

Full Screen

Close

Time evolution of Σ

(20)

Home Page

JJ II

J I

Page7

Go Back

Full Screen

Close

Quit

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:

(21)

Home Page

JJ II

J I

Page7

Go Back

Full Screen

Close

Time evolution of Σ

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

(22)

Home Page

JJ II

J I

Page7

Go Back

Full Screen

Close

Quit

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.

(23)

Home Page

JJ II

J I

Page8

Go Back

Full Screen

Close

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.

(24)

Home Page

JJ II

J I

Page8

Go Back

Full Screen

Close

Quit

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,

ρ(A) =< Ωρ, π_ρ(A)Ωρ >

Here Ω_ρ is cyclic, i.e. π_ρ(A)Ω_ρ is dense in H_ρ, and πρ is irreducible iff ρ is pure.

(25)

Home Page

JJ II

J I

Page9

Go Back

Full Screen

Close

Notice that:

1. Each (GNS) representation corresponds to a phase of the physical system. Different phases are la- beled by different values of some macroscopic observables.

(26)

Home Page

JJ II

J I

Page9

Go Back

Full Screen

Close

Quit

Notice that:

1. Each (GNS) representation corresponds to a phase of the physical system. Different phases are la- beled by different values of some macroscopic observables.

2. Under certain assumption on Σ, the dynamics in each representation πρ is hamiltonian: there exists a s.a. operator ˆH_ρ ∈ B(H_ρ) 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]. ˆH_ρ is what is of- ten called in literature the effective hamiltonian.

Physical interpretation: different phases may have different dynamical behaviors.

(27)

Home Page

JJ II

J I

Page10

Go Back

Full Screen

Close

II.2. QM_∞ [T > 0]

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

Case 1: finite system

(28)

Home Page

JJ II

J I

Page10

Go Back

Full Screen

Close

Quit

II.2. 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.

(29)

Home Page

JJ II

J I

Page11

Go Back

Full Screen

Close

Case 2: infinite system

Working on these simplifying assumptions

1. H_V₁_∪V₂− H_V₁− H_V₂ is a surface effect (short range forces);

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

we are able to define stable (i.e. minimizing a free energy density functional ) andmetastable(i.e. KMS) states also for infinite systems.

A consequence of this richer structure is that an infinite system Σ may possess more than one stable state at the same temperature (macroscopic degeneracy).

(30)

Home Page

JJ II

J I

Page12

Go Back

Full Screen

Close

Quit

Therefore this algebraic approach provides a nice framework:

(31)

Home Page

JJ II

J I

Page12

Go Back

Full Screen

Close

to study phase transitions....

(32)

Home Page

JJ II

J I

Page12

Go Back

Full Screen

Close

Quit

... for the analysis of spontaneous breaking of a symmetry....

(33)

Home Page

JJ II

J I

Page12

Go Back

Full Screen

Close

...and of the related non relativistic Goldstone’s theorem.

(34)

Home Page

JJ II

J I

Page12

Go Back

Full Screen

Close

Quit

Other results involving C*-algebras concern the rela- tion between KMS states and Tomita-Takesaki theory....

(35)

Home Page

JJ II

J I

Page12

Go Back

Full Screen

Close

....the analysis of quantum systems far from equilibrium....

(36)

Home Page

JJ II

J I

Page12

Go Back

Full Screen

Close

Quit

....the algebraic approach to quantum field theory...

(37)

Home Page

JJ II

J I

Page12

Go Back

Full Screen

Close

....the algebraic approach to quantum field theory...

...and many others.

(38)

Home Page

JJ II

J I

Page13

Go Back

Full Screen

Close

Quit

II.3. A list of problems

The above results are obtained under the require- ments that

1. The norm of the local hamiltonian H_V does not grow faster than |V |: kH_Vk ≤ c|V |;

(39)

Home Page

JJ II

J I

Page13

Go Back

Full Screen

Close

2. The interactions are short ranged

(40)

Home Page

JJ II

J I

Page13

Go Back

Full Screen

Close

Quit

2. The interactions are short ranged

3. Even in presence of unbounded operators some physically meaningful limits do exist.

(41)

Home Page

JJ II

J I

Page14

Go Back

Full Screen

Close

II.4. Algebras of unbounded operators

A possible algebraic framework: let A be a linear space, A₀ ⊂ A a ^∗-algebra with unit 11 (otherwise we can add it): A is a quasi ^∗-algebra over A₀ if

(42)

Home Page

JJ II

J I

Page14

Go Back

Full Screen

Close

Quit

A possible algebraic framework: let A be a linear space, A₀ ⊂ A a ^∗-algebra with unit 11 (otherwise we can add it): A is a quasi ^∗-algebra over A₀ if [i] the right and left multiplications of an element of A and an element of A0 are always defined and linear;

(43)

Home Page

JJ II

J I

Page14

Go Back

Full Screen

Close

[ii]x₁(x₂a) = (x₁x₂)a, (ax₁)x₂ = a(x₁x₂) and x₁(ax₂) = (x1a)x2, for each x₁, x2 ∈ A₀ and a ∈ A;

(44)

Home Page

JJ II

J I

Page14

Go Back

Full Screen

Close

Quit

[iii] an involution * (which extends the involution of A0) is defined in A with the property (ab)^∗ = b^∗a^∗ whenever the multiplication is defined.

(45)

Home Page

JJ II

J I

Page14

Go Back

Full Screen

Close

[iii] an involution * (which extends the involution of A0) is defined in A with the property (ab)^∗ = b^∗a^∗ whenever the multiplication is defined.

A quasi ^∗ -algebra (A, A₀) is locally convex (or topo- logical) if in A a locally convex topology τ is defined such that (a) the involution is continuous and the multiplications are separately continuous; and (b) A₀

(46)

Home Page

JJ II

J I

Page15

Go Back

Full Screen

Close

Quit

Example: Let H be a separable Hilbert space and N an unbounded, densely defined, self-adjoint operator.

Let D(N^k) be the domain of the operator N^k, k ∈ N , and D the domain of all the powers of N: D ≡ D^∞(N) = ∩k≥0D(N^k). This set is dense in H. Let us now introduce L^†(D), the *-algebra of all the closable operators defined on D which, together with their adjoints, map D into itself. Here the adjoint of X ∈ L^†(D) is X^† = X_D^∗ .

(47)

Home Page

JJ II

J I

Page15

Go Back

Full Screen

Close

Quit

Example: Let H be a separable Hilbert space and N an unbounded, densely defined, self-adjoint operator.

Let D(N^k) be the domain of the operator N^k, k ∈ N , and D the domain of all the powers of N: D ≡ D^∞(N) = ∩k≥0D(N^k). This set is dense in H. Let us now introduce L^†(D), the *-algebra of all the closable operators defined on D which, together with their adjoints, map D into itself. Here the adjoint of X ∈ L^†(D) is X^† = X_D^∗ .

In D the topology is defined by the following N- depending seminorms: φ ∈ D → kφk_n ≡ kNⁿφk, n ∈ N⁰, while the topology τ₀ in L^†(D) is introduced by the seminorms

X ∈ L^†(D) → kXk^{f ,k} ≡ max

kf (N)XN^kk, kN^kXf (N)k ,

where k ∈ N⁰ and f ∈ C, the set of all the positive, bounded and continuous functions on R⁺, which are decreasing faster than any inverse power of x :

(48)

Home Page

JJ II

J I

Page16

Go Back

Full Screen

Close

Quit

It contains unbounded operators, e.g. all the positive powers of N, but not e^N.

(49)

Home Page

JJ II

J I

Page16

Go Back

Full Screen

Close

It contains unbounded operators, e.g. all the positive powers of N, but not e^N.

Let furtherL(D, D⁰)be the set of all continuous maps from D into D⁰, with their topologies (in D⁰ this is the strong dual topology), and let τ denotes the topology defined by the seminorms

X ∈ L(D, D⁰) → kXk^f = kf (N)Xf (N)k,

f ∈ C. Then L(D, D⁰)[τ ] is a complete vector space and e^N ∈ L(D, D⁰).

In this case L^†(D) ⊂ L(D, D⁰) and the pair (L(D,D⁰)[τ ], L^†(D)[τ0]) is a locally convex quasi *-algebra.

(50)

Home Page

JJ II

J I

Page17

Go Back

Full Screen

Close

Quit

Other possible algebraic frameworks:

1. Partial *-algebras [J.P. Antoine, W. Karwowski (1981)]

(51)

Home Page

JJ II

J I

Page17

Go Back

Full Screen

Close

2. CQ*-algebras [Trapani, B. (≥ 1996)]

(52)

Home Page

JJ II

J I

Page17

Go Back

Full Screen

Close

Quit

2. CQ*-algebras [Trapani, B. (≥ 1996)]

Main (mathematical) references:

K. Schmüdgen, Akademie-Verlag, Berlin (1990);

J.P. Antoine, A. Inoue, C. Trapani, Kluwer (2002)

(53)

Home Page

JJ II

J I

Page18

Go Back

Full Screen

Close

Let now (A, A₀) be a quasi *-algebra, then we can define (extending our previous definitions) the notion of *-representations of (A, A0) and that of GNS construction(We use sesquilinear forms now: linear func- tionals do not work!).

The physical interpretation is analogous to that dis- cussed before: different sesquilinear forms produce different representations which can still be interpreted as different phases of the matter.

(54)

Home Page

JJ II

J I

Page19

Go Back

Full Screen

Close

Quit

II.5. A physical application: the existence of the effective hamiltonian

Definition: Let (A[τ ], A₀) be a quasi *-algebra. A

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

(i) δ(x^∗) = δ(x )^∗, ∀x ∈ A0;

(ii) δ(x y ) = x δ(y ) + δ(x )y , ∀x , y ∈ A₀.

Let now π be a *-representation of (A, A₀) such that, whenever x ∈ A0 satisfies π(x ) = 0, then π(δ(x )) = 0. Under this assumption, the linear map

δ_π(π(x )) = π(δ(x )), x ∈ A₀,

is well-defined on π(A₀) with values in π(A) and it is a *-derivation of π(A₀). We call δπ the *-derivation induced by π.

(55)

Home Page

JJ II

J I

Page20

Go Back

Full Screen

Close

Let δ be a *-derivation of A₀ and π a *-representation of (A, A₀). Then π(A0) ⊂ L^†(Dπ). We say that the

*-derivation δπ induced by π is spatial if there exists H_π = H_π^† ∈ L(D_π,D⁰_π) such that H_πξ₀ ∈ H_π and

δπ(π(x )) = i {Hπ ◦ π(x ) − π(x ) ◦ H_π}, ∀x ∈ A₀. Together with Inoue and Trapani we gave (necessary and sufficient) conditions for δπ to be spatial, analyz- ing also this problem in presence of perturbations.

(56)

Home Page

JJ II

J I

Page20

Go Back

Full Screen

Close

Quit

Let δ be a *-derivation of A₀ and π a *-representation of (A, A₀). Then π(A0) ⊂ L^†(Dπ). We say that the

*-derivation δπ induced by π is spatial if there exists H_π = H_π^† ∈ L(D_π,D⁰_π) such that H_πξ₀ ∈ H_π and

δπ(π(x )) = i {Hπ ◦ π(x ) − π(x ) ◦ H_π}, ∀x ∈ A₀. Together with Inoue and Trapani we gave (necessary and sufficient) conditions for δπ to be spatial, analyz- ing also this problem in presence of perturbations.

Also, for an infinitely extended quantum system, we gave sufficient conditions for an effective (and phase depending) hamiltonian to exist.

(57)

Home Page

JJ II

J I

Page21

Go Back

Full Screen

Close

A hard problem: the time evolution α^t

Even if δ exists, what about δ², δ³, . . .? Moreover, even if all these maps do exist, this does not mean that P_∞

k =0 t^k

k ! δ^k(x ) exists, for x ∈ A₀. Further, the effective hamiltonian H_π is symmetric but not self- adjoint, and therefore we cannot use the spectral theorem to define e^{i H}^π^t! How to define a time evolution?

(58)

Home Page

JJ II

J I

Page21

Go Back

Full Screen

Close

Quit

k =0 t^k

The existence of α^t has been proved, along these years, for different classes of models and using different techniques:

(59)

Home Page

JJ II

J I

Page21

Go Back

Full Screen

Close

k =0 t^k

• via fixed point theorems;

(60)

Home Page

JJ II

J I

Page21

Go Back

Full Screen

Close

Quit

k =0 t^k

• using explicit estimates;

(61)

Home Page

JJ II

J I

Page21

Go Back

Full Screen

Close

k =0 t^k

• working with mean field (or almost mean field) models;

(62)

Home Page

JJ II

J I

Page21

Go Back

Full Screen

Close

Quit

k =0 t^k

• working with mean field (or almost mean field) models;

• introducing convenient topologies....

(63)

Home Page

JJ II

J I

Page22

Go Back

Full Screen

Close

II.6. Work in progress

Our point of view is now slightly different: the algebraic framework is fixed. Different hamiltonians give rise to different problems:

let S be a self-adjoint, unbounded, densely defined operator on H. For simplicity we assume S = P∞

l =0slPl

Let D = D^∞(S), L^†(D) and τ constructed as usual.

Let H_L = PL

l =0 h_lP_l be ourregular hamiltonian: H_L ∈ B(H), ∀L.

In our previous attempt we always had τ − lim H_L ∈ L^†(D). We have just proven that this is not necessary, i.e., for each sequence {h_l}, if {s_l⁻¹} is in l²(N0),

(64)

Home Page

JJ II

J I

Page22

Go Back

Full Screen

Close

Quit

l =0slPl

Let H_L = PL

1. e^{i H}^L^t τ-converges to an element T_t ∈ L^†(D);

(65)

Home Page

JJ II

J I

Page22

Go Back

Full Screen

Close

l =0slPl

Let H_L = PL

1. e^{i H}^L^t τ-converges to an element T_t ∈ L^†(D);

2. ∀X ∈ L^†(D) the sequence e^{i H}^L^tXe^{−i H}^L^t τ-converges to an element α^t(X) ∈ L^†(D);

(66)

Home Page

JJ II

J I

Page22

Go Back

Full Screen

Close

Quit

l =0slPl

Let H_L = PL

1. e^{i H}^L^t τ-converges to an element Tt ∈ L^†(D);

2. ∀X ∈ L^†(D) the sequence e^{i H}^L^tXe^{−i H}^L^t τ-converges to an element α^t(X) ∈ L^†(D);

3. ∀X ∈ L^†(D) we have α^t(X) = T_tXT_−t; 4. if Q_M = PM

l =0, X ∈ L^†(D), X_M = Q_MXQ_M and

(67)

Home Page

JJ II

J I

Page23

Go Back

Full Screen

Close

δ_L(X_M) = i [H_L, X_M] then α^t(X) = τ − lim

L,M,N

XN j =0

t^j

j ! δ_L^j (XM).

(68)

Home Page

JJ II

J I

Page23

Go Back

Full Screen

Close

Quit

L,M,N

XN j =0

t^j

j ! δ_L^j (XM).

Remarks:– (1) the time evolution of each element of L^†(D) can be defined (in three different ways!) even if H_L does not define an hamiltonian of the system Σ, i.e. even if HL does not converge in any natural topology (e.g. mean field spin models).

(69)

Home Page

JJ II

J I

Page23

Go Back

Full Screen

Close

L,M,N

XN j =0

t^j

j ! δ_L^j (XM).

Remarks:– (1) the time evolution of each element of L^†(D) can be defined (in three different ways!) even if H_L does not define an hamiltonian of the system Σ, i.e. even if HL does not converge in any natural topology (e.g. mean field spin models).

(2) However, here we are assuming that S and H_L admit the same spectral projections. What if this is not the case?