Relations between Hamiltonian Cellular Automata and Quantum Mechanics of composite systems

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Relations between Hamiltonian Cellular Automata and

Quantum Mechanics of composite systems

Alessio Andreoni Pisa, 19 Ottobre 2015 UNIVERSITÀ DI PISA 1 34 3 IN S UP REMÆ DIGN_IT A T IS

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Cellular Automata

A Cellular Automaton requires:

a regular lattice of cells covering a portion of a D dimensional space; a set ψ(~r, t) = {ψ1(~r, t), ψ2(~r, t), ..., ψl(~r, t)} of discrete variables attached

to each site ~r of the lattice and giving the local state of each cell at the time t = 0, 1, ...;

a rule R = {R1, R2, ..., Rl} which specifies the time evolution of the state

ψ(~r, t): ψj(~r, t + 1) = Rj ψ(~r, t), ψ(~r + ~δ1, t), ..., ψ(~r + ~δq, t) .

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Cellular Automata

a regular lattice of cells covering a portion of a D dimensional space; a set ψ(~r, t) = {ψ1(~r, t), ψ2(~r, t), ..., ψl(~r, t)} of discrete variables attached to each site ~r of the lattice and giving the local state of each cell at the time t = 0, 1, ...;

a rule R = {R1, R2, ..., Rl} which specifies the time evolution of the state

ψ(~r, t): ψj(~r, t + 1) = Rj ψ(~r, t), ψ(~r + ~δ1, t), ..., ψ(~r + ~δq, t) .

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Cellular Automata

a rule R = {R1, R2, ..., Rl} which specifies the time evolution of the state ψ(~r, t): ψj(~r, t + 1) = Rj ψ(~r, t), ψ(~r + ~δ1, t), ..., ψ(~r + ~δq, t) .

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Cellular Automata

a rule R = {R1, R2, ..., Rl} which specifies the time evolution of the state ψ(~r, t):

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Generalized Cellular Automata (GCA)

A Generalized Cellular Automaton requires:

a denumerable set of variables ψα_{(t), where α is a multi-index which denotes} different degrees of freedom, giving the state of each cell at the time

t = 0, 1, ...;

a denumerable set of rules Rα _{which specifies the time evolution of the state} ψα_{(t) in the following way:}

ψα(t + 1) = Rα {ψβ_(t)} .

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Hamiltonian Cellular Automata (HCA)

Hamiltonian Cellular Automataare a class of GCA which updating equations derive from the variation of an action.

Its state is represented by “coordinates” xα(nl) and “conjugated momenta” pα(nl), with α an integer multi-index, l a time-scale, and nl, n ∈ Z is the discrete time.

Finite differencesfor all dynamical variables are defined by: ∆f (nl) := f (nl) − f (nl − l) . The action of anHCA:

S :=X n [(pα(nl) + pα(l(n − 1)))∆xα(nl) − 2lH(nl)] , H(nl) :=1 2Sαβ p α_(nl)pβ_{(nl) + x}α_(nl)xβ_{(nl) + Aαβp}α_(nl)xβ_{(nl) ,} with Sαβ= Sβα and Aαβ= −Aβα.

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Updating equations

The evolution of theHCAis determined by the followingpostulate.

Postulate A

TheHCA follows the discrete updating rules (equations of motions) which are determined by theaction principleδS = 0, referring to variations of all dynamical variables defined by:

δg(fn) := 1

2[g(fn+ δfn) − g(fn− δfn)] ,

where fn stands for one of the variables on which polynomial g may depend. The updating equations are:

˙ xα(nl) = c Sαβpβ(nl) + Aαβxβ(nl) , ˙ pα(nl) = −c Sαβxβ(nl) − Aαβpβ(nl) , whereO := (O(nl + l)) − O(nl − l))/2l)˙ .

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Updating equations

A change of variables

Consider the followingchange of variables(xα_{, p}α_{) → (ψ}α_{, ψ}∗α_{), where} ψα= xα+ ipαand ψ∗α is its complex conjugate.

Introducing the self-adjoint matrix ˆH = ˆS + i ˆA we can rewrite theactionas:

S :=X n i 2l[ψ ∗α_{(nl) (ψ}α_{(nl + l) − ψ}α_{(nl − l))] − ψ}∗α_(nl)H αβψβ(nl) ,

applyingPostulate Awe obtain the equations: ˙

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Solutions and conservation laws

Solutions of the updating equations given thetwo initial conditionsψα_{(0) and} ψα(l):

ψα(nl) = −in(Un−2)αβψβ(0) + i(Un−1)αβψβ(l) ,

where Un = Un(−l ˆH) are the Chebyschev polynomials of second kind. Theorem on conservation laws:

Theorem A

For any matrix ˆG that commutes with ˆH, there is a discreteconservation law: ψ∗α(ln)Gαβψ˙β(ln) + ˙ψ∗α(ln)Gαβψβ(ln) = 0 .

The correspondingconserved quantity:

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Space of states

ψα _{coefficients of a Hilbert space vector ψ. HCA space of states properties:}

States

1 _{A state of the}_HCA_{is characterized by two Hilbert vectors at}_successive

timesψ1(nl) and ψ2(nl) = ψ1(nl − l) or equivalently

ψ+(nl) = (1/2)(ψ1(nl) + ψ2(nl)) and ψ−(nl) = (1/2)(ψ1(nl) − ψ2(nl)), ψ− of order O(l) w.r.t. ψ+;

2 _{The space of states V is itself a Hilbert space, which is the}_{direct sum}_{of the}

two Hilbert spaces at point (1); a state is, thus, written as Ψ(nl) = (ψ1(nl), ψ2(nl)) or Ψ0(nl) = (ψ+(nl), ψ−(nl));

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Space of states

States

1 _{A state of the}_HCA_{is characterized by two Hilbert vectors at}_successive timesψ1(nl) and ψ2(nl) = ψ1(nl − l) or equivalently

2 _{The space of states V is itself a Hilbert space, which is the}_{direct sum}_{of the} two Hilbert spaces at point (1); a state is, thus, written as

Ψ(nl) = (ψ1(nl), ψ2(nl)) or Ψ0(nl) = (ψ+(nl), ψ−(nl));

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Space of states

States

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Space of states

States

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Observables

The observables are Hermitean operators ˆG, acting on V. We also define

restricted observablesOˆG_{, that are the ones in terms of which we can reformulate} Theorem A. Restricted observables ˆ OG:= _ˆ G 0 0 − ˆG , with ˆG an Hermitean matrix acting on Hilbert vector ψ+,−.

Conservation laws

If ˆG satisfyTheorem Awe can rewrite the conserved quantity CGˆ(nl, nl − l) as: CGˆ(nl, nl − l) = hΨ

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Observables

The observables are Hermitean operators ˆG, acting on V. We also define

restricted observablesOˆG_{, that are the ones in terms of which we can reformulate} Theorem A. Restricted observables ˆ OG:= _ˆ G 0 0 − ˆG , with ˆG an Hermitean matrix acting on Hilbert vector ψ+,−.

Conservation laws

If ˆG satisfyTheorem Awe can rewrite the conserved quantity CGˆ(nl, nl − l) as: CGˆ(nl, nl − l) = hΨ

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Loss of linearity

In Quantum Mechanics a state is a normalized Hilbert space vector.

For the states of a HCA thenormisnot

a conserved quantity. The conserved quantity corresponding to the quantum norm conservation is:

hΨ0, ˆOIΨ0i ∈ R , where the superscript I refers to the identity operator acting on ψ+, ψ−. Possible restriction of the space of states: hΨ0, ˆOIΨ0i = 1 or at least hΨ0_{, ˆ}_OI_Ψ0_{i > 0.}

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Algebra of the observables

Vector space requirements

Associativity of addition: ˆG + (ˆF + ˆH) = ( ˆG + ˆF) + ˆH; Commutativity of addition: ˆG + ˆF = ˆF + ˆG;

Identity element of addition: there exist an element 0 ∈ U , called the zero vector, such that ˆG + 0 = ˆG for all ˆG ∈ U ;

Inverse element of addition: for every ˆOGthere exists an element − ˆG, called the additive inverse of ˆG, such that ˆG + (− ˆG) = 0;

Compatibility of scalar multiplication with field multiplication: α(β ˆG) = (αβ) ˆG; Identity element of scalar multiplication: 1 ˆG = ˆG, where 1 is the usual unity in C. Distributivity of scalar multiplication with respect to vector addition:

α( ˆG + ˆF) = α ˆG + αˆF;

Distributivity of scalar multiplication with respect to field addition: (α + β) ˆG = α ˆG + β ˆG.

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Algebra of the observables

We can define a product between observables which satisfy the following requirements:

Requirements

I G(ˆˆ F ˆH) = ( ˆGˆF) ˆH;

II G(ˆˆ F + ˆH) = ˆGˆF + ˆG ˆH;

III αβ( ˆGˆF) = (α ˆG)(β ˆF); So they form analgebra.

Involution mapping †

A Gˆ††= ˆG;

B ( ˆGˆF)†= ˆF†Gˆ†;

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Algebra of the observables

Requirements

A Gˆ††= ˆG;

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Algebra of the observables

Requirements

A Gˆ††= ˆG;

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C*-algebra

The norm of an observable

Norm for an observable:

k ˆGk = sup

{(ψ+,ψ−) : |h(ψ+,ψ−),(ψ+,ψ−)i|=1}

|h(ψ+, ψ−), ˆG(ψ+, ψ−)i| .

It satisfies the following requirements:

Condition for the norm

k ˆGk ≥ 0 and k ˆGk = 0 if and only if ˆG = 0; kα ˆGk = |α|k ˆGk;

k ˆG + ˆFk ≤ k ˆGk + kˆFk; k ˆGˆFk ≤ k ˆGkkˆFk.

C*condition

k ˆG†Gk = k ˆˆ Gk2.

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Rewriting the action and the conservation laws

We can write the action and the updating equations in terms of the states Ψ0:

Action S :=X n hΨ0_{(nl), Σ} 3Ψ0(nl + l)i − hΨ0(nl), Σ3 h Σ3− il(Î + Σ1) ˆH i Ψ0(nl)i , Updating equations Ψ0(nl + l) =hΣ3− ilÎ + Σ1 ˆHiΨ0(nl) , ˆ_{I =}Î 0 0 Î Σ1=0 Î Î 0 Σ3= _Î 0 0 −Î

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Rewriting the action and the conservation laws

Theorem A: Theorem A If [ ˆG, ˆH] = 0, then ˆT†₁OˆG_T_ˆ 1= ˆOG,

which implies hΨ0_{(nl), ˆ}_OG_Ψ0_{(nl)i = const}

ˆ G = _ˆ G 0 0 Gˆ ˆ H = _ˆ H 0 0 Hˆ ˆ OG= _ˆ G 0 0 − ˆG ˆ T1= _{ˆI − il ˆ} H −il ˆH −il ˆH −ˆI − il ˆH *****Con lim*****

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Continuum limit

The continuum limit of theHCAl → 0, nl = t, when we consider the state Ψ0. The updating equations become:

Updating equation

ψ−(t) = 0 , dψ+(t)

dt = −il ˆHψ+(t) .

d

dt is the usual time derivative.

We can check that the time evolution for ψ+ becomesunitary (Un= Un(−l ˆH)):

Time evolution ψ+(nl) = −inh_(U n−2)αβ+ i(Un−1)αβ]ψ+(l) + il( ˆHUn−2)αβ i ψβ₋(l) → l→0 −→ e−i ˆHt_{ψ+(0) .}

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Continuum limit

The limit of the conserved quantities in terms of ψ+:

Conserved quantities lim l→0hΨ 0_{(t), ˆ}_OG_Ψ0_{(t)i = hψ} +(t), ˆGψ+(t)i . ˆ

G is a Hermitean matrix acting on ψ+ andcommutingwith ˆH, and ˆOG_{is the related restricted} observable.

The states become:

States

Ψ0(t) =ψ+ 0

.

Because one of the components of Ψ0 vanishes, we can keep just one of them, ψ(t) = ψ+(t)

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Single systems: Quantum Mechanics vs HCAs

Quantum Mechanics State |ψi Hamiltonian ˆ H Time evolution

|ψ(t)i = e−i ˆHt_|ψ(0)i Conservation laws [ ˆG, ˆH] = 0 ⇒ hψ(t)| ˆG|ψ(t)i = c HCAs State Ψ0= (ψ+, ψ−) Hamiltonian ˆ H Time evolution Ψ0(nl) = ( ˆT1)n−1Ψ0(l) Conservation laws [ ˆG, ˆH] ⇒ hΨ0(nl), ˆOGΨ0(nl)i = c *****(Comp)*****

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Composite HCA without interactions

We combine twoHCAsusing thetensor product structurefor the composite system.

We consider two different HCAs the states of which are

Ξ(nl) = (ξ1(nl), ξ2(nl)) ∈ V0 and Φ(nl) = (φ1(nl), φ2(nl)) ∈ V00.

They evolve following the updating equations shown before; the first with the Hamiltonian ˆH0 the second with ˆH00.

The state of the composite HCA:

Ψ = Ξ ⊗ Φ , Ψ =     ξ1⊗ φ1 ξ1⊗ φ2 ξ2⊗ φ1 ξ2⊗ φ2     .

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The action and updating equations for composite HCAs

The updating equations for composite HCAs can be written as:

Updating equation for composite HCA

Ψ(nl + l) = Ξ(nl + l) ⊗ Φ(nl + l) = ˆT0₁⊗ ˆT00₁ [Ξ(nl) ⊗ Φ(nl)] , where ˆT0₁ and ˆT00₁ are the time evolution operators for the first and the second subsystem, respectively.

We call ˆT1= ˆT0₁⊗ ˆT00₁ the time evolution operator of the composite system. It takes the form:

ˆ T1=    

−4l2_Hˆ0_{⊗ ˆ}_H00 _{−2il ˆ}_H0_{⊗ Î −2ilÎ ⊗ ˆ}_H00 _{Î ⊗ Î}

−2il ˆH0⊗ Î 0 Î ⊗ Î 0

−2ilÎ ⊗ ˆH00 Î ⊗ Î 0 0

ˆI ⊗ ˆI 0 0 0     .

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The action and updating equations for composite HCAs

ˆ T1=    

−4l2_Hˆ0_{⊗ ˆ}_H00 _{−2il ˆ}_H0_{⊗ Î −2ilÎ ⊗ ˆ}_H00 _{Î ⊗ Î}

−2il ˆH0_{⊗ Î} ₀ _{Î ⊗ Î} ₀

−2ilÎ ⊗ ˆH00 Î ⊗ Î 0 0

ˆI ⊗ ˆI 0 0 0     .

We can also rewrite theactionfor the state Ψ using ˆT1and introducing the matrix: Σ1=     0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0     It is: S :=X n hΨ(nl), Σ1Ψ(nl + l)i − hΨ(nl), Σ1T1Ψ(nl)i .ˆ

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Conservation laws for composite HCAs

Now we want to write the corresponding ofTheorem Afor composite systems. Given two restricted observables for the two subsystems, we can consider as restricted observables ˆOGfor the composite system those of the kind

ˆ

OG= ˆOG0⊗ ˆOG00 or linear combinations of them.

ˆ OG=1 4     0 0 0 Gˆ 0 0 Gˆ 0 0 Gˆ 0 0 ˆ G 0 0 0     , G = ˆˆ G0⊗ ˆG00, Ψ =     ψ1 ψ2 ψ3 ψ4     . It holds: Theorem A”

If ˆG commutes both with ˆH0_{⊗ ˆI and ˆI ⊗ ˆ}_H00_{then the value of the observable ˆ}_OG is conserved that is:

ˆ

T†₁OˆGTˆ1= ˆOG.

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Introducing the interactions

Now we want to introduce interactions between the two subsystems. Differently from Quantum Mechanics a state is made byfourHilbert space vectors

Ψ = (ψ1, ψ2, ψ3, ψ4) and the time evolution operator needstwoHamiltonians ˆH1 and ˆH2.

We can take two Hamiltonians of the kind ˆHi= bOi0⊗ bOi00, i = 1, 2. We have two cases:

[ ˆH1, ˆH2] = 0; [ ˆH1, ˆH2] 6= 0.

One-step time evolution operator

ˆ Tint₁ =      −4l2 ˆH1Hˆ2+ ˆH2Hˆ1 2 −2il ˆH1 −2il ˆH2 Iˆ −2il ˆH1 0 Iˆ 0 −2il ˆH2 Iˆ 0 0 ˆ I 0 0 0      .

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Introducing the interactions

One-step time evolution operator

ˆ Tint₁ =      −4l2 ˆH1H2+ ˆˆ H2H1ˆ 2 −2il ˆH1 −2il ˆH2 Iˆ −2il ˆH1 0 Iˆ 0 −2il ˆH2 Iˆ 0 0 ˆ I 0 0 0      . Action S :=X n

hΨ(nl), Σ1Ψ(nl + l)i − hΨ(nl), Σ1Tˆint₁ Ψ(nl)i .

Updating equation

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Conservation laws

Theorem A”

Consider the operator ˆG that acts on one of the tensor product states ψi, i = 1, ..., 4. If [ ˆG, ˆH1] = 0 and [ ˆG, ˆH2] = 0, then ˆOG_{is a conserved restricted} observable that is:

ˆ Tint†₁ OˆGTˆint₁ = ˆOG. ˆ OG=1 4     0 0 0 Gˆ 0 0 Gˆ 0 0 Gˆ 0 0 ˆ G 0 0 0     . *****(Con lim)*****

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Continuum limit

Again we study the limit l → 0, nl = t. In this limit the initial condition become: lim

l→0Ψ(l) = (ψ1(0), ψ1(0), ψ1(0), ψ1(0)) .

Also the state at time t is made of fouridenticalHilbert space vectors: lim

l→0Ψ(nl) = (ψ1(t), ψ1(t), ψ1(t), ψ1(t)) . So the information contained in Ψ isredundant.

Equation of motion for ψ = ψ1:

Dψ(t) = −ic( ˆH1+ ˆH2)ψ(t) , where D is thesymmetricderivative.

Conserved restricted observables in terms of ψ: hψ(t), ˆGψ(t)i = const , if [ ˆG, ˆH] = 0, where ˆH = ˆH1+ ˆH2.

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Continuum limit

For consistency we checked that also the solutions for ψ(t) becomeunitary, as it is in Quantum Mechanics. ψ(t) = lim l→0i n+1_(iU1 n+ U 1 n−1)i n+1_(iU2 n+ U 2 n−1)ψ(0) = e−i( ˆ H1+ ˆH2)t_{ψ(0) ,} where U1 n = Un(−l ˆH1) and Un2= Un(−l ˆH2). *****(Sum)*****

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Composite systems: Quantum Mechanics vs HCAs

Quantum Mechanics State |ψi Hamiltonian ˆ H Time evolution

|ψ(t)i = e−i ˆHt_|ψ(0)i Conservation laws [ ˆG, ˆH] = 0 ⇒ hψ(t)| ˆG|ψ(t)i = const HCAs State Ψ = (ψ1, ψ2, ψ3, ψ4) Hamiltonians ˆ H1 , Hˆ2 Time evolution Ψ(nl) = ( ˆT1)n−1Ψ(l) Conservation laws [ ˆG, ˆH1] & [ ˆG, ˆH2] ⇒ hΨ(nl), ˆOGΨ(nl)i = const

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Numerical studies

Some numerical results for the time evolution of single and composite HCAs: For simplicity, in all numerical evaluations we show forsingle systems, we study the behaviour in time of ψ+ for initial conditions Ψ0(l) = (ψ+(l), ψ−(l)) = (ψ+(l), 0), where ψ+= (1/2)(ψ1+ ψ2) and ψ−= (1/2)(ψ1− ψ2).

We show for single systems that we get very different behaviours depending on the maximum eigenvalue maxof the Hamiltonian.

Then we show the behaviour of ψ+ in the case lmax< 1. Finally we considercomposite systemsand studyentanglement. We study systems with initial conditions

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Single System

We want to show that for ρmax= lmax≥ 1 themainpart of the Hilbert space vector ψ+(nl), for n 1 is the eigenstate corresponding to max.

To do this we study the quantity P+(n, x = ρ; ∆) = Un−2(x−∆)+iUn−1(x−∆)−i(x−∆)Un−2(x−∆) 2 Un−2(x)+iUn−1(x)−iρUn−2(x) 2 as a function of x.

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Single System

While, if ρmax< 1 we have that the eigenstates corresponding to each eigenvalue as the following qualitative behaviour in time.

20 40 60 80 100 120 140 n 0.50001 0.50002 0.50003 0.50004 0.50005 y ÈRΑ +HnL 2 for Ρ1=-Ρ2=0.01 0.8 1.0 1.2 1.4 y ÈRΑ_+HnL2 for Ρ1=0.8 and Ρ2=0.6

In these figures we plotted the quantity |Rα +(nl)|2=

Un−2(ρα) + iUn−1(ρα) − iραUn−2(ρα)(1/2) 2

, for two different values of ρα.

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Single System

We also show the behaviour of the real and imaginary part of the eigenstate both for small (ρ = 0.1) and big (ρ = 0.8) eigenvalues:

10 20 30 40 50 60 70n - 0.4 - 0.2 0.2 0.4 y

Im@R1HnLD and Re@R1HnLD for Ρ1=0.1

10 20 30 40 50n - 1.0 - 0.5 0.5 1.0 y

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Single System

And the difference between the time evolution of a quantum eigenstate ( = 0.1) and an HCA eigenstate corresponding to the same eigenvalue (ρ = 0.1).

10 20 30 40 50n - 1.0 - 0.5 0.5 1.0 y

Quantum Hamiltonian state vs. HCA Hilbert vector

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Composite system

We study a composite system made by two two-dimensional HCAs. Time evolution operator built starting from two interacting Hamiltonians

ˆ

H1 and ˆH2, with [ ˆH1, ˆH2] = 0.

We are interested in the behaviour of the Hilbert space vector ψ++ which is four-dimensional.

We will work in a factored basis in which the two Hamiltonians are ˆ H1= ˆH2= 1₂( ˆK1⊗ ˆK2+ ˆQ1⊗ ˆQ2). ˆ K1=k 0 1 0 0 k100 , K2ˆ =k 0 2 0 0 k002 , Q1ˆ = 0 q1 q1∗ 0 , Q2ˆ = 0 q2 q∗2 0 . We can distinguish betweenentangledandnon-entangledstates:

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Composite system

ˆ

We will work in a factored basis in which the two Hamiltonians are ˆ

H1= ˆH2= 1₂( ˆK1⊗ ˆK2+ ˆQ1⊗ ˆQ2).

We can distinguish betweenentangledandnon-entangledstates:

Non-entangled states     a 0 0 0     ,     0 a 0 0     ,     0 0 a 0     ,     0 0 0 a     ,     a b 0 0     ,     a 0 b 0     ,     0 a 0 b     ,     0 0 a b     ,     a b c d     ,

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Composite system

ˆ

We will work in a factored basis in which the two Hamiltonians are ˆ

H1= ˆH2= 1₂( ˆK1⊗ ˆK2+ ˆQ1⊗ ˆQ2).

We can distinguish betweenentangledandnon-entangledstates:

Entangled states     0 a b 0     ,     a 0 0 b     ,     a b c 0     ,     a b 0 c     ,     a 0 b c     ,     0 a b c     ,     a b c d     ,

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Composite system

We start with thenon-entangledstate ψ++(l) = (1, 0, 0, 0). Theone-step time evolution operator acting on ψ++ is of the kind:

    A2+ |E|2 0 0 E(A + D) 0 B2+ |F |2 F (B + C) 0 0 F∗(B + C) C2+ |F |2 0 E∗(A + D) 0 0 D2+ |E|2     ,

where A, B, C, D, E, F ∈ C. Thus, at time 2l we get the state:

ψ++(2l) =     A2_{+ |E|}2 0 0 E∗(A + D)     , where A2+ |E|2=1 − ilk 0 1k 0 2 2 2 − l2 |q1|2₄|q2|2 and E∗(A + D) = −ilq₁∗q₂∗− l2_q∗ 1q2∗ k0₁k0₂+k₁00k00₂

4 . Thus, ψ++(2l) is one of theentangled states.

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Conclusions & perspective

In this thesis we have seen that:

HCAs are Cellular Automata the updating equations of which go into the Schr¨odinger equation in the limit l → 0, nl = t;

Their space of states is doubled w.r.t. the corresponding quantum space of states, so maybe there is “space” for hidden variables.

They have conservation laws that are in a 1:1 correspondence with their quantum counterparts, with the difference that the conserved quantities are the real part of correlation functions between states at consecutive times;

We can build on HCAs a C*-algebraic structure similar to that of Quantum Mechanics with the main difference that the normalization of the states is not possible, since their norm is not conserved;

We can combine two HCAs using the tensor product structure between their spaces of states;

We can find non-interacting time evolution operators for composite systems that do not introduce spurious correlations between factored initial states, while interacting time evolution operator can produce entanglement in a way similar to that of Quantum Mechanics.

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Conclusions & perspective

They have conservation laws that are in a 1:1 correspondence with their quantum counterparts, with the difference that the conserved quantities are the real part of correlation functions between states at consecutive times;

We can build on HCAs a C*-algebraic structure similar to that of Quantum Mechanics with the main difference that the normalization of the states is not possible, since their norm is not conserved;

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Conclusions & perspective

They have conservation laws that are in a 1:1 correspondence with their quantum counterparts, with the difference that the conserved quantities are the real part of correlation functions between states at consecutive times; We can build on HCAs a C*-algebraic structure similar to that of Quantum Mechanics with the main difference that the normalization of the states is not possible, since their norm is not conserved;

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Conclusions & perspective

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Conclusions & perspective

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Conclusions & perspective

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Conclusions & perspective

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Conclusions & perspective

Perspectives:

We think that it is worth to study the case in which a composite system has two non-commuting Hamiltonians. This study might show very peculiar behaviours (for example the restricted observable corresponding to the sum of the two Hamiltonians will not be conserved);

It would be useful try to reconsider the space of states of the HCAs as a Krein spaces and build for the observables a slightly different algebraic structure; It would be worth to explore the effect of finite l on entanglement as higher order corrections to Quantum Mechanics;