Stochastic Dynamics of Quantum Physical Systems

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1. Introduction

As known [1] a quantum physical system can be represented by a couple, where is some -algebra which are hermit’s elements that are called observables and some set of positive functional with norm one, called the quantum states of this physical system [2] .

We say what the functional majorizes functional if is a positive functional [2] .

The state quantum physical system is called the pure state if it majorizes only functional type [2] . Denote the set of all pure states on -algebra by.

In the set of all linear continuous functional on -algebra we have topological structure, called as weakly topological structure [2] and defined by pre-basis:

,

where; according to this in the set we have the topological structure induced from this topological structure.

It is very known if commutative -algebra, then every positive linear functional defines complex valued measure on the which is separable and locally compact space of pure states under the ^{*} weekly topological structure. This measure is

defined by corresponding, , for all continuous

function. In non-commutative case we cannot define the measure on in this way. In the work [3] for each linear functional, with norm one, we define the probability measure for commutative and non-commutative cases in other ways. It gives us the opportunity to present a quantum physical system as a statistical structure [4] . Representation of quantum physical system in this form, in our opinion, is more comfortable for the solution of problem of quantum system, for example for testing hypotheses [4] . In this paper, using this representation, we have tried to consider the dynamic of quantum physical system as a random process.

2. Quantum Physical System as a Statistical Structure

Denote by the set of Hermit’s elements of -algebra.

Easy to show that every linear functional on the -algebra uniquely will be defined by its values on Banach subspace of Hermit’s elements,as it’s known [2] that every element of -algebra uniquely represented as, where and are Hermit’s.

Every a Hermit’s element in the -algebra any can be represented by integral

,

where projectors and represents the partition of unityof Hermit’s element [5] .

Correspond to projector the family of elements of the - algebra which has the condition:, if, , if and, if, where is the unit element in the algebra. It is clear that.

.

If, then, where and are Hermit’s elements. The representation such will be

,

Obviously, if some linear continuous functional on -algebra, then from the last equality we will have

,

where and are Hermit’s elements.

Let, be the set of all one dimensional projectors on -algebra and pure state, then the last equality follows that this state has the non zero meaning only on some projector and the meaning 0 on the other one dimensional projectors. Otherwise, we can always construct such functional which will not have the type and well majorized by the pure state. So how, if functional is a pure state, then for all Hermit’s elements.

Let, and such functional, which has the non zero meaning only on some projector and the meaning 0 on the other one dimensional projectors. It is clear, if we take sufficiently small, thenwe can achieve, that for every will have place inequality

.

It means, that the pure state majorize the functional which does not have the type, but this is impossible. It follows that the pure states are sach functional which satisfy the condition.

An integral representation of Hermit’s elements follows that for the pure states has a place of equality, where is some element of spectrum of Hermit element. It gives opportunity to identify every pre state with the set of number, where.

Consider the Tikhonov’s product, where spectrum of element.

It is clear, , because the set of such elements in product which represents linear continue maps with respect to the topological structure in which is defined by the norm:

.

Consequently in the set we have, induced from Tikhonov’s product topological structure. This topological structure coincide with theinduced topological structure from weakly topological structure on set of functionals on algebra.

We can also identify the set with the set of one dimensional projectors. We call as physical space of quantum system.

In the work [3] we have proved:

Theorem 1. Every state in space with weakly topological structure, defined on the Borel -algebra of a probability measure.

This measure constructed as: for the subset of, measure is norm of positive functional, , if this exists, which values on the elements of this subset are coincide to corresponding to values of the state. i.e..

Every measure describes distribution elementary particle in physical space of quantum system in the state.

If -algebra has a unit, then in the space with weakly topological structure the set of all state is convex compact set and represent convex linear combination of pure states from the set:

or limit of sequence, where [2] .

This means, that elements of set are the extreme points of set [2] .

Because each state defines a probability measure on couple, where is borel -algebra therefore it is easy to show, that every represent convex linear combination

of Dirak measures where or limit of sequense, where [2] .

For every state we have therefore it is easy that the value of

quantum state on observable is the middle value of this observable. The value is called the middle value of observable of quantum physical system in the state.

All told above follows that a quantum physical system is an object, so-called statistical structure [4] :

,

where some -algebra, Hermit element of which are called observables of this system, is the space of quantum system, Borel -algebra in, the probability measure defined by state and which describes distribution elementary particle in physical space of quantum system in the state.

3. A Stochastic Dynamics of Quantum System

Theorem 2. Every Hermit’s element, in -algebra defines probability measure on the set of states.

Proof: It is well-known that the map defined by formula is isometric embedding as Banach space in the double conjugate space [3] . If is a state then [2] ; it follows that if is positive element, then and.

Thus if is positive element then for each state on. Because is isometric, and therefore.

.

If is hermit’s element, because, for such elements and

.

Let, be hermit’s positive element in, then spectrum. Let be the set of all states on, if is a set of states; we assume the measure of this set is, if consists for all such element for which

Since, 0 and 1 are elements of [5] , and

therefore. It is clear that, if then

.

If we assume, then we get a measure on.

The sets for which we define measure, make -algebra in. This is not a Borel’s -algebra in space whit the weekly topology. Denote it by. Thus, we define on probability measure. The theorem is proved.

Consider the family of measures defined above, where is the set of positive hermit’s elements whit norm 1, is corresponding to hermit’s element, -algebra in.

Let statistical structure represent a quantum physical system,. For each we can define the measure on the set of states of given quantum physical system such:

if,.

Literally, we have defined measure on the set of measures, of which each element describes distribution elementary particles in physical space of quantum system in the state.

If is strongly one parametric group of maps of -algebra whit unity and for all, then following conditions are equivalent [2] :

1) All automorphisms of;

2) for all;

3) , where is the set of positive elements in;

4) , for all.

Each defined measure describes distribution of states in relatively to middle value of observable over states in, or distribution of elementary particles in physical space of quantum system in the states relatively to middle value of observable over states in.

It follows: If strongly continuous one parametric group of automorphisms describes dynamic of structure of observables, according to this, we have a picture of evolution of distribution of states quantum system relatively to each observable.

Such, the representation of quantum physical system as a statistical structure allows formalizing the dynamics of the quantum system as a random process.

References

[1] Bogolyubov, Н.Н., Logunov, A.А., Oksak, A.I. and Todorov, I.T. (1987) General Principles of the Quantum Field Theories. Nauka, Moscow. (In Russian)

[2] Bratteli, O. and Robinson, D. (1982) Operator Algebras and Quantum Statistical Mechanics. Nauka, Moscow. (In Russian)

[3] Mumladze, M. and Zerakidze, Z. (2015) The Criteria of Checking Hypotheses of Quantum States of Quantum Physical System. Journal of Algebra, Number Theory: Advances and Applications, 14, 57-67.

[4] Zerakidze, Z. and Mumladze, M. (2015) Statistical Structures and Consistent Criteria for Checking Hypotheses. LAMBERT Academic Publishing, Saarbrucken.

[5] Liusterniik, L.A. and Sobolev, V.I. (1965) Elements of Functional Analyses. Nauka, Moscow. (In Russian)