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arXiv:physics/0403077

Repeated dynamic quantization

Yuri A. Rylov

Published Mar 13, 2004• 0 citations
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Abstract

In conventional quantum mechanics the quantum particle is a special object, whose properties are described by special concepts and quantum principles. The quantization is a special procedure, which is accompanied by introduction of special concepts, and this procedure cannot be repeated. In the model conception of quantum phenomena (MCQP) the quantum particle is a partial case of a stochastic particle, and the quantum dynamics is a special case of the stochastic particle dynamics. In MCQP the quantization is a dynamical procedure, where a special quantum term is added to the Lagrangian of the statistical ensemble of (classical) particles. This procedure can be repeated many times, because it is not accompanied by introduction of new concepts. It is very convenient from the formal viewpoint, because the set of dynamic quantizations forms a one-parameter group, which allows one to separate the dynamical and statistical (stochastic) properties.

DOI
10.48550/arxiv.physics/0403077

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In conventional quantum mechanics the quantum particle is a special object, whose properties are described by special concepts and quantum principles. The quantization is a special procedure, which is accompanied by introduction of special concepts, and this procedure cannot be repeated. In the model conception of quantum phenomena (MCQP) the quantum particle is a partial case of a stochastic particle, and the quantum dynamics is a special case of the stochastic particle dynamics. In MCQP the quantization is a dynamical procedure, where a special quantum term is added to the Lagrangian of the statistical ensemble of (classical) particles. This procedure can be repeated many times, because it is not accompanied by introduction of new concepts. It is very convenient from the formal viewpoint, because the set of dynamic quantizations forms a one-parameter group, which allows one to separate the dynamical and statistical (stochastic) properties.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
Classical model S_dc of Dirac particle S_D is constructed. S_D is the dynamic system described by the Dirac equation. Its classical analog S_dc is described by a system of ordinary differential equations, containing the quantum constant h as a parameter. Dynamic equations for S_dc are determined by the Dirac equation uniquely. Both dynamic systems S_D and S_dc appear to be nonrelativistic. One succeeded in transforming nonrelativistic dynamic system S_dc into relativistic one S_dcr. The dynamic system S_dcr appears to be a two-particle structure (special case of a relativistic rotator). It explains freely such properties of S_D as spin and magnetic moment, which are strange for pointlike structure. The rigidity function f_r(a), describing rotational part of total mass as a function of the radius aaa of rotator, has been calculated for S_dcr. For investigation of S_D and construction of S_dc one uses new dynamic methods: dynamic quantization and dynamic disquantization. These relativistic pure dynamic procedures do not use principles of quantum mechanics (QM). They generalize and replace conventional quantization and transition to classical approximation. Totality of these methods forms the model conception of quantum phenomena (MCQP). Technique of MCQP is more subtle and effective, than conventional methods of QM. MCQP relates to conventional QM, much as the statistical physics relates to thermodynamics.
One can construct the model conception of quantum phenomena (MCQP) which relates to the axiomatic conception of quantum phenomena (ACQP), (i.e. to the conventional quantum mechanics) in the same way, as the statistical physics relates to thermodynamics. Such a possibility is based on a new conception of geometry, which admits one to construct such a deterministic space-time geometry, where motion of free particles is primordially stochastic. The space-time geometry can be chosen in such a way that statistical description of random particle motion coincides with the quantum description. Methods of MCQP in investigation of quantum phenomena appear to be more subtle and effective than, that of ACQP. For instance, investigation of the free Dirac equation in framework of MCQP shows that the Dirac particle is in reality a rotator, i.e. two particles rotating around their common center of inertia. In the framework of MCQP one can discover the force field, responsible for pair production, that is impossible in the framework of ACQP.
A series of successive quantizations is considered, starting with the quantization of a non-relativistic or relativistic point particle: (1) quantization of a particle’s position, (2) quantization of wave function, (3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first-order equation in time derivative for complex wave function (or functional), the equivalent second-order equation for its real part was used. In such a way, at each level of quantization, the equation a quantum state satisfies is just like that of a harmonic oscillator, and wave function(al) is composed in terms of the pair of its canonically conjugated variables.
Quantum dynamics can be regarded as a generalization of classical finite-state dynamics. This is a familiar viewpoint for workers in quantum computation, which encompasses classical computation as a special case. Here this viewpoint is extended to mechanics, where classical dynamics has traditionally been viewed as a macroscopic approximation of quantum behavior, not as a special case. When a classical dynamics is recast as a special case of quantum dynamics, the quantum description can be interpreted classically. For example, sometimes extra information is added to the classical state in order to construct the quantum description. This extra information is then eliminated by representing it in a superposition as if it were unknown information about a classical statistical ensemble. This usage of superposition leads to the appearance of Fermions in the quantum description of classical lattice-gas dynamics and turns continuous-space descriptions of finite-state systems into illustrations of classical sampling theory. A direct mapping of classical systems onto quantum systems also allows us to determine the minimum possible energy scale for a classical dynamics, based on a localized rate of state change. We use a partitioning description of dynamics to define locality, and discuss the ideal energy of two model systems.