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arXiv:quant-ph/9909025

Structure behind Mechanics I: Foundation

Ono, Toshihiko

Published Sep 8, 1999• 0 citations
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Abstract

This paper proposes a basic theory on physical reality and a new foundation for quantum mechanics and classical mechanics. It presents a scenario not only to solve the problem of the arbitrariness on the operator ordering for the quantization procedure, but also to clarify how the classical-limit occurs. This paper is the first of the three papers into which the previous paper quant-ph/9906130 has been separated for readability.

DOI
10.48550/arxiv.quant-ph/9909025

Related papers

Structure behind Mechanics

Ono, Toshihiko

This paper proposes a basic theory on physical reality, and a new foundation for quantum mechanics and classical mechanics. It does not only solve the problem of the arbitrariness on the operator ordering for the quantization procedure, but also clarifies how the classical-limit occurs. It further compares the new theory with the known quantization methods, and proposes a self-consistent interpretation for quantum mechanics. It also provides the internal structure inducing half-integer spin of a particle, the sense of the regularization in the quantum field theory, the quantization of a phenomenological system, the causality in quantum mechanics and the origin of the thermodynamic irreversibility under the new insight.

arXiv:quant-ph/9906130•Jun 30, 1999•0 citations

Overview of "Structure behind Mechanics"

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This letter proposes a new scenario to solve the structural or conceptual problems remained in quantum mechanics, and gives an overview of the theory proposed in quant-ph/9906130 (including quant-ph/9909025 and quant-ph/0001015).

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Structure behind Mechanics II: Deduction

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This paper proves that protomechanics, previously introduced in quant-ph/9909025, deduces both quantum mechanics and classical mechanics. It does not only solve the problem of the arbitrariness on the operator ordering for the quantization procedure, but also that of the analyticity at the exact classical-limit of $\hbar =0 $. In addition, proto-mechanics proves valid also for the description of a half-spin.

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This paper proposes a basic theory on physical reality and a new foundation for quantum mechanics and classical mechanics. It presents a scenario not only to solve the problem of the arbitrariness on the operator ordering for the quantization procedure, but also to clarify how the classical-limit occurs. This paper is the first of the three papers into which the previous paper quant-ph/9906130 has been separated for readability.
This paper proposes a basic theory on physical reality, and a new foundation for quantum mechanics and classical mechanics. It does not only solve the problem of the arbitrariness on the operator ordering for the quantization procedure, but also clarifies how the classical-limit occurs. It further compares the new theory with the known quantization methods, and proposes a self-consistent interpretation for quantum mechanics. It also provides the internal structure inducing half-integer spin of a particle, the sense of the regularization in the quantum field theory, the quantization of a phenomenological system, the causality in quantum mechanics and the origin of the thermodynamic irreversibility under the new insight.
In this thesis we explore the mathematical foundations that unite physics at a quantum scale, quantum mechanics, with a macroscopic scale, classical mechanics. We seek to understand the mathematical motivation behind the quantum-classical correspondence and how it unites two seemingly different theories of the physical world. We show how this correspondence binds the Hamiltonian theory of classical physics to the Hilbert space theory in quantum mechanics, and establish a way to translate between classical observables and quantum operators, using the Fourier transform. These approaches to “quantizing” a physical state can be applied generally to a wide variety of observable quantities in classical mechanics.
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)].