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arXiv:quant-ph/0406079

Quantum mechanics as kinetics

Прохоров Лев Васильевич

Published Jun 12, 2004• 1 citation
arXivHTMLPDF

Abstract

Relations between Hamiltonian mechanics and quantum mechanics are studied. It is stressed that classical mechanics possesses all the specific features of quantum theory: operators, complex variables, probabilities (in case of ergodic systems). The Planck constant and the Fock space appear after putting a dynamical system in a thermal bath. For harmonic oscillator in a thermal bath, the probability amplitudes can be identified with the complex valued phase functions $f(q+ip)$ describing small deviations from the equilibrium state, when the time of relaxation is large. A chain of such oscillators models both the one-dimensional space (or string), and one-dimensional quantum field theory.

DOI
10.48550/arxiv.quant-ph/0406079

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Relations between Hamiltonian mechanics and quantum mechanics are studied. It is stressed that classical mechanics possesses all the specific features of quantum theory: operators, complex variables, probabilities (in case of ergodic systems). The Planck constant and the Fock space appear after putting a dynamical system in a thermal bath. For harmonic oscillator in a thermal bath, the probability amplitudes can be identified with the complex valued phase functions f(q+ip)f(q+ip)f(q+ip) describing small deviations from the equilibrium state, when the time of relaxation is large. A chain of such oscillators models both the one-dimensional space (or string), and one-dimensional quantum field theory.
The nature of quantum description is clarified. It is shown that complex-valued probability amplitudes are admissible within classical Hamiltonian mechanics. According to standard probability theory, such a description is always possible. The case of a spherical phase space is considered. It is shown that, in such a classical theory, there appears a universal constant that has dimensions of action ( h ), as well as Fock space and all attributes of quantum mechanics. Excitations of a chain of such systems are described by the equations of quantum mechanics with a correct normalization condition. It is shown that an answer to the question of what a particle and its wave function are is provided by quantum field theory (these are a single-particle field excitation and a function that describes it). Experiments are proposed that would make it possible to solve the problem of the physical nature of the wave function.
The Heisenberg uncertainty postulates are a convenient way to describe the qualitative properties of quantum mechanics. The response from Heisenberg’s uncertainty principle is that one can create states that require a certain amount of momentum and position localization. The numerical quantities used by the old Newtonian mechanics, position, momentum, and energy are mere “shadows” of what nature describes: operators. The energy value contradicts the Newtonian notion of any energy level being necessary for a harmonic oscillator. The basic postulates of classical statistical mechanics are now applied to quantum statistical mechanics. In classical mechanics, the variables characterize a microstate that evolves according to the Hamiltonian equations. If the standard deviation relative to what we are interested in is very high, it is probably safe to use the expected value as a classical value. A quantum field theory is a relativistic invariant quantum theory where there is a particular form of quantity operators connected with the classical fields.
Abstract For present purposes, the first thing to recall about quantum mechanics is that all dynamical quantities, instead of being functions of the location of a system in phase space as in classical mechanics, are now represented by operators in a Hilbert space of quantum states. In particular, the quantum analog of the classical phase space distribution function is the quantum mechanical density matrix. Since this is seldom discussed in introductory texts on either quantum mechanics or statistical mechanics, it may be helpful to summarize some of its basic properties.
Dirac's Poisson-bracket-to-commutator analogy for the transition from classical to quantum mechanics assures that for many systems, the classical and quantum systems share the same algebraic structure. The quantum side of the analogy (involving operators on Hilbert space with commutators scaled by Planck's constant ℏ) not only gives the algebraic structure but also dictates the average values of physical quantities in the quantum ground state. On the other hand, the Poisson brackets of nonrelativistic mechanics, which give only the classical canonical transformations, do not give any values for physical quantities. Rather, one must go outside nonrelativistic classical mechanics in order to obtain a fundamental phase space distribution for classical physics. We assume that the values of physical quantities in classical theory at any temperature depend on the phase space probability distribution that arises from thermal radiation equilibrium including classical zero-point radiation with the scale set by Planck's constant ℏ. All mechanical systems in thermal radiation will inherit the constant ℏ from thermal radiation. Here, we note the connections between classical and quantum theories (agreement and contrasts) at all temperatures for the harmonic oscillator in one and three spatial dimensions.
The generator aspect of observables in classical mechanics leads naturally to a generalized classical mechanics, of which quantum mechanics is shown to be a particular case. Basic features of quantum mechanics follow, such as the identification of observables with self-adjoint operators, and canonical quantization rules. This point of view also gives a new insight on the geometry of quantum theory: Planck's constant is related for instance to the curvature of the quantum-mechanical space of states, and the uniqueness of quantum mechanics can be proved. Finally, the origin of the probabilistic interpretation is discussed.