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arXiv:quant-ph/0401035

On the classical limit of the hyperbolic quantum mechanics

Andrei Yu. Khrennikov

Published Jan 8, 2004• 3 citations
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Abstract

We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation -- so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit $h\to 0$ not only of the ordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus there are two different deformations of classical phase-space: complex Hilbert space and hyperbolic Hilbert space (module over a so called hyperbolic algebra -- the two dimensional Clifford algebra). To prove the correspondence principle we use the calculus over the hyperbolic algebra similar to functional superanalysis of Vladimirov-Volovich. Ordinary (complex) and hyperbolic quantum mechanics are characterized by two types of interference perturbation of the classical formula of total probability: ordinary $\cos$-interference and hyperbolic $\cosh$-interference.

DOI
10.48550/arxiv.quant-ph/0401035

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We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation -- so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit h→0h\to 0h→0 not only of the ordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus there are two different deformations of classical phase-space: complex Hilbert space and hyperbolic Hilbert space (module over a so called hyperbolic algebra -- the two dimensional Clifford algebra). To prove the correspondence principle we use the calculus over the hyperbolic algebra similar to functional superanalysis of Vladimirov-Volovich. Ordinary (complex) and hyperbolic quantum mechanics are characterized by two types of interference perturbation of the classical formula of total probability: ordinary cos⁡\coscos-interference and hyperbolic cosh⁡\coshcosh-interference.
Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.
The paper develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise product of observables to an appropriate noncommutative ⋆\star⋆-product and the deformation of the Poisson bracket to an appropriate Lie bracket is the key element in introducing the quantization of classical Hamiltonian systems. The considered class of deformations and the corresponding ⋆\star⋆-products contains as a special cases all deformations which can be found in the literature devoted to the subject of the phase space quantum mechanics. Fundamental properties of ⋆\star⋆-products of observables, associated with the considered deformations are presented as well. Moreover, a space of states containing all admissible states is introduced, where the admissible states are appropriate pseudo-probability distributions defined on the phase space. It is proved that the space of states is endowed with a structure of a Hilbert algebra with respect to the ⋆\star⋆-multiplication. The most important result of the paper shows that developed formalism is more fundamental then the axiomatic ordinary quantum mechanics which appears in the presented approach as the intrinsic element of the general formalism. The equivalence of two formulations of quantum mechanics is proved by observing that the Wigner-Moyal transform has all properties of the tensor product. This observation allows writing many previous results found in the literature in a transparent way, from which the equivalence of the two formulations of quantum mechanics follows naturally.
Quantum and classical mechanics share a common algebraic formalism which is expressed naturally in the language of category theory. A third realization of this formalism is the so-called hyperbolic quantum mechanics where split-complex numbers replace the usual complex numbers. We introduce and explore the corresponding generalization of C*-algebras and prove that hyperbolic quantum mechanics is not a viable candidate for describing Nature. Quantum and classical mechanics are the only acceptable theories of Nature which are invariant under tensorial composition.
We start to develop the quantization formalism in a hyperbolic Hilbert space. Generalizing Born's probability interpretation, we found that unitary transformations in such a Hilbert space represent a new class of transformations of probabilities which describe a kind of {\it hyperbolic interference}. The most interesting problem which was generated by our investigation is to find experimental evidence of hyperbolic interference. The hyperbolic quantum formalism can also be interesting as a new theory of probabilistic waves that can be developed parallely to the standard quantum theory. Comparative analysis of these two wave theories could be useful for understanding of the role of various structures of the standard quantum formalism. In particular, one of distinguishing feature of the hyperbolic quantum formalism is the restricted validity of the superposition principle.
Quantum mechanics (QM) provides a variety of ideas that can assist in developing Artificial Intelligence for healthcare, and opens the possibility of developing a unified system of Best Practice for inference that will embrace both QM and classical inference. Of particular interest is inference in the hyperbolic-complex plane, the counterpart of the normal i-complex plane of basic QM. There are two reasons. First, QM appears to rotate from i-complex Hilbert space to hyperbolic-complex descriptions when observations are made on wave functions as particles, yielding classical results, and classical laws of probability manipulation (e.g. the law of composition of probabilities) then hold, whereas in the i-complex plane they do not. Second, i-complex Hilbert space is not the whole story in physics. Hyperbolic complex planes arise in extension from the Dirac-Clifford calculus to particle physics, in relativistic correction thereby, and in regard to spinors and twisters. Generalization of these forms resemble grammatical constructions and promote the idea that probability-weighted algebraic elements can be used to hold dimensions of syntactic and semantic meaning. It is also starting to look as though when a solution is reached by an inference system in the hyperbolic-complex, the hyperbolic-imaginary values disappear, while conversely hyperbolic-imaginary values are associated with the un-queried state of a system and goal seeking behavior.