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arXiv:1301.5476

A covering probability amplitude formalism for classical and quantum mechanics, and possible new modes of non-classical behaviour

Ram K. Varma

Published Jan 23, 2013• 0 citations
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Abstract

A generalized dynamics is postulated in a product space ${\cal R}^{3}\times {\cal S}^{1}$ with ${\cal R}^3$ representing the configuration space of a one particle system to which is attached the U(1) fibre bundle represented by the manifold ${\cal S}^{1}$. The manifold ${\cal S}^{1}$ is chosen to correspond to the action phase, with the action being the principal function corresponding to the associated classical dynamical system. A Hilbert space representation of the flow equation representing the dynamics in the above product space is then obtained, which is found to yield a probability amplitude formalism in terms of a generalized set of equations of the Schrödinger form which constitutes a covering formalism for both quantum and classical mechanics. Being labelled by an integral index $n$, the equation corresponding to $n=1$ is identified with the Schrödinger equation, while equations corresponding to $n\rightarrow large$ yield the classical dynamics through the WKB limit. Equations with lower indices n=2,3,4,... predict the existence of new modes of non-classical behaviour, the observational implications of which are discussed. Some preliminary experimental results are presented which point to the existence of these modes of non-classical behaviour.

DOI
10.48550/arxiv.1301.5476

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It is pointed out that the behaviour of certain appropriate ensembles of a class of Hamiltonian systems can be described in terms of a set of probability amplitudes obeying Schrodinger like equations. Such a description leads to a new concept of described by the different equations. Such have in fact been observed experimentally recently in a certain classical mechanical system and thus testify to the validity of the probability amplitude description for the classical deterministic system. The possibility of the existence of new modes of quantum behaviour corresponding to the new ensemble modes other than the Schrodinger mode is also pointed out.
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We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced by an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper. In the second part of this investigation are derived several forms of the bundle (analogue of the) Schrödinger equation governing the time evolution of state liftings of paths or sections along paths. We prove that up to a constant the matrix-bundle Hamiltonian, entering the bundle analogue of the matrix form of the conventional Schrödinger equation, coincides with the matrix of coefficients of the evolution transport. This allows us to interpret the Hamiltonian as a gauge field. We apply the bundle approach to the description of observables. It is shown that to any observable there corresponds a unique Hermitian lifting of paths or morphism along paths in corresponding bundles.
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A possible generalization of quantum mechanics is examined by showing that the motion in phase space of a classical Brownian particle may be described by a complex probability amplitude depending on the phase-space coordinates and the time, and obeying a Schrödinger-like equation. However formal this result may seem, the usual dynamical operators may be defined whose physical meaning stems directly from the theory. An outstanding feature of the formalism is that ordinary quantum mechanics in configuration space may be recovered in a limiting process whereby the velocity variable, defined now through a statistical distribution, is eliminated. Therefore, it plays the role of a hidden variable. This result supports recent reinterpretations of von Neumann's theorem on the nonexistence of such variables in quantum mechanics and serves as a counterexample of the usual interpretation of his theorem.