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arXiv:1303.1735

Fibre bundle formulation of time-dependent mechanics

Gennadi Aleksandrovich Sardanashvily

Published Mar 7, 2013• 0 citations
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Abstract

We address classical and quantum mechanics in a general setting of arbitrary time-dependent transformations. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over a time axis. Connections on these bundles describe reference frames. Quantum time-dependent mechanics is phrased in geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is leafwise geometric quantization.

DOI
10.48550/arxiv.1303.1735

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We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the bundle description the wavefunctions are replaced with (state) sections (covariant approach) or liftings of paths (equivalently: sections along paths) (time-dependent approach) of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation of the state sections/liftings in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations. Connections between the (retarded) Green functions of these equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.
We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present, second, part of our investigation, we consider a covariant approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) sections of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the transport along the identity map of the space\nobreakdash-time) of the state sections in the (total) bundle space. Connections between the (retarded) Green functions of the relativistic wave equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.
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