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arXiv:quant-ph/9803083 · Journal of Physics A Mathematical and General

Quantum mechanics from a geometric-observer's viewpoint

Bozhidar Zakhariev Iliev

Published Jan 30, 1998• 6 citations
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Abstract

We propose a version of non-relativistic quantum mechanics in which the pure states of a quantum system are described as sections of a Hilbert (generally infinitely-dimensional) fibre bundle over spacetime. Evolution is governed via a (kind of) parallel transport in this bundle. Some problems concerning observables are considered. The equations of motion for the state sections and observables are derived. We show that up to a constant the matrix of the coefficients of the evolution operator (transport) coincides with the matrix of the Hamiltonian of the investigated quantum system.

DOI
10.1088/0305-4470/31/4/017

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We propose a version of non-relativistic quantum mechanics in which the pure states of a quantum system are described as sections of a Hilbert (generally infinitely-dimensional) fibre bundle over spacetime. Evolution is governed via a (kind of) parallel transport in this bundle. Some problems concerning observables are considered. The equations of motion for the state sections and observables are derived. We show that up to a constant the matrix of the coefficients of the evolution operator (transport) coincides with the matrix of the Hamiltonian of the investigated quantum system.
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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 with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or sections 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. The present, first, part of this investigation is devoted to the introduction of basic concepts on which the fibre bundle approach to quantum mechanics rests. We show that the evolution of pure quantum mechanical states can be described as a suitable linear transport along paths, called evolution transport, of the state liftings in the Hilbert bundle of states of a considered quantum system.
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.
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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 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.
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 with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or sections 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 third part of our series we investigate the bundle analogues of the conventional pictures of motion. In particular, we find the state liftings and observable liftings corresponding to state vectors and observables respectively in the different pictures of motion. The equations of motion for these quantities are derived. Using the results obtained, problems concerning the integrals of motion are considered from the bundle viewpoint. Necessary and sufficient invariant bundle conditions for a dynamical variable to be an integral of motion are found.