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arXiv:0908.4543

Quantum Geometry of a Configuration Space in a Covariant Dynamical Theory

Natalia N. Gorobey, Alexander S. Lukyanenko, Inna Lukyanenko

Published Aug 31, 2009• 0 citations
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Abstract

A quantum version of the action principle in a simple covariant dynamical theory of two relativistic particles is formulated. The central object of this new formulation of quantum theory is a stationary eigenvalue of the quantum action. This quantity defines a quantum geometry in a configuration space. In the presence of "probe" fields it plays the role of a generation function of observables.

DOI
10.48550/arxiv.0908.4543

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A quantum version of the action principle in a simple covariant dynamical theory of two relativistic particles is formulated. The central object of this new formulation of quantum theory is a stationary eigenvalue of the quantum action. This quantity defines a quantum geometry in a configuration space. In the presence of "probe" fields it plays the role of a generation function of observables.
Hamilton's principle of stationary action is formulated in classical mechanics using the Lie algebra of Schouten concomitants of symmetric contravariant tensor fields on the configuration space of the system. Such a formulation is global and coordinate free. It is shown that a directly parallel formulation holds in quantum mechanics so long as all the Poisson brackets involved can be replaced in the quantum version by commutators in a canonical way. An example (where the Hamiltonian possesses a velocity dependent potential) in which this cannot be done is discussed and concluded that in this case the action is stationary only for a subclass of variations, namely those corresponding to Killing vector fields on the configuration manifold.
In its usual presentation, classical mechanics appears to give time a very special role. But it is well known that mechanics can be formulated so as to treat the time variable on the same footing as the other variables in the extended configuration space. Such covariant formulations are natural for relativistic gravitational systems, where general covariance conflicts with the notion of a preferred physical-time variable. The standard presentation of quantum mechanics, in turn, again gives time a very special role, raising well known difficulties for quantum gravity. Is there a covariant form of (canonical) quantum mechanics? We observe that the preferred role of time in quantum theory is the consequence of an idealization: that measurements are instantaneous. Canonical quantum theory can be given a covariant form by dropping this idealization. States prepared by noninstantaneous measurements are described by ``spacetime smeared states.'' The theory can be formulated in terms of these states, without making any reference to a special time variable. The quantum dynamics is expressed in terms of the propagator, an object covariantly defined on the extended configuration space.
For pt.II, see abstr. A66629 of 1973. The authors show that the Frechet derivative, which they have previously used to formulate a quantum-mechanical version of Hamilton's principle of stationary action, suffers from certain defects. They set up a simpler, coordinate-free, formulation of quantum mechanics based on the Lie algebra of symmetric contravariant tensor fields on the configuration manifold of the system and re-express the principle of stationary action using this formalism. They find the same relation between Hamiltonian and Lagrangian as before, but a much larger class of allowable variations, namely those associated with and C infinity vector fields.
The world geometry underlying the theory of elementary particles is investigated. The elementary, non-relativistic geometry of the physical space is given a general formulation, sufficient to yield the quanta of momentum and energy when the group quantization process is applied to the physical space. The elementary geometry is represented by a commutative (finite or continuous) group of translations to which Pontrjagin's theorem concerning the duality between a commutative group and its character group applies. The principle of uncertainty is given a group theoretical formulation in terms of the Pontrjaginian duality. This formulation allows a consequent and meaningful extension of quantum theory to a finite geometry. The role of the relativistic (finite or continuous) geometry of space-time in the theory of elementary particles is investigated. A formulation of the problem of elementary particles is given, and an eventual connection between the observed elementary particles and a geometry of the Lorentz group space is suggested. (auth)