Group quantization of parametrized systems I. Time levels

TL;DR

This paper develops a gauge-invariant quantization method for parametrized systems, addressing the problem of time and the scarcity of perennials, by generalizing Dirac's ideas to all finite-dimensional first-class systems.

Contribution

It introduces a generalized framework based on elementary perennials and symmetry groups, enabling construction of quantum evolution and compatibility across different gauges.

Findings

Conditions for constructing time evolution in quantum mechanics are identified.

The framework ensures compatibility of quantum theories from different gauge choices.

A practical method for solving problems without explicit perennials is proposed.

Abstract

A method of quantizing parametrized systems is developed that is based on a kind of ``gauge invariant'' quantities---the so-called perennials (a perennial must also be an ``integral of motion''). The problem of time in its particular form (frozen time formalism, global problem of time, multiple choice problem) is met, as well as the related difficulty characteristic for this type of theory: the paucity of perennials. The present paper is an attempt to find some remedy in the ideas on ``forms of relativistic dynamics'' by Dirac. Some aspects of Dirac's theory are generalized to all finite-dimensional first-class parametrized systems. The generalization is based on replacing the Poicar\'{e} group and the algebra of its generators as used by Dirac by a canonical group of symmetries and by an algebra of elementary perennials. A number of insights is gained; the following are the main results. First, conditions are revealed under which the time evolution of the ordinary quantum mechanics, or a generalization of it, can be constructed. The construction uses a kind of gauge and time choice and it is described in detail. Second, the theory is structured so that the quantum mechanics resulting from different choices of gauge and time are compatible. Third, a practical way is presented of how a broad class of problems can be solved without the knowledge of explicit form of perennials.