Group quantization of parametrized systems I. Time levels

TL;DR

This paper develops a gauge-invariant framework for quantizing finite-dimensional parametrized systems using perennials (functions constant on constraint orbits) and Dirac-inspired time surfaces. It builds a dual quantization path—algebraic via an algebra of elementary perennials and group-based via first-class canonical symmetry groups—and shows how projections to transversal surfaces preserve both Poisson and group structures, enabling calculations without explicit perennials. Time evolution is reinterpreted as a symmetry-driven motion of transversal surfaces, with Hamiltonians defined as perennial generators that move time levels; this yields a practical, gauge-independent route to Schrödinger and Heisenberg dynamics on reduced phase spaces. The approach addresses time-related issues such as the global time problem and multiple time-choice ambiguities by providing compatible dynamics across a family of gauges and time surfaces, with explicit theorems linking perennials, symmetries, and their projections. Overall, the work offers a structured, versatile toolkit for quantization of constrained systems and lays groundwork for applications to theories like gravity where time and gauge choices are intricate.

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 Poicaré 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.