Coordinate-free Quantization and its Application to Constrained Systems1

Canonical quantization entails using Cartesian coordinates, and Cartesian coordinates exist only in flat spaces. This situation can either be questioned or accepted. In this paper we offer a brief and introductory overview of how a flat phase space metric can be incorporated into a covariant, coordinate-free quantization procedure involving a continuous-time (Wiener measure) regularization of traditional phase space path integrals. Additionally we show how such procedures can be extended to incorporate systems with constraints and illustrate that extension for special systems. Introduction In order to quantize a system with constraints it is of course first necessary to have a quantization procedure for systems without constraints. Although the quantization of systems without constraints would seem to be well in hand due to the pioneering work of Heisenberg, Schrödinger, and Feynman, it is a less appreciated fact that all of the standard methods of quantization are consistent only in Cartesian coordinates [1]. As a consequence it follows that the usual quantization procedures depend—or at least seem to depend—on choosing the right set of coordinates before promoting c-numbers to qnumbers. This circumstance gives rise to an apparently unwanted coordinate dependence on the very process of quantization. For systems without constraints this is generally not a major problem because an underlying Euclidean space expressed in terms of Cartesian coordinates can generally be assumed. However, for systems with constraints, the configuration space—let alone the frequently more complicated phase space—are generally incompatible with a flat Euclidean structure needed to carry Cartesian coordinates. Hence, before we can properly quantize systems with constraints it will be necessary for us to revisit the quantization of systems without constraints in order to present a coordinate-free procedure for such cases. Only then will we be able to undertake the program represented by the title to this contribution. All individuals engaged in quantization have a natural inclination to seek a quantization procedure that is as coordinate independent as possible, and in particular, does not To appear in the Proceedings of the 2nd Jagna Workshop, Jagna, Bohol, Philippines, January, 1998. On leave from Laboratory of Theoretical Physics, JINR, Dubna, Russia.