Effective approach to the problem of time: general features and examples

TL;DR

This paper presents an effective approach to quantum dynamics that reformulates the Dirac quantization for constrained systems, enabling local internal times and relational evolution to be understood and managed through gauge transformations, with practical examples.

Contribution

It introduces a method to handle the problem of time in quantum systems using local internal times and gauge transformations, supported by illustrative quantum mechanical examples.

Findings

Local internal times can be generalized to quantum systems.

Relational evolution can be described via gauge choices.

The approach is demonstrated with two quantum mechanical examples.

Abstract

The effective approach to quantum dynamics allows a reformulation of the Dirac quantization procedure for constrained systems in terms of an infinite-dimensional constrained system of classical type. For semiclassical approximations, the quantum constrained system can be truncated to finite size and solved by the reduced phase space or gauge-fixing methods. In particular, the classical feasibility of local internal times is directly generalized to quantum systems, overcoming the main difficulties associated with the general problem of time in the semiclassical realm. The key features of local internal times and the procedure of patching global solutions using overlapping intervals of local internal times are described and illustrated by two quantum mechanical examples. Relational evolution in a given choice of internal time is most conveniently described and interpreted in a corresponding choice of gauge at the effective level and changing the internal clock is, therefore, essentially achieved by a gauge transformation. This article complements the conceptual discussion in arXiv:1009.5953.