An algebraic approach to the "frozen formalism" problem of time

TL;DR

This paper introduces an algebraic framework for addressing the problem of time in quantum gravity, enabling consistent dynamical reduction and evolution without relying on complete observables or semiclassical assumptions.

Contribution

It presents a novel algebraic formulation of symplectic reduction that bypasses traditional difficulties and extends semiclassical results to fully quantum regimes.

Findings

Provides a consistent algebraic approach to the problem of time

Imposes new conditions on deparameterization methods

Shows internal time can be defined without semiclassical approximation

Abstract

The long-standing problem of time in canonical quantum gravity is the source of several conceptual and technical issues. Here, recent mathematical results are used to provide a consistent algebraic formulation of dynamical symplectic reduction that avoids difficult requirements such as the computation of a complete set of Dirac observables or the construction of a physical Hilbert space. In addition, the new algebraic treatment makes it possible to implement a consistent realization of the gauge structure off the constraint surface. As a consequence, previously unrecognized consistency conditions are imposed on deparameterization -- the method traditionally used to unfreeze evolution in completely constrained systems. A detailed discussion of how the new formulation extends previous semiclassical results shows that an internal time degree of freedom need not be semiclassical in order to define a consistent quantum evolution.