Canonical Quantum Gravity and the Problem of Time

TL;DR

The paper surveys the problem of time in quantum gravity, contrasting canonical and covariant routes and organizing approaches into time-identified-before-quantisation, time-identified-after-quantisation, and timeless schemes. It provides an in-depth canonical-GR framework (ADM formalism, constraint algebra, and internal-time reductions) and outlines numerous concrete strategies for recovering time, such as internal Schrödinger, matter clocks, unimodular gravity, and semiclassical WKB methods, as well as timeless perspectives like conditional probabilities and consistent histories. A central theme is the tension between diffeomorphism invariance and the need for a dynamical time parameter, with substantial technical and interpretational challenges (global obstructions, operator ordering, Hilbert-space definitions, and the problem of functional evolution). The work emphasizes that no single solution yet resolves how time arises in quantum gravity, but it clarifies the landscape, highlighting promising avenues such as semiclassical regimes, relational observables, and decoherence-based accounts of emergent classical spacetime. Overall, it delineates the significant conceptual and technical hurdles that continue to shape the search for a coherent quantum theory of gravity and time.

Abstract

This is the write-up of my lectures at the NATO Summer School held in Salamanca in June 1992. The paper deals with the problem of time in quantum gravity. All the major schemes are reviewed. Please note that the paper is in two parts for ease of email transmission; this is part 1. The mailer from gr-qc may further subdivide these two sections.

Figures (2)

• Figure 1: The flow lines of the foliation of ${\cal M}$
• Figure 2: The lapse function and shift vector of a foliation