Implement the arbitrariness of simultaneity conventions in canonical quantum gravity

Develop a canonical quantum gravity framework that faithfully implements the arbitrariness of simultaneity conventions (i.e., the freedom to choose foliations or time functions) within Dirac’s constrained Hamiltonian formalism, so that physical predictions are independent of these conventions and the resulting theory remains mathematically tractable and empirically adequate.

Background

Dirac’s approach to gauge symmetry in constrained Hamiltonian systems underlies many canonical quantization programs. In relativistic field theories, however, at-a-time symmetries require simultaneity conventions that are arbitrary. Enforcing empirical irrelevance of these conventions is technically difficult, and in practice the canonical program has struggled to realize this invariance in a complete and rigorous way.

In the standard model of particle physics the Dirac strategy is supplanted by BRST methods; in canonical approaches to quantum gravity, it remains central but incomplete. The thesis explicitly identifies the implementation of arbitrary simultaneity conventions as an outstanding open problem in canonical quantum gravity, highlighting a key obstacle to a fully satisfactory gauge-invariant quantization.