Discrete Hamiltonian evolution and quantum gravity

TL;DR

This paper explores how different time discretization methods affect the preservation of canonical structures and constraints in Hamiltonian systems, with implications for quantum gravity models and gauge fixing procedures.

Contribution

It introduces conditions for explicit discretizations that preserve Poisson brackets and presents implicit discretizations that do so universally, revealing their impact on constraint algebra in quantum gravity.

Findings

Explicit discretizations impose strong restrictions on Hamiltonians.

Implicit discretizations preserve Poisson brackets for any Hamiltonian.

Time discretization acts as a complex gauge fixing in quantum gravity.

Abstract

We study constrained Hamiltonian systems by utilizing general forms of time discretization. We show that for explicit discretizations, the requirement of preserving the canonical Poisson bracket under discrete evolution imposes strong conditions on both allowable discretizations and Hamiltonians. These conditions permit time discretizations for a limited class of Hamiltonians, which does not include homogeneous cosmological models. We also present two general classes of implicit discretizations which preserve Poisson brackets for any Hamiltonian. Both types of discretizations generically do not preserve first class constraint algebras. Using this observation, we show that time discretization provides a complicated time gauge fixing for quantum gravity models, which may be compared with the alternative procedure of gauge fixing before discretization.