Reduced Hamiltonians

TL;DR

This paper revisits a classical method to derive canonical Hamiltonian systems from known dynamical equations, enabling quantization of constrained systems like gauge theories and gravity, including cases with closed spatial manifolds.

Contribution

It demonstrates how to construct a canonical Hamiltonian formalism for systems with degenerate brackets, extending the method to gravity on closed manifolds and providing explicit perturbative examples.

Findings

Constructed a positive semidefinite Hamiltonian for perturbations around flat space.

The Hamiltonian matches ADM energy in the appropriate limit.

Method applies to gauge theories and gravity with closed spatial manifolds.

Abstract

We resurrect a standard construction of analytical mechanics dating from the last century. The technique allows one to pass from any dynamical system whose first order evolution equations are known, and whose bracket algebra is not degenerate, to a system of canonical variables and a non-zero Hamiltonian that generates their evolution. We advocate using this method to infer a canonical formalism, as a prelude to quantization, for systems in which the naive Hamiltonian is constrained to vanish. The construction agrees with the usual results for gauge theories and can be applied as well to gravity, {\it even when the spatial manifold is closed.} As an example, we construct such a reduced Hamiltonian in perturbation theory around a flat background on the manifold $T^3 \times R$. The resulting Hamiltonian is positive semidefinite and agrees with the A.D.M. energy in the limit that deviations from flat space remain localized as the toroidal radii become infinite.