Formulation of singular theories in a partial Hamiltonian formalism using a new bracket and multi-time dynamics

TL;DR

This paper introduces a novel partial Hamiltonian formalism and a new antisymmetric bracket to analyze singular classical theories without constraints, providing a unified approach to multi-time dynamics and gauge classification.

Contribution

It proposes a partial Hamiltonian framework with an arbitrary number of momenta and introduces a new bracket, advancing the understanding of singular theories and their quantization.

Findings

Equations of motion are first-order and align with multi-time dynamics under specific conditions.

A classification scheme for singular theories into gauge and nongauge types is developed.

The new bracket generalizes the Poisson and Dirac brackets for singular theories.

Abstract

A formulation of singular classical theories (determined by degenerate Lagrangians) without constraints is presented. A partial Hamiltonian formalism in the phase space having an initially arbitrary number of momenta (which can be smaller than the number of velocities) is proposed. The equations of motion become first-order differential equations, and they coincide with those of multi-time dynamics, if a certain condition is imposed. In a singular theory, this condition is fulfilled in the case of the coincidence of the number of generalized momenta with the rank of the Hessian matrix. The noncanonical generalized velocities satisfy a system of linear algebraic equations, which allows an appropriate classification of singular theories (gauge and nongauge). A new antisymmetric bracket (similar to the Poisson bracket) is introduced, which describes the time evolution of physical quantities in a singular theory. The origin of constraints is shown to be a consequence of the (unneeded in our formulation) extension of the phase space. In this case the new bracket transforms into the Dirac bracket. Quantization is briefly discussed.