Classical field theories from Hamiltonian constraint: Local symmetries and static gauge fields

TL;DR

This paper develops a Hamiltonian constraint approach to classical field theories using geometric algebra, introducing gauge fields to maintain local symmetries and exploring their role in gravitational and Yang-Mills interactions.

Contribution

It presents a novel Hamiltonian framework for classical fields with local symmetries, employing geometric algebra and defining a generic gauge field form for gravitational and Yang-Mills interactions.

Findings

Gauge fields can be specialized to realize gravitational interactions.

Gauge fields can be adapted for Yang-Mills theories.

The approach maintains local invariance in the Hamiltonian formalism.

Abstract

We consider the Hamiltonian constraint formulation of classical field theories, which treats spacetime and the space of fields symmetrically, and utilizes the concept of momentum multivector. The gauge field is introduced to compensate for non-invariance of the Hamiltonian under local transformations. It is a position-dependent linear mapping, which couples to the Hamiltonian by acting on the momentum multivector. We investigate symmetries of the ensuing gauged Hamiltonian, and propose a generic form of the gauge field strength. In examples we show how a generic gauge field can be specialized in order to realize gravitational and/or Yang-Mills interaction. Gauge field dynamics is not discussed in this article. Throughout, we employ the mathematical language of geometric algebra and calculus.