Classical field theories from Hamiltonian constraint: Symmetries and conservation laws

TL;DR

This paper develops a Hamiltonian framework for classical field theories that emphasizes symmetries and conservation laws, utilizing geometric algebra to derive a generalized Noether theorem and relate it to traditional formulations.

Contribution

It introduces a Hamiltonian constraint approach to classical field theories, deriving a generalized Noether theorem using geometric algebra and linking new currents to traditional ones.

Findings

Derived a Hamiltonian version of Noether's theorem for field theories

Identified generalized Noether currents with momentum contracted with symmetry vectors

Established the relation between generalized and traditional Noether currents

Abstract

We discuss the relation between symmetries and conservation laws in the realm of classical field theories based on the Hamiltonian constraint. In this approach, spacetime positions and field values are treated on equal footing, and a generalized multivector-valued momentum is introduced. We derive a field-theoretic Hamiltonian version of the Noether theorem, and identify generalized Noether currents with the momentum contracted with symmetry-generating vector fields. Their relation to the traditional vectorial Noether currents is then established. Throughout, we employ the mathematical language of geometric algebra and calculus.