Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton-Jacobi theory

TL;DR

This paper develops a geometric algebra-based framework for classical field theories using Hamiltonian constraints, deriving local equations of motion and Hamilton-Jacobi equations, with applications to mechanics, scalar fields, and string theory.

Contribution

It introduces a coordinate-free, geometric algebra approach to formulating classical field theories via Hamiltonian constraints, deriving local equations of motion and Hamilton-Jacobi equations.

Findings

Derived local equations of motion for classical surfaces and momenta.

Formulated a local Hamilton-Jacobi equation for field theories.

Illustrated the method with examples from mechanics, scalar fields, and string theory.

Abstract

Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined by a (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton-Jacobi equation applicable in the field theory then follows readily. The general method is illustrated with three examples: non-relativistic Hamiltonian mechanics, De Donder-Weyl scalar field theory, and string theory. Throughout, we use the mathematical formalism of geometric algebra and geometric calculus, which allows to perform completely coordinate-free manipulations.