Hamiltonian constraint formulation of classical field theories

TL;DR

This paper develops a Hamiltonian constraint framework for classical field theories using geometric algebra, deriving local equations of motion, a Hamilton-Jacobi equation, and a Hamiltonian Noether theorem, with applications to scalar and string theories.

Contribution

It introduces a novel Hamiltonian constraint formulation for classical fields employing geometric algebra, unifying spacetime and field variables symmetrically.

Findings

Derived local equations of motion for classical surfaces and momenta.

Established a Hamilton-Jacobi equation for field theories.

Formulated a Hamiltonian version of Noether's theorem linking symmetries and conservation laws.

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 via the (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive the 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. In addition, we discuss the relation between symmetries and conservation laws, and derive a Hamiltonian version of the Noether theorem, where the Noether currents are identified as the classical momentum contracted with the symmetry-generating vector fields. The general formalism is illustrated by two examples: the scalar field theory, and the string theory. Throughout the article, we employ the mathematical formalism of geometric algebra and calculus, which allows us to perform completely coordinate-free manipulations.