Covariant reduction of classical Hamiltonian Field Theories: From D'Alembert to Klein-Gordon and Schr\"odinger

TL;DR

This paper introduces a covariant reduction method for classical field theories, enabling the derivation of lower-dimensional theories like Klein-Gordon and Schrödinger from higher-dimensional frameworks using multisymplectic geometry.

Contribution

It presents a new covariant reduction procedure for classical field theories based on the geometry of solution manifolds, extending symplectic reduction to a covariant setting.

Findings

Reduced D'Alembert theory to Klein-Gordon in 4D

Derived Schrödinger equation from higher-dimensional theory

Utilized multisymplectic geometry for covariant reduction

Abstract

A novel reduction procedure for covariant classical field theories, reflecting the generalized symplectic reduction theory of Hamiltonian systems, is presented. The departure point of this reduction procedure consists in the choice of a submanifold of the manifold of solutions of the equations describing a field theory. Then, the covariance of the geometrical objects involved, will allow to define equations of motion on a reduced space. The computation of the canonical geometrical structure is performed neatly by using the geometrical framework provided by the multisymplectic description of covariant field theories. The procedure is illustrated by reducing the D'Alembert theory on a five-dimensional Minkowski space-time to a massive Klein-Gordon theory in four dimensions and, more interestingly, to the Schr\"odinger equation in 3+1 dimensions.