Covariant canonical formulations of classical field theories

TL;DR

This paper reviews covariant canonical formulations of classical relativistic field theories, comparing approaches, discussing conservation laws, boundary effects, and providing geometric background to make the concepts accessible.

Contribution

It provides a clear comparison of covariant and non-covariant Hamiltonian formulations, highlighting their relationships and implications for gauge theories and general relativity.

Findings

Clarifies relationships between different covariant approaches

Analyzes the role of boundaries in covariant formulations

Emphasizes conservation laws linked to geometric symmetries

Abstract

We review in simple terms the covariant approaches to the canonical formulation of classical relativistic field theories (in particular gauge field theories and general relativity) and we discuss the relationships between these approaches as well as the relation with the standard (non-covariant) Hamiltonian formulation. Particular attention is paid to conservation laws (notably related to geometric symmetries) within the different approaches. Moreover, for each of these approaches, the impact of space-time boundaries is also addressed. To make the text accessible to a wider audience, we have included an outline of Poisson and symplectic geometry for both classical mechanics and field theory.