Momentum Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories

TL;DR

This paper develops the canonical formulation of classical field theories from a covariant perspective, exploring the relation between multisymplectic geometry, constraints, gauge freedom, and energy-momentum maps.

Contribution

It bridges covariant and canonical formulations of field theories, elucidating the role of constraints, gauge transformations, and momentum maps in the Hamiltonian framework.

Findings

Covariant multisymplectic geometry induces instantaneous symplectic structures.

Constraints and gauge freedoms are characterized in Hamiltonian field dynamics.

Energy-momentum maps relate covariant momentum maps to Hamiltonian energy functions.

Abstract

With the covariant formulation in hand from the first paper of this series (physics/9801019), we begin in this second paper to study the canonical (or ``instantaneous'') formulation of classical field theories. The canonical formluation works with fields defined as time-evolving cross sections of bundles over a Cauchy surface, rather than as sections of bundles over spacetime as in the covariant formulation. In Chapter 5 we begin to relate these approaches to classical field theory; in particular, we show how covariant multisymplectic geometry induces the instantaneous symplectic geometry of cotangent bundles of sections of fields over a Cauchy surface. In Chapter 6, we proceed to consider field dynamics. A crucial feature of our discussion here is the degeneracy of the Lagrangian functionals for the field theories of interest. As a consequence of this degeneracy, we have constraints on the choice of initial data, and gauge freedom in the evolution of the fields. Chapter 6 considers the role of initial value constraints and gauge transformations in Hamiltonian field dynamics. In Chapter 7, we then describe how covariant momentum maps defined on the multiphase space induce "energy-momentum maps'' on the instantaneous phase spaces. We show that for a group action which leaves the Cauchy surface invariant, this energy-momentum map coincides with the usual notion of a momentum map. We also show, when the gauge group"includes'' the spacetime diffeomorphism group, that one of the components of the energy-momentum map corresponding to spacetime diffeomorphisms can be identified (up to sign) with the Hamiltonian for the theory.