Equivalence of Lagrangian and Multisymplectic Hamiltonian Formalisms for the Description of Classical Fields Interacting with Point Particles

TL;DR

This paper proves the equivalence between Lagrangian and multisymplectic Hamiltonian formalisms for classical fields interacting with point particles, enabling a covariant canonical approach to field theory.

Contribution

It demonstrates the equivalence in the momentum representation and proposes a Lorentz-covariant Poisson bracket for classical fields.

Findings

Explicit derivation for scalar, electromagnetic, and Dirac fields.

Proposal of a Lorentz-covariant bilinear Poisson bracket.

Establishes a foundation for covariant canonical quantization.

Abstract

The multisymplectic Hamiltonian formalism is a generalization of the Hamiltonian formalism that is known to manifestly preserve Lorentz covariance in the description of fields and has been suggested as a possible path for developing a covariant canonical quantization process. However, since this formalism allows for multiple definitions of Poisson brackets, its practical use in field theory has been very limited. In this paper, we demonstrate the equivalence between the Lagrangian and the multisymplectic Hamiltonian formalisms in the momentum representation for describing classical fields interacting with point particles by deriving the image in that representation of the de Donder-Weyl function for a general tensor representation of the Lorentz group. The calculation is performed explicitly for the complex scalar field, the electromagnetic field, and the classical Dirac field. The obtained results allow us to propose a Lorentz-covariant bilinear form as Poisson bracket with respect to the canonical variables of the fields, thus opening the possibility of studying the complete system in a canonical and relativistic coherent way.