Lagrangian and Multisymplectic Descriptions of Classical Fields: A Connection in the Momentum Representation

The multisymplectic Hamiltonian formalism is a generalization of the Hamiltonian formalism that manifestly preserves covariance in the description of fields and that has been proposed as a possible framework for developing a Lorentz-covariant, canonical quantization scheme. However, the possibility of defining multiple Poisson brackets within this formalism has significantly limited its practical use in field theory. In this paper, we establish a connection between the Lagrangian and multisymplectic descriptions of classical fields interacting with point particles in the field's momentum representation by deriving the image of the de Donder-Weyl function for a general tensor representation of the Lorentz group. The calculation is carried out explicitly for the complex scalar field, the electromagnetic field, and the classical Dirac field. On this basis, we propose a Lorentz-covariant Poisson bracket with respect to the canonical variables of the fields, thereby opening the possibility of studying the complete system within a consistent relativistic and canonical framework.