Covariant Hamiltonian field theory. Path integral quantization

TL;DR

This paper proposes a path integral quantization method for covariant Hamiltonian field theory, demonstrating equivalence with Lagrangian quantization in the case of almost-regular quadratic Lagrangians.

Contribution

It introduces a novel path integral quantization approach for covariant Hamiltonian field theory, linking it to Lagrangian systems on phase space.

Findings

Path integral quantization of covariant Hamiltonian field theory is feasible.

Quantizations of Hamiltonian and Lagrangian theories are equivalent for almost-regular quadratic Lagrangians.

The approach extends the quantization framework to covariant Hamiltonian systems.

Abstract

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field theories are equivalent.