  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.
