Secondary calculus and the covariant phase space

The covariant phase space of a lagrangian field theory is the solution space of the associated Euler–Lagrange equations. It is, in principle, a nice environment for covariant quantization of a lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) “presymplectic structure” ω (as first noticed by Zuckerman in 1986) whose degeneracy (functional) distribution is naturally interpreted as the Lie algebra of gauge transformations. We propose a fully rigorous approach to the covariant phase space in the framework of jet spaces and (A. M. Vinogradov’s) secondary calculus. In particular, we describe the degeneracy distribution of ω. As a byproduct we rederive the existence of a Lie bracket among gauge invariant functions on the covariant phase space.