Title: Poisson Bracket and Symplectic Structure of Covariant Canonical Formalism of Fields,
Abstract: The covariant canonical formalism is a covariant extension of the traditional
canonical formalism of fields. In contrast to the traditional canonical theory,
it has a remarkable feature that canonical equations of gauge theories or
gravity are not only manifestly Lorentz covariant but also gauge covariant or
diffeomorphism covariant. A mathematical peculiarity of the covariant canonical
formalism is that its canonical coordinates are differential forms on a
manifold. In the present paper, we find a natural Poisson bracket of this new
canonical theory, and study symplectic structure behind it. The phase space of
the theory is identified with a ringed space with the structure sheaf of the
graded algebra of "differentiable" differential forms on the manifold. The
Poisson and the symplectic structure we found can be even or odd, depending on
the dimension of the manifold. Our Poisson structure is an example of physical
application of Poisson structure defined on the graded algebra of differential
forms.