  It has been a long standing question how to extend the canonical Poisson
bracket formulation from classical mechanics to classical field theories, in a
completely general, intrinsic, and canonical way. In this paper, we provide an
answer to this question by presenting a new completely canonical bracket
formulation of Hamiltonian Classical Field Theories of first order on an
arbitrary configuration bundle. It is obtained via the construction of the
appropriate field-theoretic analogues of the Hamiltonian vector field and of
the space of observables, via the introduction of a suitable canonical Lie
algebra structure on the space of currents (the observables in field theories).
This Lie algebra structure is shown to have a representation on the affine
space of Hamiltonian sections, which yields an affine analogue to the Jacobi
identity for our bracket. The construction is analogous to the canonical
Poisson formulation of Hamiltonian systems although the nature of our
formulation is linear-affine and not bilinear as the standard Poisson bracket.
This is consistent with the fact that the space of currents and Hamiltonian
sections are respectively, linear and affine. Our setting is illustrated with
some examples including Continuum Mechanics and Yang-Mills theory.
