  Starting with the well-defined product of quantum fields at two spacetime
points, we explore an associated Poisson structure for classical field theories
within the deformation quantization formalism. We realize that the induced
star-product is naturally related to the standard Moyal product through an
appropriate causal Green's functions connecting points in the space of
classical solutions to the equations of motion. Our results resemble the
Peierls-DeWitt bracket analyzed in the multisymplectic context. Once our
star-product is defined we are able to apply the Wigner-Weyl map in order to
introduce a generalized version of Wick's theorem. Finally, we include some
examples to explicitly test our method: the real scalar field, the bosonic
string and a physically motivated nonlinear particle model. For the field
theoretic models we have encountered causal generalizations of the
creation/annihilation relations, and also a causal generalization of the
Virasoro algebra for the bosonic string. For the nonlinear particle case, we
use the approximate solution in terms of the Green's function in order to
construct a well-behaved causal bracket.
