  In the case of gauge theories, which are ruled by an infinite-dimensional
invariance group, various choices of antisymmetric bilinear maps on field
functionals are indeed available. This paper proves first that, within this
broad framework, the Peierls map (not yet the bracket) is a member of a larger
family. At that stage, restriction to gauge-invariant functionals of the
fields, with the associated Ward identities and geometric structure of the
space of histories, make it possible to prove that the new map is indeed a
Poisson bracket in the simple but relevant case of Maxwell theory. The building
blocks are available for gauge theories only: vector fields that leave the
action functional invariant; the invertible gauge-field operator, and the Green
function of the ghost operator.
