# Optimal space of linear classical observables for Maxwell $k$-forms via spacelike and timelike compact de Rham cohomologies

January 29, 2014

Being motivated by open questions in gauge field theories, we consider
non-standard de Rham cohomology groups for timelike compact and spacelike
compact support systems. These cohomology groups are shown to be isomorphic
respectively to the usual de Rham cohomology of a spacelike Cauchy surface and
its counterpart with compact support. Furthermore, an analog of the usual
PoincarĂ© duality for de Rham cohomology is shown to hold for the case with
non-standard supports as well. We apply these results to find optimal spaces of
linear observables for analogs of arbitrary degree $k$ of both the vector
potential and the Faraday tensor. The term optimal has to be intended in the
following sense: The spaces of linear observables we consider distinguish
between different configurations; in addition to that, there are no redundant
observables. This last point in particular heavily relies on the analog of
PoincarĂ© duality for the new cohomology groups.

Keywords:

Classical field theory on curved spacetimes, Maxwell field, De Rham cohomology