  Axioms for nonunital spectral triples, extending those introduced in the
unital case by Connes, are proposed. As a guide, and for the sake of their
importance in noncommutative quantum field theory, the spaces $\R^{2N}$ endowed
with Moyal products are intensively investigated. Some physical applications,
such as the construction of noncommutative Wick monomials and the computation
of the Connes--Lott functional action, are given for these noncommutative
hyperplanes.
