Moyal algebra: relevant properties, projective limits and applications in noncommutative field theory

From the definition of the Moyal ∗-product in terms of projective limits of the ring of polynomials of vector fields, the Moyal configuration space of Schwartzian functions, equipped with the ∗-product, is built as a formal power series ring with elements assimilated to free indeterminates. We then define the projector on the ideal depending on a fixed indeterminate, which allows to use the definition of algebraic derivations with respect to any order of field derivative. As a consequence and in a direct manner, Euler-Lagrange equations of motion, in the framework of both the noncommutative scalar and gauge induced Dirac fields, are deduced from the nonlocal Lagrange function. A connection of this theory to a generalized Ostrogradski's formalism is also discussed here.