Deformation theory and quantization. I. Deformations of symplectic structures

Abstract

We present a mathematical study of the differentiable deformations of the algebras associated with phase space. Deformations of the Lie algebra of C ∞ functions, defined by the Poisson bracket, generalize the well-known Moyal bracket. Deformations of the algebra of C ∞ functions, defined by ordinary multiplication, give rise to noncommutative, associative algebras, isomorphic to the operator algebras of quantum theory. In particular, we study deformations invariant under any Lie algebra of “distinguished observables”, thus generalizing the usual quantization scheme based on the Heisenberg algebra.