Noncommutative geometry and its relation to stochastic calculus and symplectic mechanics

Abstract

In the context of a noncommutative differential calculus on the algebra of real valued functions of an $n$-dimensional manifold $M$, a commutative and associative product of 1-forms is naturally defined. Ordinary differential calculus corresponds to this product being trivially zero. We consider the minimal generalization in which the algebra of 1-forms with this product is nilpotent of degree 3. Basic tensor analysis and differential geometry are developped in this context and applied to the formulation of symplectic geometry and Hamiltonian dynamics. It is shown that the corresponding Liouville equation can take the form of a generalized Fokker-Planck equation, well known in statistical mechanics of open systems, as well as in stochastic calculus. Specifically it may be the same with that obtained via Ito stochastic calculus, when solving a linear stochastic differential equation with a Wiener process as the stochastic term. The close connection between noncommutative differential structures of this kind, with semimartingale stochastic processes on manifolds thus suggested, is further explored.