On the quantization of $C^{\infty}(\mathbb R^d)$

TL;DR

This paper proves the Basic Universal Deformation Formula and explores how Weyl algebras serve as effective deformations of polynomial rings, revealing the role of uncertainty in stability and the nature of Poisson structures on smooth functions.

Contribution

It establishes the Basic Universal Deformation Formula, demonstrates the effectiveness of Weyl algebras as deformations, and links Poisson structures to infinitesimal deformations with vanishing primary obstructions.

Findings

Weyl algebras encode Heisenberg's uncertainty principle.

Uncertainty is necessary for the stability of deformations.

Certain Poisson structures can be integrated to full deformations.

Abstract

The Basic Universal Deformation Formula is proven and applied to show that Weyl algebras, which encode Heisenberg's uncertainty principle, are effective deformations of polynomial rings, and that uncertainty is necessary for stability. Deformation problems may have associated modular groups an algebraic example of which is given. Poisson structures on $C^{\infty}(\mathbf R^d)$, shown by Kontsevich to be infinitesimal deformations integrable to full deformations, here are shown to be the skew forms of those infinitesimal deformations of $C^{\infty}(\mathbf R^d)$ with vanishing primary obstructions. In dimension 3 any smooth multiple of such an infinitesimal again has vanishing primary obstruction. This exceptional property suggests that in our universe a large disturbance like the Big Bang can be confined to an arbitrarily small interval in time and almost completely confined to an arbitrarily small region in space.