Geometry, stochastic calculus and quantum fields in a non-commutative space-time

TL;DR

This paper explores how stabilizing the algebra of relativistic quantum mechanics through non-commutative space-time leads to new geometric and quantum stochastic calculus insights, impacting quantum field theory.

Contribution

It introduces a stabilized algebra for relativistic quantum mechanics with non-commutative space-time, revealing new geometric and quantum stochastic calculus structures.

Findings

Non-commutative space-time stabilizes the quantum mechanics algebra.

New geometric structures emerge from the stabilized algebra.

Quantum stochastic calculus is extended to the non-commutative setting.

Abstract

The algebras of non-relativistic and of classical mechanics are unstable algebraic structures. Their deformation towards stable structures leads, respectively, to relativity and to quantum mechanics. Likewise, the combined relativistic quantum mechanics algebra is also unstable. Its stabilization requires the non-commutativity of the space-time coordinates and the existence of a fundamental length constant. The new relativistic quantum mechanics algebra has important consequences on the geometry of space-time, on quantum stochastic calculus and on the construction of quantum fields. Some of these effects are studied in this paper.