Noncommutative Geometry and Symplectic Field Theory

TL;DR

This paper develops a formalism connecting noncommutative geometry and symplectic field theory, deriving fundamental equations and symmetries in phase space with applications to relativistic quantum fields.

Contribution

It introduces a novel approach to representing the Poincare group over symplectic manifolds and derives key relativistic equations within this framework.

Findings

Derived Klein-Gordon and Dirac equations in phase space.

Established a phase space version of Noether's theorem.

Discussed an interacting gauge field approach in this formalism.

Abstract

In this work we study representations of the Poincare group defined over symplectic manifolds, deriving the Klein-Gordon and the Dirac equation in phase space. The formalism is associated with relativistic Wigner functions; the Noether theorem is derived in phase space and an interacting field, including a gauge field, approach is discussed.