Clifford Algebras in Symplectic Geometry and Quantum Mechanics

TL;DR

This paper explores the role of Clifford algebras, especially quaternion Clifford algebra C(0,2), in symplectic geometry and quantum mechanics, revealing their connection to symplectic structures, Heisenberg algebras, and quantum dynamics.

Contribution

It introduces a Poisson Clifford algebra framework for finite-dimensional phase spaces, linking classical symplectic structures with quantum operator realizations.

Findings

Clifford algebras encode rotational and symplectic structures.

A Poisson Clifford algebra H(F) models phase space dynamics.

Quantum dynamics are realized through Hermitian operators within this algebra.

Abstract

The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C(0,2). This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional sub-space, Fa of the Euclidean three-space. This enables us to construct a Poisson Clifford algebra, H(F), of a finite dimensional phase space which will carry the dynamics. The quantum dynamics appears as a realization of H(F) in terms of a Clifford algebra consisting of Hermitian operators.