The Metaplectic Group, the Symplectic Spinor and the Gouy Phase

TL;DR

This paper introduces a simplified algebraic framework for symplectic spinors and the metaplectic group, providing new insights into the Gouy phase and its relation to the Berry phase in optical systems.

Contribution

It extends the Heisenberg algebra with a new idempotent element, simplifying the representation of symplectic spinors and offering a novel perspective on the Gouy phase.

Findings

New algebraic approach to symplectic spinors

Connection between Gouy phase and symplectic covering group

Relation of Gouy phase to Berry phase

Abstract

In this paper we discuss a simplified approach to the symplectic Clifford algebra, the symplectic Clifford group and the symplectic spinor by first extending the Heisenberg algebra. We do this by adding a new idempotent element to the algebra. It turns out that this element is the projection operator onto the Dirac standard ket. When this algebra is transformed into the Fock algebra, the corresponding idempotent is the projector onto the vacuum. These additional elements give a very simple way to write down expressions for symplectic spinors as elements of the symplectic Clifford algebra. When this algebra structure is applied to an optical system we find a new way to understand the Gouy effect. It is seen to arise from the covering group, which in this case is the symplectic Clifford group, also known as the metaplectic group. We relate our approach to that of Simon and Mukunda who have shown the relation of the Gouy phase to the Berry phase.