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Mathematics and Mechanics of Complex Systems

Symplectic projection in generalized Dirac dynamics

Tatiana Vladimirovna Salnikova, Е. И. Кугушев

Published Nov 29, 2025• 1 citation
Publisher
DOI
10.2140/memocs.2026.14.59

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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.
A covariant, symplectic, classical dynamical system is presented whose quantization, by replacement of the Poisson brackets with commutators, gives precisely the Dirac electron theory. For the classical system the velocity and momentum are independent dynamical variables, as in Dirac electron; it undergoes a real Zitterbewegung so that the spin is the orbital angular momentum of the Zitterbewegung. Thus the longstanding problem of representing the configuration space of quantum spin by a classical model is solved. Three different forms of the dynamical equations are given.