A Chiral Spin Theory in the Framework of an Invariant Evolution Parameter Formalism

TL;DR

This paper develops a covariant, invariant evolution parameter formalism for describing particles with spin, resulting in a chiral first order equation that unifies transverse and longitudinal components, and incorporates electromagnetic interactions with correct gyromagnetic ratio.

Contribution

It introduces a novel covariant framework with a chiral first order equation for spinning particles, extending the Dirac formalism within an invariant evolution parameter approach.

Findings

Derives a consistent quantized formalism for off-shell particles with spin.

Obtains a chiral form of the first order equation compatible with second order Schrödinger-Stueckelberg type.

Incorporates electromagnetic interactions yielding the correct gyromagnetic ratio.

Abstract

We present a formulation for the construction of first order equations which describe particles with spin, in the context of a manifestly covariant relativistic theory governed by an invariant evolution parameter; one obtains a consistent quantized formalism dealing with off-shell particles with spin. Our basic requirement is that the second order equation in the theory is of the Schr\"{o}dinger-Stueckelberg type, which exhibits features of both the Klein-Gordon and Schr\"{o}dinger equations. This requirement restricts the structure of the first order equation, in particular, to a chiral form. One thus obtains, in a natural way, a theory of chiral form for massive particles, which may contain both left and right chiralities, or just one of them. We observe that by iterating the first order system, we are able to obtain second order forms containing the transverse and longitudinal momentum relative to a time-like vector $t_{\mu}t^{\mu}=-1$ used to maintain covariance of the theory. This time-like vector coincides with the one used by Horwitz, Piron, and Reuse to obtain an invariant positive definite space-time scalar product, which permits the construction of an induced representation for states of a particle with spin. We discuss the currents and continuity equations, and show that these equations of motion and their currents are closely related to the spin and convection parts of the Gordon decomposition of the Dirac current. The transverse and longitudinal aspects of the particle are complementary, and can be treated in a unified manner using a tensor product Hilbert space. Introducing the electromagnetic field we find an equation which gives rise to the correct gyromagnetic ratio, and is fully Hermitian under the proposed scalar product. Finally, we show that the original structure of Dirac's