Spin 1 Particle in a 15-Component Formalism, Interaction with Electromagnetic and Gravitational Fields

TL;DR

This paper develops a 15-component formalism for spin 1 particles, exploring their electromagnetic and gravitational interactions, and introduces a new electromagnetic characteristic called polarizability, with implications for massless particles and curved spacetime.

Contribution

It introduces a novel 15-component wave function formalism for vector particles, including a new electromagnetic parameter and covariant equations in curved spacetime.

Findings

A basis where only one parameter $\sigma$ remains.

The $C$-operation reduces to complex conjugation.

Massless case and curved spacetime extensions are formulated.

Abstract

A generalized vector particle theory with the use of an extended set of Lorentz group irredicible representations, including scalar, two 4-vectors, and antisymmetric 2-rang tensor, is investigated. Initial equations depend upon four complex parameters $\lambda_{i}$, obeying two supplementary conditions, so restriction of the model to the case of electrically neutral vector particle is not a trivial task. A special basis in the space of 15-component wave functions is found where instead of four $\lambda_{i}$ only one real-valued quantity $\sigma$, a bilinear combination of $\lambda_{i}$, is presented. This $\lambda$-parameter is interpreted as an additional electromagnetic characteristic of a charged vector particle, polarizability. It is shown that in this basis $C$-operation is reduced to the complex conjugation only, without any accompanying linear transformation. Restriction to a massless vector particle is determined. Extension of the whole theory to the case of Riemannian space-time is accomplished. Two methods of obtaining corresponding generally covariant wave equations are elaborated: of tensor- and of tetrad-based ones. Their equivalence is proved. It is shown that in case of pure curved space-time models without Cartan torsion no specific additional interaction terms because of non-flat geometry arise. The conformal symmetry of a massless generally covariant equation is demonstrated explicitly. A canonical tensor of energy-momentum $T_{\beta \alpha}$ is constructed, its conservation law happens to involves the Riemann curvature tensor. Within the framework of known ambiguity of any energy-momentum tensor, a new tensor $\bar{T}_{\beta \alpha}$ is suggested to be used, which obeys a common conservation law.