Invariant bilinear forms and the discrete symmetries for relativistic arbitrary-spin fields

Abstract

The existence of a Hermitianizing matrix $\ensuremath{\eta}$ is usually assumed in the study of first-order relativistic wave equations because it provides for an invariant scalar product, bilinear densities (e.g., Lagrangian), and parity realization in a canonical way. However, an $\ensuremath{\eta}$ will exist only if the representation of $\mathrm{SL}(2,C)$ which governs the transformation of the wave function is self-conjugate. The drawbacks of this fact for theories with $s>1$ are discussed and a class of relativistic wave equations which avoids these drawbacks and which does not allow for the existence of an $\ensuremath{\eta}$ matrix is set aside for study. It is shown that a dual space may be defined (or, equivalently, a metric operator may be introduced) such that all of the above $\ensuremath{\eta}$-matrix benefits may be maintained without an $\ensuremath{\eta}$ matrix. The discrete symmetries are defined for these equations and it is shown that the realization of parity in terms of an antilinear operator naturally emerges. The locality, positive-definite metric, and positive-definite energy of the second-quantized version of the formulation are described. These considerations apply to a class of wave equations which provide a simple and uniform description of a massive, spin-$s$ relativistic particle and which remain consistent and causal in the presence of a minimally coupled external electromagnetic field.