Bhabha first-order wave equations: I.C,P, andT

Abstract

We discuss properties of Bhabha first-order wave equations for arbitrary spin, of which the Dirac and Duffin-Kemmer-Petiau (DKP) equations are special examples. The $C$, $P$, and $T$ transformation matrices for the Dirac field are reviewed in various representations, and the $C$, $P$, and $T$ transformation matrices for the DKP and general Bhabha cases are then derived. The Bhabha transformation matrices are polynomials of order $2\mathcal{S}$ in the algebra matrices, where $\mathcal{S}$ is the maximum spin of a particular Bhabha algebra. For the cases $\mathcal{S}=1\mathrm{} \mathrm{and} \frac{1}{2}$ they reduce to the DKP and Dirac transformation matrices. We also discuss $C$, $P$, and $T$ for the Sakata-Taketani (ST) reduction of the DKP equation, and explicitly exhibit the "subsidiary component" ST Hamiltonian equation, as well as the known "particle component" ST equation. Throughout we emphasize that physical insight which can be gained from the use of the first-order Bhabha formalism, including a possible connection between meson nonconservation and $\mathrm{CP}$ violation.