Bhabha first-order wave equations. II. Mass and spin composition, Hamiltonians, and general Sakata-Taketani reductions

Abstract

Beginning with the Bhabha first-order wave equation of maximum spin 1 [the Duffin-Kemmer-Petiau (DKP) equation], where Sakata and Taketani (ST) separated out the "particle components" from the built in "subsidiary components," we derive for the first time the Hamiltonian equation for the "subsidiary components," and show that its solution is an identity in terms of the particle-components solution. We then derive a set of general inverse and ST operators for arbitrary-spin Bhabha fields. With these generalized operators we can discuss and understand the mass and spin composition of a general Bhabha so(5) field, it being a particular sum of [$2\times \mathrm{}(2S+1)\mathrm{}$] components for each particular mass and spin ($S$) state, as well as built in "subsidiary components" for integer spin. We then can use these general inverse and ST operators to (a) derive the general Bhabha Hamiltonian for arbitrary spin, (b) decouple the "particle components" from the "subsidiary components" in the Hamiltonian equations for integer spin (where, as was the case for DKP, we find that the Hamiltonian "subsidiary components" solution is an identity in terms of the particle-components solution), and (c) decouple the $\mathcal{S}+\frac{1}{2} \left(\mathcal{S}\right)$ different mass states for half-integer (integer) spin. We discuss the physical implications of this observation and other aspects of our results.