Theory of Neutral Particles: Mclennan-Case Construct for Neutrino, Its Generalization, and a New Wave Equation

Continuing our recent argument where we constructed a FNBWW-type spin-$1$ boson having opposite relative intrinsic parity to that of the associated antiparticle, we now study eigenstates of the Charge Conjugation operator. Based on the observation that if $\phi_{_{L}}(p^\mu)$ transforms as a $(0,\,j)$ spinor under Lorentz boosts, then $\Theta_{[j]}\,\phi_{_{L}}^\ast(p^\mu)$ transforms as a $(j,\,0)$ spinor (with a similar relationship existing between $\phi_{_{R}}(p^\mu)$ and $\Theta_{[j]}\,\phi_{_{R}}^\ast(p^\mu)$; where $ \Theta_{[j]}\,{\bf J}\,\Theta_{[j]}^{-1}\,=\,-\,{\bf J}^\ast $ with $\Theta_{[j]}$ the well known Wigner matrix involved in the operation of time reversal) we introduce McLennan-Case type $(j,\,0)\oplus(0,\,j)$ spinors. Relative phases between $\phi_{_{R}}(p^\mu)$ and $\Theta_{[j]}\,\phi_{_{R}}^\ast(p^\mu)$, and $\Theta_{[j]}\,\phi_{_{L}}^\ast(p^\mu)$ and $\phi_{_{L}}(p^\mu)$, turn out to have physical significance and are fixed by appropriate requirements. Explicit construction, and a series of physically relevant properties, for these spinors are obtained for spin-$1/2$ and spin-$1$ culminating in the construction of a new wave equation and introduction of Dirac-like and Majorana-like quantum fields.