Extended Phase Space. III. Unified spin-1/2 fields

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Abstract
In extended phase space V8, the classical field equation for spin‐1/2 elementary particles is written as [vk∂/∂qk+ak∂/∂pk−imI] ×ψ(q,p)=0. The 16×16 matrices vk,ak stand for the instantaneous 4‐velocity and 4‐acceleration. The equation is called the Boltzmann–Dirac–Yukawa (in short BDY) equation. This equation treats q,p variables on equal footings and is covariant under the extended Poincaré group P8. The spin‐1/2 field ψ is decomposed into plane wave solutions, and integral constants such as the total energy, the total momentum, the average position, etc., are computed. These integral constants become meaningful provided the modulus squared of each amplitude is interpreted as the statistical distribution function. Next the ψ field is subject to the Hankel transform ψ( ρ,ϑ)∼∑∞t=−∞∑8 R=1∑2α=1∫∞0dκ κ [β(Rα)u(Rα) +ggr̄(Rα)v(Rα)]Jt(κρ)eitϑ=∑∞t=−∞ψ(t); ρ=(q2+p2)1/2, θ=arctan (p/q). The integral constants are constructed from a single (t)‐mode ψ(t). These turn out to be physically meaningful for eight spin‐1/2 particles. Specially the total charge can be identified with Gell‐Man–Nishijima’s expression for the baryon provided the quantum number t3 corresponds to the isotopic spin, 2t4 is identified with the strangeness, and the baryon number b is allowed to take 0,1,2. With each of the (t)‐mode ψ(t), eight baryon fields can be associated so that ψ stands for the unified baryon fields. Finally some brief comments are made on the possibility of treating lepton fields within the framework of the BDY equation.

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