arXiv subjects
Publications and source records attributed to Lester C. Welch.
By considering Dirac's equation using quaternions ($\mathbb{H}$) with their greater degree of freedom in imaginaries, it is shown that a model can be created with oscillations among flavors even if the particles, are massless. Furthermore the solutions are spin $\frac{1}{2}$ and have helicities depending on whether their energy is positive or negative.
The relationship between mathematics and physics has long been an area of interest and speculation. Subscribing to the recent definition by Tegmark, we present a mathematical structure involving the only division rings - the real, $\mathbb{R}$, the complex, $\mathbb{C}$, and the quaternion, $\mathbb{H}$ - known to meet the requirements of a mathematical description of our universe. The known forces and particle structure, in terms of families and color, can be associated with natural relationships within that structure. A "toy" scientific theory - based on the proposed mathematical structure - is presented to illustrate its applicability. It is shown how parity and time reversal invariance are broken. Two relevant predictions come from the formalism. First, there are no gravity waves and second, matter comes in three varieties, one of which is "dark" in the sense that it is oblivious to electromagnetic interactions.
Dirac's equations are formulated in a consistent way by using only elements from each of R, C, and H. In H, the quaternions, the symmetry resulting from a "single" conjugation (i, j, or k) results in three conserved currents - possibly associated with "color." The role that the numerical field plays is discussed and speculated on.
By starting with the simpliest expression of the first order linear wave equation (Dirac's equation) and by confining the elements of the coefficients (matrices) to the quaternions, it is shown that the association of the imaginary bases of the quaternions with three "colors" and their conserved currents results in quasi-particles with charges of 2/3 and -1/3 with a minimum of assumptions. It is shown how color neutral particles are required by Q covariance, first discussed by Finkelstein, et. al. in 1963.