Quaternion/vector dual space algebras applied to the Dirac equation and its extensions

Abstract

The paper re-applies the 64-part algebra discussed by P. Rowlands in a series of (FERT and other) papers in recent years. It demonstrates that the original introduction of the α algebra by Dirac to "the quantum theory of the electron" can be interpreted with the help of quaternions: both the α matrices and the Pauli (σ) matrices in Dirac's original interpretation can be rewritten in quaternion forms. This allows the construction of the Dirac γ matrices in two (quaternion-vector) matrix product representations - in accordance with the double vector algebra introduced by P. Rowlands. The paper attempts to demonstrate that the Dirac equation in its form modified by P. Rowlands, essentially coincides with the original one. The paper shows that one of these representations a_ects the γ₄ and γ₅ matrices, but leaves the vector of the Pauli spinors intact; and the other representation leaves the γ matrices intact, while it transforms the spin vector into a quaternion pseudovector. So the paper concludes that the introduction of quaternion formulation in QED does not provide us with additional physical information. These transformations affect all gauge extensions of the Dirac equation, presented by the author earlier, in a similar way. This is demonstrated by the introduction of an additional parameter (named earlier isotopic field-charge spin) and the application of a so-called tau algebra that governs its invariance transformation. The invariance of the extended Dirac equation is subject to the convolution of the Lorentz transformation and this invariance group. It is shown that additional physical information can be expected by extending these investigations to the Finsler-like metric proposed by Dirac in 1962 for the theory of the electron.