Note on a New Quaternionic Approach to Relativity and the Dirac Theory

The purpose of this note is to provide a convincing argument that the quaternionic spinor analysis1) leads to the existence of an extra imaginary unit commuting with Hamilton’s imaginary units of quaternions so that the conventional quaternionic approach2) to relativity, which makes use of complex quaternions, is derived from our new quaternionic approach1) defined over the field of quaternions, H. The proof is based on the fact that the Dirac representation is the only fundamental representation of the quaternionic spinor group Spin(2,H) ⊂ SL(2,H) but is defined as a direct sum of two fundamental representations (two inequivalent Weyl representations) of SL(2,C). It is explicitly shown that 2-component quaternionic Dirac spinor when transformed into 2-component complex quaternionic Dirac spinor via complexification is decomposed into 4-component Dirac spinor and its Dirac adjoint in the Weyl representation of Dirac matrices linearly. Let us first summarize the quaternionic spinor analysis1) in which we associate an arbitrary space-time point xμ with an hermitian quaternionic matrix