Adjunctation and Scalar Product in the Dirac Equation - I

Abstract

Part-I Dima (Int. J. Theor. Phys. 55, 949, 2016) of this paper showed in a representation independent way that γ0 is the Bergmann-Pauli adjunctator of the Dirac {γμ} set. The distiction was made between similarity (MATH) transformations and PHYS transformations - related to the (covariant) transformations of physical quantities. Covariance is due solely to the gauging of scalar products between systems of reference and not to the particular action of γ0 on Lorentz boosts - a matter that in the past led inadvertently to the definition of a second scalar product (the Dirac-bar product). Part-II shows how two scalar products lead to contradictions and eliminates this un-natural duality in favour of the canonical scalar product and its gauge between systems of reference. What constitutes a proper observable is analysed and for instance spin is revealed not to embody one (except as projection on the boost direction - helicity). A thorough investigation into finding a proper-observable current for the theory shows that the Dirac equation does not possess one in operator form. A number of problems with the Dirac current operator are revealed - its Klein-Gordon counterpart being significantly more physical. The alternative suggested is finding a current for the Dirac theory in scalar form \(j^{\mu } = \langle \, \rho \,\rangle _{_{\psi }}v^{\mu }_{\psi }\).