Linear Representation of Spinors by Tensors

Abstract

A linear representation of spinors in n-dimensional space by tensors is proposed. In particular, in three-dimensional space a set composed by a scalar and a vector is associated to any two-component spinor, while in four-dimensional space the set corresponding to a four-component spinor is composed by a scalar, a pseudoscalar, a vector, a pseudovector, and an antisymmetrical tensor of second order. The resulting formalism is then applied to Schrödinger's and Dirac's equations. In three-dimensional space it turns out that the proposed procedure automatically assigns an intrinsic magnetic moment to an electron in a magnetic field without introducing any relativistic ideas or ad hoc assumptions. In four-dimensional space we can write the Dirac equation in a generally covariant fashion, without introducing new concepts with respect to the usual tensor analysis. The zero-mass Dirac equation splits into two sets of equations, describing respectively the neutrino and the photon. The possible bearing of the proposed approach upon the theories of elementary particles is briefly discussed.