Dirac Redux - Dirac’s Equation in Physical Spacetime without Internal Degrees of Freedom

Abstract The Dirac equation (DE) is one of the cornerstones of quantum physics. We prove in the present contribution that the notion of internal degrees of freedom of the electron represented by Dirac’s matrices is superfluous. One can write down a coordinate-free manifestly covariant vector equation by direct quantization of the 4-momentum vector with modulus m: Pψ=mψ (no slash!), the spinor ψ taking care of the different vector grades at the two sides of the equation. Electron spin and all the standard DEproperties emerge from this equation. In coordinate representation, the four orthonormal time-space frame vectors Xµ appear instead of Dirac’s Yµ-matrices, the two sets obeying to the same Clifford algebra. The present formalism expands Hestenes’ spacetime algebra (STA) by adding a reflector vector x5 that is defining for the C (particle-antiparticle) symmetry and the CPT symmetry of DE, as well as for left- and right-handed rotors and spinors. In 3D it transforms a parity-odd vector x into a parity-even vector σ=x5x and vice versa. STA augmented by the reflector will be referred to as STAR, which operates on a real vector space of same dimension as the equivalent real dimension of Dirac’s complex 4×4 matrices. There are no matrices in STAR and the complex character springs from the signature and dimension of spacetime-reflection. This appears most clearly by first showing that STAR comprises two isomorphic subspaces, one for generators of polar vectors and boosts and the other for generators of axial vectors and rotors, including Pauli spin vectors. These then help to discuss the symmetries, probability current, transformation properties and nonrelativistic approximation of STAR DE. We prove here that the information from γ-matrices is contained in spacetime-reflection, which makes the matrices redundant. Therefore, it becomes relevant to reexamine those areas of quantum physics that take the γ-matrices and their generalizations as fundamental.