Dirac Equation Redux by Direct Quantization of the 4-Momentum Vector

Abstract Dirac equation (DE) is a cornerstone of quantum physics. We prove that direct quantization of the 4-momentum vector p with modulus ???????? (???? is rest mass) yields a coordinate-free and manifestly covariant equation. In coordinate representation, this is equivalent to DE with spacetime frame vectors xμ replacing Dirac’s γμ -matrices. Remember that standard DE is not manifestly covariant. The two sets {xμ}, {γμ} obey to the same Clifford algebra. Adding an independent Hermitian vector x5 to the spacetime basis {xμ} allows to accommodate the momentum operator in a real vector space with a complex structure generated alone by vectors and multivectors. The real vector space arising from the action of the Clifford product onto the quintet {xμ , x5 } has dimension 32, the same as the equivalent real dimension for the space of Dirac matrices. x5 proves defining for the combined CPT symmetry, axial vs. polar vectors, left and right handed rotors & spinors, etc.; therefore, we name it reflector and {xμ , x5 } – a basis for spacetime-reflection (STR). The pentavector ???? ≡ x05123 in STR substitutes the imaginary unit i. We develop the formalism by deriving all the essential results from the novel STR DE: conserved probability currents, symmetries, nonrelativistic approximation and spin 1/2 magnetic angular momentum. It will become clear that key symmetries follow more directly and with clearer geometric interpretation in STR than in the standard approach. In simple terms, we demonstrate how Dirac matrices are a redundant representation of spacetime-reflection directors.