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

Abstract The Dirac equation (DE) is a cornerstone of quantum physics. We prove that direct quantization of the 4-momentum vector p with modulus m (rest energy, c = 1) yields a coordinate-free and manifestly covariant equation. Remember that standard DE is not manifestly covariant. In coordinate representation, this is equivalent to DE with spacetime frame vectors replacing Dirac’s gamma-matrices. The two sets obey to the same Clifford algebra. Adding an independent Hermitian vector x5 to the spacetime basis {xu} allows accommodating the momentum operator in a real vector space with a complex structure arising alone from vectors and multivectors. The real vector space generated from the action of the Clifford product onto the quintet {x0,x1,x2,x3,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, distinction of axial vs. polar vectors, left and right handed rotors & spinors, etc. Therefore, we name it reflector and {x0,x1,x2,x3,x5} – a basis for spacetime-reflection (STR). The pentavector I = x05123 commutes with all elements of STR and it depicts the pseudoscalar of STR. 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. All key symmetries of DE have a clear and rich geometric interpretation in STR. In simple terms, we demonstrate how Dirac matrices are a redundant representation of spacetime-reflection directors.