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 equal to the rest energy m (c =1) yields a manifestly covariant equation with the frame-free operator p. In coordinate representation, this is equivalent to DE with spacetime frame vectors xu replacing Dirac’s gamma-matrices. Recall that standard DE is not manifestly covariant. Adding an independent Hermitian vector x5 to the spacetime basis {xu} allows to accommodate 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 or geometric 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 C& CPT symmetries, 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 depicts the pseudoscalar in STR. We develop the formalism by deriving all essential results from the novel STR DE: spin ½ magnetic angular momentum, conserved probability currents, symmetries and nonrelativistic approximation. In simple terms, we demonstrate how Dirac matrices are a redundant representation of spacetime-reflection frame vectors.