Pauli Terms Must Be Absent In Dirac Equation

TL;DR

This paper investigates the algebraic structure of Dirac's equation, demonstrating that Pauli terms violate fundamental algebraic axioms and are thus absent from the Dirac operator, which is spanned only by specific basis elements.

Contribution

It proves that Pauli terms cannot be part of Dirac's equation due to algebraic constraints, refining the understanding of its algebraic basis.

Findings

Pauli terms violate *-algebra axioms in Dirac's equation

Dirac operator is spanned only by 10 specific basis elements

Pauli terms are excluded from the Dirac algebra

Abstract

It should be of interest, whether Dirac's equation involves all 16 basis elements of his Clifford algebra $Cl_D.$ These include the 6 `tensorial' $\sigma^{\mu\nu}$ with which the `Pauli terms' are formed. We find that these violate a basic axiom of any *-algebra, when Dirac's $\Psi$ is canonical. Then the Dirac operator is spanned only by the 10 elements $1,i\gamma_5,\gamma^\mu,\gamma^\mu\gamma_5$ (which don't form a basis of $Cl_D$ because the $\sigma^{\mu\nu}$ are excluded).