Negative-norm states, superselection rules, and the lepton family

Abstract

Field theories containing states of both positive and negative norm are considered. With the correct definition of the number operators for the quantum fields, all physical quantities are rendered canonically normalized. If the theory admits a global symmetry leading to a superselection rule which forbids transitions between positive- and negative-norm states, then the negative-norm states are allowed to be physical. Specifically, a spinor theory with higher-order field equations and multiple excitations is considered and applied to the charged lepton system: $e$, $\ensuremath{\mu}$, $\ensuremath{\tau}$. In this model, the negative norm of the muon state allows us to understand the nonexistence of $\ensuremath{\mu}\ensuremath{\rightarrow}e\ensuremath{\gamma}$ decay. For minimal coupling, the theory is renormalizable and equivalent to three separate fermion electrodynamics with the additional prediction of equal charge for the leptons. A further anomalous magnetic moment coupling can only allow one of the decays $\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\mu}\ensuremath{\gamma}$ or $\ensuremath{\tau}\ensuremath{\rightarrow}e\ensuremath{\gamma}$.