Dirac Hamiltonian with Imaginary Mass and Induced Helicity-Dependence by Indefinite Metric

TL;DR

This paper explores the properties of the imaginary-mass Dirac equation, revealing its pseudo-Hermitian nature, spectrum sum rules, and the helicity-dependent indefinite metric in the quantized field, contributing to the understanding of superluminal matter wave equations.

Contribution

It provides an explicit analysis of the imaginary-mass Dirac Hamiltonian, demonstrating its pseudo-Hermitian properties and the helicity-dependent indefinite metric in the quantum field.

Findings

Spectrum sum rules for real and complex energies

Pseudo-Hermitian and modified pseudo-Hermitian properties of the Hamiltonian

Helicity-dependent indefinite metric in the quantized field

Abstract

It is of general theoretical interest to investigate the properties of superluminal matter wave equations for spin one-half particles. One can either enforce superluminal propagation by an explicit substitution of the real mass term for an imaginary mass, or one can use a matrix representation of the imaginary unit that multiplies the mass term. The latter leads to the tachyonic Dirac equation, while the equation obtained by the substitution m->i*m in the Dirac equation is naturally referred to as the imaginary-mass Dirac equation. Both the tachyonic as well as the imaginary-mass Dirac Hamiltonians commute with the helicity operator. Both Hamiltonians are pseudo-Hermitian and also possess additional modified pseudo-Hermitian properties, leading to constraints on the resonance eigenvalues. Here, by an explicit calculation, we show that specific sum rules over the spectrum hold for the wave functions corresponding to the well-defined real energy eigenvalues and complex resonance and anti-resonance energies. In the quantized imaginary-mass Dirac field, one-particle states of right-handed helicity acquire a negative norm ("indefinite metric") and can be excluded from the physical spectrum by a Gupta--Bleuler type condition.