February 2012
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Day February 1, 2012

Dirac Equation with Imaginary Mass and Helicity-Dependence

U. D. Jentschura

In the matter wave equations describing 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 thetachyonic 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. The spectrum is found to consist of well-defined real energy eigenvalues and complex resonance and anti-resonance energies. The quantization of the tachyonic Dirac field has recently been discussed, and we here supplement a discussion of the quantized imaginary-mass Dirac field. Just as for the tachyonic Dirac Hamiltonian, we find that one-particle states of right-handed helicity acquire a negative norm and can be excluded from the physical spectrum by a Gupta–Bleuler type condition. This observation may indicate a deeper, general connection of superluminal propagation and helicity-dependent interactions.

High Energy Physics – Phenomenology (hep-ph); High Energy Physics – Theory (hep-th)