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.
Highlights
Theory and ExperimentThe superluminal propagation of matter waves is a highly intriguing subject which is not without controversy
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 im in the Dirac equation is naturally referred to as the imaginary-mass Dirac equation
It has been argued that the tachyonic Dirac equation [1,2] provides for a convenient framework for the description of tachyonic particles; in this equation, the mass is multiplied by a matrix representation of the imaginary unit
Summary
The superluminal propagation of matter waves is a highly intriguing subject which is not without controversy. Starting from the Dirac Hamiltonian, we explore a Dirac equation where the mass is explicitly multiplied by the imaginary unit and we find certain fundamental relations for the corresponding spin-1/2 field theory. The best estimate for the neutrino mass square has been determined as negative in all experiments [10,11,12,13,14,15,16], but the result has been consistent with a vanishing neutrino mass within experimental error bars. JENTSCHURA thought by analyzing a theory where the imaginary mass is used explicitly, rather than a matrix representation thereof. The latter has been used in Refs.
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