Symmetries of the Tachyonic Dirac Equation

U. D. Jentschura, B. J. Wundt

We show that it is possible to construct a tachyonic version of the Dirac equation, which contains the fifth current and reads (i gamma^\mu partial_\mu – gamma^5 m) \psi = 0. Its spectrum fulfills the dispersion relation E^2 = p^2 – m^2, where E is the energy, p is the spatial momentum, and m is the mass of the particle. The tachyonic Dirac equation is shown to be CP invariant, and T invariant. The Feynman propagator is found. In contrast to the covariant formulation, the tachyonic Hamiltonian H_5 = alpha.p + beta gamma^5 m breaks Lorentz covariance (as does the Hamiltonian formalism in general, because the time variable is singled out and treated differently from space). The tachyonic Dirac Hamiltonian H_5 breaks parity but is found to be invariant under the combined action of parity and a noncovariant time reversal operation T’. In contrast to the Lorentz-covariant T operation, T’ involves the Hermitian adjoint of the Hamiltonian. Thus, in the formalism developed by Bender et al., primarily in the context of quantum mechanics, H_5 is PT’ symmetric. The PT’ invariance (in the quantum mechanical sense) is responsible for the fact that the energy eigenvalues of the tachyonic Dirac Hamiltonian are real rather than complex. The eigenstates of the Hamiltonian are shown to approximate the helicity eigenstates of a Dirac neutrino in the massless limit.
High Energy Physics – Phenomenology (hep-ph)

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