Pseudo-Hermitian Quantum Dynamics of Tachyonic Spin-1/2 Particles

Abstract

We investigate the spinor solutions, the spectrum and the symmetry properties of a matrix-valued wave equation whose plane-wave solutions satisfy the superluminal (tachyonic) 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 equation reads (i gamma^mu partial_mu - gamma^5 m) psi = 0, where gamma^5 is the fifth current. The tachyonic equation is shown to be CP invariant, and T invariant. The tachyonic Hamiltonian H_5 = alpha.p + beta gamma^5 m breaks parity and is non-Hermitian but fulfills the pseudo-Hermitian property H_5(r) = P H^+_5(-r) P^{-1} = PP H^+_5(-r) PP^{-1} where P is the parity matrix and PP is the full parity transformation. The energy eigenvalues and eigenvectors describe a continuous spectrum of plane-wave solutions (which correspond to real eigenvalues for |p|>=m and evanescent waves, which constitute resonances and antiresonances with complex-conjugate pairs of resonance eigenvalues (for |p|<=m) . In view of additional algebraic properties of the Hamiltonian which supplement the pseudo-Hermiticity, the existence of a resonance energy eigenvalues E implies that E^*, -E, and -E^* also constitute resonance energies of H_5.