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 = \vec{p}^{\,2} - m^2, where E is the energy, \vec{p} is the spatial momentum and m is the mass of the particle. The equation reads (iγμ ∂μ - γ5 m)ψ = 0, where γ5 is the fifth current. The tachyonic equation is shown to be {CP} invariant and T invariant. The tachyonic Hamiltonian H_5 = \vec{\alpha }\cdot \vec{p} + \beta \, \gamma ^5 \, m breaks parity and is non-Hermitian but fulfils the pseudo-Hermitian property H_5(\vec{r}) = P \, H^{+}_5(-\vec{r}) \, P^{-1} = {P}\, H^{+}_5(\vec{r}) \, {P}^{-1}, where P is the parity matrix and {P} is the full parity transformation. The energy eigenvalues and eigenvectors describe a continuous spectrum of plane-wave solutions (which correspond to real eigenvalues for |\vec{p}| \ge m) and evanescent waves, which constitute resonances and anti-resonances with complex-conjugate pairs of resonance eigenvalues (for |\vec{p}| < m). In view of additional algebraic properties of the Hamiltonian which supplement the pseudo-Hermiticity, the existence of a resonance energy eigenvalue E implies that E*, -E and -E* also constitute resonance energies of H5.
This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.- Publication:
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Journal of Physics A Mathematical General
- Pub Date:
- November 2012
- DOI:
- 10.1088/1751-8113/45/44/444017
- arXiv:
- arXiv:1110.4171
- Bibcode:
- 2012JPhA...45R4017J
- Keywords:
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- High Energy Physics - Phenomenology;
- Mathematical Physics;
- Quantum Physics
- E-Print:
- 17 pages