J. H. Noble, U. D. Jentschura
The PT-symmetric cubic oscillator with Hamiltonian \(H_3 = – (1/2) d_x^2 + (1/2) x^2 + i G x^3\) is a paradigmatic example of a pseudo-Hermitian, or PT-symmetric Hamiltonian with a purely real spectrum when endowed with \(L^2(R)\) boundary conditions. Eigenfunctions of the stationary Schr\”{o}dinger equation \(H_3 \psi_n(x) = E_n \psi_n(x)\) are manifestly complex, while the energy eigenvalues \(E_n\) are real and positive. Although \(H_3\) does not commute with the parity operator, we find that for a natural choice of the global complex phase of the eigenstate wave function, the real and imaginary parts of the eigenfunctions [i.e., Re \(\psi_n(x)\) and Im \(\psi_n(x)\)] are eigenstates of parity. Both the real and imaginary parts of the eigenfunctions are found to have an infinite number of zeros, even for the ground state and even for infinitesimal G, but the real and imaginary parts of psi_n(x) never vanish simultaneously when \(G>0\). Furthermore, we find that the eigenfunctions are “concentrated” in an “allowed” region where the energy \(E_n\) is larger than the complex modulus of the complex potential. PT-symmetric Hamiltonians constitute natural generalizations of Hermitian Hamiltonians as time-evolution operators. Our results suggest that PT-symmetric (pseudo-Hermitian) time evolution can naturally be interpreted as time evolution in a situation where manifestly complex “gain” and “loss” terms mutually compensate and lead to the real energy eigenvalues.
http://arxiv.org/abs/1301.5758
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Atomic Physics (physics.atom-ph)
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.
http://arxiv.org/abs/1201.6300
High Energy Physics – Phenomenology (hep-ph); High Energy Physics – Theory (hep-th)
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.
http://arxiv.org/abs/1110.4171
High Energy Physics – Phenomenology (hep-ph)