A non-symmetric transition probability in quantum mechanics

Stefan Wojciechowski

Gerd Niestegge

Kazimierz Urbanik

Roman Frigg, Stephan Hartmann

Alexander Fagerlund, Chistopher Ekman

In this report, we first briefly summarize Hermitian quantum mechanics before moving on to the non-Hermitian case. We then review PT-symmetric quantum mechanics with a focus on finite-dimensional systems, and include a novel generalization of a perturbative calculation of the C-operator. After briefly covering the basics of neutrino oscillations, we perturbatively examine a PT-symmetric addition to the neutrino oscillation Hamiltonian. We examine the effects of the addition with two different definitions of transition probabilities. However, probability is not conserved to first order with either definition. Further, we note that the effect of the chosen perturbation is to shift the transition probabilities by some phase, and to change the amplitudes of the transition probabilities.

Kh. Saaidi

In this article, we discussed certain properties of non-Hermitian $\CP$-symmetry Hamiltonian, and it is shown that a consistent physical theory of quantum mechanics can be built on a ${\cal C} \CP$-symmetry Hamiltonian. In particular, we show that these theories have unitary time evolution, and conservation probability. Furthermore, transition from quantum mechanics to classical mechanics is investigate and it is found that the Ehrenfest theorem is satisfied.