Non-Hermitian $\mathcal{PT}$-Symmetric Relativistic Quantum Theory in an Intensive Magnetic Field

Abstract

We develop relativistic non-Hermitian quantum theory and its application to neutrino physics in a strong magnetic field. It is well known, that one of the fundamental postulates of quantum theory is the requirement of Hermiticity of physical parameters. This condition not only guarantees the reality of the eigenvalues of Hamiltonian operators, but also implies the preservation of the probabilities of the considered quantum processes. However as it was shown relatively recently (Bender and Boettcher Phys Rev Lett 80:5243, 1998), Hermiticity is a sufficient but it is not a necessary condition. It turned out that among non-Hermitian Hamiltonians it is possible to allocate a number of such which have real energy spectra and can ensure the development of systems over time with preserving unitarity. This type of Hamiltonians includes so-called parity-time ($\mathcal{PT}$) symmetric models which is already used in various fields of modern physics. The most developed in this respect are models, which used in the field of $\mathcal{PT}$-symmetric optics, where for several years produced not only theoretical but experimental studies.