The formulation of non-Hermitian PT-symmetric Hamiltonians and pseudo-Hermiticity

In recent years there has been much interest in non-Hermitian Hamiltonians with real eigenvalues. In particular, PT-symmetric Hamiltonians H, i.e., , where P is the parity operator and T is the time reversal operator, have been conjectured to have a real spectrum except when the symmetry is spontaneously broken. It is also known that PT symmetry is not a necessary condition for the occurrence of real eigenvalues. Then, -pseudo-Hermiticity, i.e., , was introduced to establish the mathematical relation with PT symmetry and explored the basic structure responsible for the reality of the spectrum of non-Hermitian Hamiltonians. Because of intrinsic interest and their applications in condensed matter, quantum optics and others, we study the pseudo-Hermitian interactions in both non-relativistic and relativistic quantum mechanics by considering several examples about delocalization transition in superconductors and spin-orbit coupling in semiconductors. In addition, the pseudo-supersymmetry and boson sea are studied. Finally, the PT-symmetric quantum field theory, Klein-Gordon-type field and quantum cosmology are discussed, too. Keywords: Pseudo unitarity; Pseudo-Hyper-virial theorem; Pseudo-probability current density; Pseudo-supersymmetry; Boson sea; Abnormal pseudo-Hermitian fermion; PT-symmetric quantum field theory; Klein-Gordon-type field.1.Introduction 2 2.Basic properties of pseudo-Hermiticity and PT symmetry 6 3.Generalized Ehrenfest theorem in -pseudo-Hermitian Q.M. 11 4.The application of the pseudo-Hellman-Feynman theorem to the Dirac equation with Rashba interaction term 13 5.The application of the pseudo-Hyper-virial theorem 19 6.Positive definite metric operators and quasi-Hermitian operators 22 7.Conservation of probability current density for non-Hermitian systems 29 8.The Equations-of-motion (EOM) methodfor the PT-symmetric Hamiltonian, pseudo-supersymmetry and the boson sea 36 9.The relation between the EOM method and pseudo-Hermitian quantum theory 51 10.The application of the pseudo-Hermiticity to the localization- delocalization transition 55 11.PT-symmetric quantum field theory, Klein-Gordon-type field and quantum cosmology 58 12.Conclusion 6