A non-Hermitian version of Rashba Hamiltonian has been introduced motivated from the Levy-leblond type linearisation of Schrodinger equation in a Galilean invariant frame-work. The said Hamiltonian is found to be pseudo-Hermitian under fermionic time-reversal and its eigen-spinors show bi-orthogonality. The whole discussion has been cast in a Clifford algebraic frame-work and the said biorthogonality has been variously understood in terms of time-reversal operator and Clifford involutions. The possibility of an equivalent version of Kramers' degeneracy theorem has been explored and the super-symmetric and pseudo-super-symmetric structures relating to the present system are also discussed.
A set of two-parameter bi-orthogonal eigen-spinors has been constructed from a deformed pseudo- Hermitian extension of Pauli Hamiltonian and its Hermitian conjugate. The Hamiltonians thus obtained are iso-spectral to the original Pauli Hamiltonian. A pair of spin-projection operators has been constructed as an essential ingredient of a possible bi-orthogonal quantum mechanics. An analogue of Kramers theorem in pseudo-Hermitian setting has also been inferred in a conjectural sense. The properties of time reversal and bi-orthogonality have been elaborated in the frame work of Clifford algebra Cl3, where the spinors have been viewed as elements of left ideal and the relevant inner-products are understood in terms of different involutions leading to elements of a division ring. The whole process of present construction is found to be based on both direct and time reversed Cl3 generators. A new variant of Kustaanheimo-Stiefel transformation has been introduced with the help of spinor operator.
In a recent paper [M. Sato, K. Hasebe, K. Esaki, and M. Kohmoto, Prog. Theor. Phys. 127, 937 (2012)] Sato and his collaborators established a generalization of the Kramers degeneracy structure to pseudo-Hermitian Hamiltonian systems, admitting even time-reversal symmetry, |
T2=1|. This extension is achieved using the mathematical structure of split-quaternions instead of quaternions, usually adopted in the case of Hermitian Hamiltonians with odd time-reversal symmetry, |
T2=−1|. Here we find that the metric operator for the pseudo-Hermitian Hamiltonian |
H| that allows the realization of the generalized Kramers degeneracy is necessarily indefinite. We show that such |
H| with real spectrum also possesses odd antilinear symmetry induced from the existing odd time-reversal symmetry of its Hermitian counterpart |
h|, so that the generalized Kramers degeneracy of |
H| is in fact crypto-Hermitian Kramers degeneracy. We study in greater detail a new example of the pseudo-Hermitian split-quaternionic four-level Hamiltonian system, which admits an indefinite metric operator and time-reversal symmetry and, as a consequence, a generalized Kramers degeneracy structure. We provide a complete solution of the eigenvalue problem, construct pseudo-Hermitian ladder operators closing the normal and abnormal pseudo-fermionic algebras, and show that this system fulfills a crypto-Hermitian degeneracy.
We formulate quantum mechanics for non-Hermitian Hamiltonians that are invariant under
PT, where
P is the parity and
T denotes time reversal, for the case that time-reversal symmetry is odd (${\mathcal{T}}^{2}=\ensuremath{-}1$), generalizing prior work for the even case (
T2=1). We discover an analog of Kramer's theorem for
PT quantum mechanics, present a prototypical example of a
PT quantum system with odd time reversal, and discuss potential applications of the formalism.
For ordinary Hermitian Hamiltonians, the states support the mathematical structure of quaternion and show the Kramers degeneracy when the system has a half-odd-integer spin and the time-reversal operator obeys Θ 2 = -1, while no such structure exists when Θ 2 = +1.Here, we point out that for non-Hermitian systems, there exists a similar mathematical structure called split-quaternion even when Θ 2 = +1.It is also found that a degeneracy similar to the Kramers degeneracy follows from the split-quaternion structure if the Hamiltonian is pseudo-Hermitian with the metric operator η satisfying {η, Θ} = 0. Furthermore, we show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split-quaternion in a class of non-Hermitian Hamiltonians.As concrete examples, we examine in detail N × N Hamiltonians with N = 2 and 4, which are non-Hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.