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.
Construction of a few Quantum mechanical Hamiltonians via L
eˊvi-Leblond type Linearization:Spinor states and Supersymmetry
A number of new L
eˊvi-Leblond type equations admitting four component spinor solutions have been proposed. The pair of linearized equations thus obtained in each case lead to Hamiltonians with characteristic features like L-S coupling and supersymmetry. The relevant momentum operators have often been understood in terms of Clifford algebraic bases producing Schr
o¨dinger Hamiltonians with L-S coupling. As for example Hamiltonians representing Rashba effect or three dimensional harmonic oscillator have been constructed. The supersymmetric nature of one dimensional oscillator has also been appreciated.
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 study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (𝒫), time-reversal (𝒯), and charge-conjugation (𝒞) operators and establish the 𝒫𝒯- and 𝒞𝒫𝒯-invariance of H.
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.