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arXiv:2212.02069

Bi-orthogonal eigen-spinors related to T-pseudo-Hermitian Pauli Hamiltonian : Time reversal and Clifford Algebra

Arindam Chakraborty

Published Dec 5, 2022• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2212.02069

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In this thesis, real quantum mechanics (QM) is constructed over the field ${\\cal F}\\sb{P\\sb0\\oplus P\\sb3}$ in the Pauli algebra, where the trivector $i = \\sigma\\sb1\\sigma\\sb2\\sigma\\sb3$ arises naturally as a product of the unit basis vectors in 3-dimensional Euclidean space. In this scheme, i is the volume element of the algebra with $i\\sp2 = -1$ and has geometrical meaning. Components of the state vector in Hilbert space take the form of a real element in the field ${\\cal F}\\sb{P\\sb0\\oplus P\\sb3}$ and the hermitean conjugate of the state vector is just its transpose, so that all the observables, represented by hermitean operators, are symmetric; i itself is an anti-hermitean operator. In this construction, the time reversal operator ${\\cal T}$ is a combination of the transpose "+" and the unitary operator which changes the sign of t. Since ${\\cal T}$ is an anti-automorphism the interpretation of its action must be carefully analyzed in terms of the state vectors and their inner products. Furthermore, the parity transformation is not represented by the spatial inversion $\\tilde\\Pi$ (an antilinear, anti-unitary transformation). Instead, the spatial reversal $\\bar\\Pi$ (an anti-automorphic, linear transformation) is proposed as a parity operator. It is unitary as well as hermitean, and it commutes with the Hamiltonian, H, of a system, thus, it is a constant of motion. This thesis mainly deals with non-relativistic QM for spinless particles.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1991 .W444. Source: Masters Abstracts International, Volume: 31-01, page: 0331. Chair: W. E. Baylis. Thesis (M.Sc.)--University of Windsor (Canada), 1991.

•Jan 1, 1991•0 citations
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.
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.
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=1T^{2}=1T2=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=−1T^{2}=-1T2=−1|⁠. Here we find that the metric operator for the pseudo-Hermitian Hamiltonian |HHH| that allows the realization of the generalized Kramers degeneracy is necessarily indefinite. We show that such |HHH| with real spectrum also possesses odd antilinear symmetry induced from the existing odd time-reversal symmetry of its Hermitian counterpart |hhh|⁠, so that the generalized Kramers degeneracy of |HHH| 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.
The recently introduced by us, two- and three-parameter (p, q)- and (p, q, [Formula: see text])-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite [Formula: see text](N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of “almost free” (but essentially nonlinear) Hamiltonian.
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.
In this thesis, real quantum mechanics (QM) is constructed over the field calFsbPsb0oplusPsb3{\\cal F}\\sb{P\\sb0\\oplus P\\sb3}calFsbPsb0oplusPsb3 in the Pauli algebra, where the trivector i=sigmasb1sigmasb2sigmasb3i = \\sigma\\sb1\\sigma\\sb2\\sigma\\sb3i=sigmasb1sigmasb2sigmasb3 arises naturally as a product of the unit basis vectors in 3-dimensional Euclidean space. In this scheme, i is the volume element of the algebra with isp2=−1i\\sp2 = -1isp2=−1 and has geometrical meaning. Components of the state vector in Hilbert space take the form of a real element in the field calFsbPsb0oplusPsb3{\\cal F}\\sb{P\\sb0\\oplus P\\sb3}calFsbPsb0oplusPsb3 and the hermitean conjugate of the state vector is just its transpose, so that all the observables, represented by hermitean operators, are symmetric; i itself is an anti-hermitean operator. In this construction, the time reversal operator calT{\\cal T}calT is a combination of the transpose "+" and the unitary operator which changes the sign of t. Since calT{\\cal T}calT is an anti-automorphism the interpretation of its action must be carefully analyzed in terms of the state vectors and their inner products. Furthermore, the parity transformation is not represented by the spatial inversion tildePi\\tilde\\PitildePi (an antilinear, anti-unitary transformation). Instead, the spatial reversal barPi\\bar\\PibarPi (an anti-automorphic, linear transformation) is proposed as a parity operator. It is unitary as well as hermitean, and it commutes with the Hamiltonian, H, of a system, thus, it is a constant of motion. This thesis mainly deals with non-relativistic QM for spinless particles.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1991 .W444. Source: Masters Abstracts International, Volume: 31-01, page: 0331. Chair: W. E. Baylis. Thesis (M.Sc.)--University of Windsor (Canada), 1991.