Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 088, 10 pages      arXiv:0709.1198      https://doi.org/10.3842/SIGMA.2007.088
Contribution to the Proceedings of the 3-rd Microconference Analytic and Algebraic Methods III

Complex Projection of Quasianti-Hermitian Quaternionic Hamiltonian Dynamics

Giuseppe Scolarici
Dipartimento di Fisica dell'Università del Salento, and INFN, Sezione di Lecce, I-73100 Lecce, Italy

Received July 05, 2007, in final form September 03, 2007; Published online September 08, 2007

Abstract
We characterize the subclass of quasianti-Hermitian quaternionic Hamiltonian dynamics such that their complex projections are one-parameter semigroup dynamics in the space of complex quasi-Hermitian density matrices. As an example, the complex projection of a spin-½ system in a constant quasianti-Hermitian quaternionic potential is considered.

Key words: pseudo-Hermitian Hamiltonians; quaternions.

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References

  1. Birkhoff G., von Neumann J., The logic of quantum mechanics, Ann. Math. 37 (1936), 823-843.
  2. Finkelstein D., Jauch J.M., Schiminovich S., Speiser D., Foundations of quaternion quantum mechanics, J. Math. Phys. 3 (1962), 207-220.
    Finkelstein D., Jauch J.M., Schiminovich S., Speiser D., Principle of general Q-covariance, J. Math. Phys. 4 (1963), 788-796.
    Finkelstein D., Jauch J.M., Speiser D., Quaternionic representations of compact groups, J. Math. Phys. 4 (1963), 136-140.
  3. Adler S.L., Quaternionic quantum mechanics and quantum fields, Oxford University Press, New York, 1995.
  4. Kossakowski A., Remarks on positive maps of finite-dimensional simple Jordan algebras, Rep. Math. Phys. 46 (2000), 393-397.
  5. Scolarici G., Solombrino L., Complex entanglement and quaternionic separability, in The Foundations of Quantum Mechanics: Historical Analysis and Open Questions (Cesena, 2004), Editors C. Garola, A. Rossi and S. Sozzo, World Scientific, Singapore, 2006, 301-310.
  6. Asorey M., Scolarici G., Complex positive maps and quaternionic unitary evolution, J. Phys. A: Math. Gen. 39 (2006), 9727-9741.
  7. Asorey M., Scolarici G., Solombrino L., Complex projections of completely positive quaternionic maps, Theoret. and Math. Phys. 151 (2007), 735-743.
  8. Asorey M., Scolarici G., Solombrino L., The complex projection of unitary dynamics of quaternionic pure states, Phys. Rev. A 76 (2007), 012111-012117.
  9. Proceedings of the Ist, IInd, IIIrd, IVth and Vth International Workshops on "Pseudo-Hermitian Hamiltonians in Quantum Physics" in Czech. J. Phys. 54 (2004), no. 1, no. 10, Czech. J. Phys. 55 (2005), no. 9, J. Phys. A: Math. Gen. 39 (2006), no. 32, and Czech. J. Phys. 56 (2006), no. 9, respectively.
  10. Scolarici G., Pseudoanti-Hermitian operators in quaternionic quantum mechanics, J. Phys. A: Math. Gen. 35 (2002), 7493-7505.
  11. Blasi A., Scolarici G., Solombrino L., Alternative descriptions in quaternionic quantum mechanics, J. Math. Phys. 46 (2005), 42104-42111, quant-ph/0407158.
  12. Scolarici G., Solombrino L., Time evolution of non-Hermitian quantum systems and generalized master equations, Czech. J. Phys. 56 (2006), 935-941.
  13. Mostafazadeh A., Batal A., Physical aspects of pseudo-Hermitian and PT-symmetric quantum mechanics, J. Phys. A: Math. Gen. 37 (2004), 11645-11680, quant-ph/0408132.
  14. Mostafazadeh A., Time-dependent pseudo-Hermitian Hamiltonians defining a unitary quantum system and uniqueness of the metric operator, Phys. Lett. B 650 (2007), 208-212, arXiv:0706.1872.
  15. Zhang F., Quaternions and matrices of quaternions, Linenar Algebra Appl. 251 (1997), 21-57.
  16. Sholtz F.G., Geyer H.B., Hahne F.J.W., Quasi-Hermitian operators in quantum mechanics and the variational principle, Ann. Phys. 213 (1992), 74-101.
  17. Blasi A., Scolarici G., Solombrino L., Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries, J. Phys. A: Math. Gen. 37 (2004), 4335-4351, quant-ph/0310106.
  18. Mostafazadeh A., Exact PT-symmetry is equivalent to Hermiticity, J. Phys. A: Math. Gen. 36 (2003), 7081-7092, quant-ph/0304080.
  19. Gorini V., Kossakowski A., Sudarshan E.C.G., Completely positive dynamical semigroups and N-level systems, J. Math. Phys. 17 (1976), 821-825.
  20. Lindblad G., On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48 (1976), 119-130.

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