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Complex Projection of Quasianti-Hermitian Quaternionic Hamiltonian Dynamics
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

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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-$\frac12$ system in a constant quasianti-Hermitian quaternionic potential is considered.
Keywords: pseudo-Hermitian Hamiltonians; quaternions. 
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     author = {Giuseppe Scolarici},
     title = {Complex {Projection} of {Quasianti-Hermitian} {Quaternionic} {Hamiltonian} {Dynamics}},
     journal = {Symmetry, integrability and geometry: methods and applications},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a87/}
}
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Giuseppe Scolarici. Complex Projection of Quasianti-Hermitian Quaternionic Hamiltonian Dynamics. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a87/

[1] Birkhoff G., von Neumann J., “The logic of quantum mechanics”, Ann. Math., 37 (1936), 823–843 | DOI | MR

[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 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl

[3] Adler S. L., Quaternionic quantum mechanics and quantum fields, Oxford University Press, New York, 1995 | MR | Zbl

[4] Kossakowski A., “Remarks on positive maps of finite-dimensional simple Jordan algebras”, Rep. Math. Phys., 46 (2000), 393–397 | DOI | MR | Zbl

[5] Scolarici G., Solombrino L., “Complex entanglement and quaternionic separability”, The Foundations of Quantum Mechanics: Historical Analysis and Open Questions (Cesena, 2004), eds. C. Garola, A. Rossi and S. Sozzo, World Scientific, Singapore, 2006, 301–310 | Zbl

[6] Asorey M., Scolarici G., “Complex positive maps and quaternionic unitary evolution”, J. Phys. A: Math. Gen., 39 (2006), 9727–9741 | DOI | MR | Zbl

[7] Asorey M., Scolarici G., Solombrino L., “Complex projections of completely positive quaternionic maps”, Theoret. and Math. Phys., 151:3 (2007), 735–743 | DOI | MR | Zbl

[8] Asorey M., Scolarici G., Solombrino L., “The complex projection of unitary dynamics of quaternionic pure states”, Phys. Rev. A, 76 (2007), 012111–012117 | DOI

[9] Proceedings of the 1st International Workshop Pseudo-Hermitian Hamiltonians in Quantum Physics, Prague, 2003, Czech. J. Phys., 54:1 (2004), 1–156 ; Proceedings of the 2nd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, Prague, 2004, 54:10 (2004), 1005–1148 ; Proceedings of the 3nd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, Prague, 2005, 55:9 (2005), 1049–1192 ; Proceedings of the 4nd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, Prague, 2005, J. Phys. A, 39 (2006); Proceedings of the 5nd International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, Prague, 2006, Czech. J. Phys., 56:9 (2006), respectively | DOI | MR | Zbl | DOI | MR | DOI

[10] Scolarici G., “Pseudoanti-Hermitian operators in quaternionic quantum mechanics”, J. Phys. A: Math. Gen., 35 (2002), 7493–7505 | DOI | MR | Zbl

[11] Blasi A., Scolarici G., Solombrino L., “Alternative descriptions in quaternionic quantum mechanics”, J. Math. Phys., 46 (2005), 42104–42111 ; quant-ph/0407158 | DOI | MR

[12] Scolarici G., Solombrino L., “Time evolution of non-Hermitian quantum systems and generalized master equations”, Czech. J. Phys., 56 (2006), 935–941 | DOI | MR | Zbl

[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 | DOI | MR

[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 | DOI | MR

[15] Zhang F., “Quaternions and matrices of quaternions”, Linenar Algebra Appl., 251 (1997), 21–57 | DOI | MR | Zbl

[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 | DOI | MR

[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 | DOI | MR | Zbl

[18] Mostafazadeh A., “Exact PT-symmetry is equivalent to Hermiticity”, J. Phys. A: Math. Gen., 36 (2003), 7081–7092 ; quant-ph/0304080 | DOI | MR

[19] Gorini V., Kossakowski A., Sudarshan E. C. G., “Completely positive dynamical semigroups and $N$-level systems”, J. Math. Phys., 17 (1976), 821–825 | DOI | MR

[20] Lindblad G., “On the generators of quantum dynamical semigroups”, Comm. Math. Phys., 48 (1976), 119–130 | DOI | MR | Zbl