Geodesic
Hannay angles and Grassmannian action-angle quantum states
Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 2, pp. 278-289

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We show how to derive the Hannay angles of Grassmannian classical mechanics from the evolution of Grassmannian action–angle quantum states. Just as in the commutative case, this evolution defines a geometric transport, which can also be obtained from a quantum canonical transformation or a variational principle. As examples, we explicitly construct the quantum states for the classical counterparts of a first- and second-quantized $N$-level system. In the latter case, these states reduce to standard fermionic coherent states and the classical Hannay angles coincide with the quantum Berry phases.
Keywords: Berry phase, Hannay angle, action–angle fermionic coherent state. 
H. Lakehal; M. Maamache. Hannay angles and Grassmannian action-angle quantum states. Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 2, pp. 278-289. http://geodesic.mathdoc.fr/item/TMF_2020_202_2_a6/
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     language = {ru},
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[1] M. V. Berry, “Quantal phase factors accompanying adiabatic changes”, Proc. Roy. Soc. London Ser. A, 392:1802 (1984), 45–57 | DOI | MR

[2] J. H. Hannay, “Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian”, J. Phys. A, 18:2 (1985), 221–230 | DOI | MR

[3] B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry's phase”, Phys. Rev. Lett., 51:24 (1983), 2167–2170 ; J. Anandan, “Geometric angles in quantum and classical physics”, Phys. Lett. A, 129:4 (1988), 201–207 | DOI | MR | DOI | MR

[4] Raul Bott, Chzhen Shen-shen, “Ermitovy vektornye rassloeniya i ravnoraspredelennost nulei ikh golomorfnykh sechenii”, Matematika, 14:2 (1970), 117–154 | MR

[5] R. Montgomery, “The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case”, Commun. Math. Phys., 120:2 (1988), 269–294 ; S. Golin, A. Knauf, S. Marmi, “The Hannay angles: Geometry, adiabaticity, and an example”, 123:1 (1989), 95–122 | DOI | MR | DOI | MR

[6] M. Maamache, J.-P. Provost, G. Vallée, “Berry's phase, Hannay's angle and coherent states”, J. Phys. A: Math. Gen., 23:24 (1990), 5765–5775 | DOI | MR

[7] M. Maamache, J.-P. Provost, G. Vallée, “Berry's phase and Hannay's angle from quantum canonical transformations”, J. Phys. A: Math. Gen., 24:3 (1991), 685–688 | DOI | MR

[8] G. Giavarini, E. Gozzi, D. Rohrlich, W. D. Thacker, “Some connections between classical and quantum anholonomy”, Phys. Rev. D, 39:10 (1989), 3007–3015 | DOI | MR

[9] R. Casalbuoni, “On the quantization of systems with anticommuting variables”, Nuovo Cimento A, 33:1 (1976), 115–125 ; “The classical mechanics for bose-fermi systems”, 33:3 (1976), 389–431 ; F. A. Berezin, M. S. Marinov, “Particle spin dynamics as the grassmann variant of classical mechanics”, Ann. Phys. (N. Y.), 104:2 (1977), 336–362 | DOI | DOI | DOI

[10] E. Gozzi, W. D. Thacker, “Classical adiabatic holonomy in a Grassmannian system”, Phys. Rev. D, 35:8 (1987), 2388–2397 | DOI | MR

[11] E. Gozzi, D. Rohrlich, W. D. Thacker, “Classical adiabatic holonomy in field theory”, Phys. Rev. D, 42:8 (1990), 2752–2762 | DOI | MR

[12] S. Abe, “Adiabatic holonomy and evolution of fermionic coherent state”, Phys. Rev. D, 39:8 (1989), 2327–2331 | DOI | MR

[13] M. Maamache, J. P. Provost, G. Vallée, “Comment on ‘Adiabatic holonomy and evolution of fermionic coherent state’ ”, Phys. Rev. D, 46:2 (1992), 873–875 | DOI | MR

[14] A. Barducci, F. Buccella, R. Casalbuoni, L. Lusanna, E. Sorace, “Quantized Grassmann variables and unified theories”, Phys. Lett. B, 67:3 (1977), 344–346 | DOI

[15] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford Univ. Press, Oxford, 1993 | DOI | MR

[16] O. Cherbal, M. Drir, M. Maamache, D. A. Trifonov, “Fermionic coherent states for pseudo-Hermitian two-level systems”, J. Phys. A: Math. Theor., 40:8 (2007), 1835–1844, arXiv: quant-ph/0608177 | DOI | MR

[17] G. Najarbashi, M. A. Fasihi, H. Fakhri, “Generalized Grassmannian coherent states for pseudo-Hermitian $n$-level systems”, J. Phys. A: Math. Theor., 43:32 (2010), 325301, 10 pp. | DOI | MR

[18] M. Combescure, D. Robert, “Fermionic coherent states”, J. Phys. A: Math. Theor., 45:24 (2012), 244005, 27 pp. | DOI | MR

[19] D. A. Trifonov, “Nonlinear fermions and coherent states”, J. Phys. A: Math. Theor., 45:24 (2012), 244037, 14 pp., arXiv: 1207.6242 | DOI | MR