Abstract
We first introduce theWigner–Weyl–Moyal formalism for a theorywhose phase space is an arbitrary Lie algebra. We alsogeneralize to quantum Lie algebras and to supersymmetrictheories. It turns out that the noncommutativity leads to a deformation ofthe classical phase space: instead of being a vectorspace, it becomes a manifold, the topology of which isgiven by the commutator relations. It is shown in fact that the classical phase space, for asemisimple Lie algebra, becomes a homogeneous symplecticmanifold. The symplectic product is also deformed. Wefinally make some comments on how to generalise to C*-algebras and other operator algebras,too.
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REFERENCES
Abe, S. (1992). Journal of Mathematical Physics, 35, 1690.
Alekseev, A. Yu., and Malkin, A. Z. (1994). Communication s in Mathematical Physics, 162, 147.
Amiet, J.-P., and Ciblis, M. B. (1991). Journal of Physics A, 24, 1515.
Berezin, F. A. (1975). Communication s in Mathematical Physics, 40, 153.
Bratteli, O., and Robinson, D. W. (1979). Operator Algebras and Quantum Statistical Mechanics I, Springer, New York.
Carter, R. W. (1977). Simple Groups of Lie Type, Wiley, New York.
Cohen, L. (1966). Journal of Mathematical Physics, 7, 781.
Dahl, J.-P. (1982). Physica Scripta, 25, 499.
Dahl, J.-P. (1991). The dual nature of phase-space representations. In Proceedings Second International Wigner Symposium, Goslar, Germany.
Dahl, J.-P. (1992). Theoretica Chimica Acta, 81, 329.
Davidović, D. M., Lalović, D., and Tančić, A. R. (1995). Physics Letters A, 201, 393.
DeWitt, B. (1984). Supermanifolds, Cambridge University Press, Cambridge.
Etingof, P., and Kazhdan, D. (1995a). Quantization of Lie Bialgebras, I, q-alg/9506005.
Etingof, P., and Kazhdan, D. (1995b). Quantization of Poisson algebraic groups and Poisson homogeneous spaces, q-alg/9510020.
Flato, M., and Sternheimer, D. (1980). Deformation of Poisson brackets. In Harmonic Analysis and Representations o f Semisimple Lie Groups, J. A. Wolf, M. Cahen, and M. DeWilde, eds., Reidel, Dordrecht.
Fuchs, J. (1992). Affine Lie Algebras and Quantum Groups, Cambridge University Press, Cambridge.
Gamkrelidze, R. V., ed. (1991). Geometry I, Encyclopedia of Mathematical Sciences, No. 28, Springer-Verlag, Berlin.
Grossman, A. (1976). Communications in Mathematical Physics, 48, 191.
Göckeler, M., and Schücker, T. (1987). Differential Geometry, Gauge Theories and Gravity, Cambridge University Press, Cambridge.
Isham, C. (1984). In Relativity, Groups and Topology II, B. S. DeWitt and R. Stora, eds., North-Holland, Amsterdam.
Isham, C., and Kakas, A. C. (1984). Classical and Quantum Gravity, 1 621–650.
Jacobson, N. (1962). Lie Algebras, Dover, New York.
Kac, V. (1985). Infinite Dimensional Lie Algebras, 2nd ed., Cambridge University Press, Cambridge.
Kasperkowiz, P. (1993). Group-related phase-space formalisms: Existence, general structure, and equivalence, preprint.
Kasperowiz, P., and Peev, M. (1994). Annals of Physics, 230, 21.
Hui Li (1994a). Physics Letters A, 188, 107.
Hui Li (1994b). Physics Letters A, 190, 370.
Merzbacher, E. (1970). Quantum Mechanics, 2nd ed., Wiley, New York.
Moreno, (1986). Letters in Mathematical Physics, 12, 217.
Murphy, G. J. (1990). C*-Algebras and Operator Theory, Academic Press, London.
Nash, C. (1991). Differential Topology and Quantum Field Theory, Academic Press, London.
Niehto, L. M. (1991). Journal of Physics A, 24, 1579.
Omishchik, A. L., ed. (1993). Lie Groups and Lie Algebras I, Encyclopedia of Mathematical Sciences, No. 20, Springer-Verlag, Berlin.
Omishchik, A. L., ed. (1994). Lie Groups and Lie Algebras III, Encyclopedia of Mathematical Sciences, No. 41, Springer-Verlag, Berlin.
Royer, A. (1977). Physical Review A, 15, 449.
Springborg, M. (1983). Journal of Physics A, 16, 535.
Sunder, V. S. (1987). An Invitation to von Neumann Algebras, Springer, New York.
Unterberger, A., and Upmai, H. (1994). Communications in Mathematical Physics, 164, 563.
Várilly, J. C., and Gracia-Bondía, J. M. (1989). Annals of Physics, 190, 107.
Wegge-Olsen, N. E. (1993). K-Theory and C*-Algebras, Oxford University Press, Oxford.
Wess, J., and Bagger, K. (1992). Supersymmetry and Supergravity, 2nd ed., Princeton University Press, Princeton, New Jersey.
Wess, J., and Zumino, B. (1990). Nuclear Physics B, 18B, 302.
Woodhouse, N. M. J. (1992). Geometric Quantization, 2nd ed., Oxford University Press, Oxford.
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Antonsen, F. Wigner–Weyl–Moyal Formalism on Algebraic Structures. International Journal of Theoretical Physics 37, 697–757 (1998). https://doi.org/10.1023/A:1026612428446
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DOI: https://doi.org/10.1023/A:1026612428446