Deformation quantization in the teaching of quantum mechanics

American Journal of Physics

Volumn 70, Issue 5, 2002, Pages 537-547

  Hirshfeld, Allen C ^a   Henselder, Peter ^a  

^a TU DORTMUND UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23044532427     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1450573     Document Type: Article

References (42)

1. P. A. M. Dirac, Tlie Principles of Quantum Mechanics (Clarendon, Oxford, 1930).
2. J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer-Verlag, Berlin, 1932); Math. Ann. 104, 570-578 (1931).
3. R. P. Feynman, "Space-time approach to non-relativistic quantum mechanics," Rev. Mod. Phys. 20, 367-387 (1947).
4. J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, MA, 1994).
5. M. E. Peskin and D. Schroeder, An Introduction to Quantum Field Theory (Perseus, Reading, MA, 1995).
6. E. P. Wigner, "Quantum corrections for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
7. H. Weyl, Quantenmechanik und Gruppentheorie (Hirzel Verlag, Leipzig, 1928); Z. Phys. 46, 1-46 (1927).
8. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, "Deformation theory and quantization I," Ann. Phys. (N.Y.) 111, 61-110 (1978);
9. "Deformation theory and quantization II," 111, 111-151 (1978).
10. G. Dito and S. Stermheimer, "Deformation quantization: genesis, developments and metamorphoses," in Deformation quantization (G. Halbout, ed.), IRMA Lectures in Math. Phys. I, 9-54 (Walter de Gruyter, Berlin, 2002).
11. S. Gutt, "Variations on deformation quantization," presented at the Moshe Flato Conference, Dijon, 1999; math.DG/0003107 (2000).
12. A. Weinstein, "Deformation quantization," Semin. Bourbaki, Asterique 789, 389-409 (1995).
13. C. Zachos, "Deformation quantization: quantum mechanics lives and works in phase space," hep-th/0110114 (2001).
14. R. Haag, Local Quantum Physics (Springer-Verlag, Berlin, 1992).
15. M. Diitsch and K. Fredenhagen, "Perturbative algebraic field theory and deformation quantization," hep-th/0101079 (2001).
16. M. Kontsevich, "Deformation quantization of Poisson manifolds," q-alg/9709040 (1997).
17. J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, Vol. 17 (Springer-Verlag, Berlin, 1994).
18. I. Vaisman, Geometry of Poisson Manifolds (Birkhäuser Verlag, Basel, 1994).
19. A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras (Am. Math. Soc. ISBN 0-8218-0952-0, 1999).
20. H. J. Groenewold, "On the principles of elementary quantum mechanics," Physica (Amsterdam) 12, 405-460 (1946).
21. M. Gerstenhaber, "On the deformation of rings and algebras," Ann. Math. 79, 59-103 (1964).
22. M. DeWilde and P. B. A. Lecomte, "Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds," Lett. Math. Phys. 7, 487-496 (1983).
23. J. E. Moyal, "Quantum mechanics as a statistical theory," Proc. Cambridge Philos. Soc. 45, 99-124 (1949).
24. M. Bertelson, M. Cahen, and S. Gutt, "Equivalence of star products," Class. Quantum Grav. 14, A93-A107 (1997).
25. F. Hansen, "Quantum mechanics in phase space," Rep. Math. Phys. 19, 361-381 (1984).
26. M. A. Rieffel, "Deformation quantization of Heisenberg manifolds," Commun. Math. Phys. 122, 531-562 (1989).
27. C. Zachos, "A survey of star product geometry," hep-th/000801002. (2000).
28. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
29. G. S. Agarwal and E. Wolf, "Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics I," Phys. Rev. D 10, 2161-2186 (1970).
30. R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766-2788 (1963).
31. M. Bordemann and S. Waldmann, "Formal GNS construction and states in deformation quantization," Commun. Math. Phys. 195, 549-583 (1998);
32. M. Bordemann, N. Neumaier, and S. Waldmann, "Homogeneous Fedosov star products on cotangent bundles I. Weyl and standard ordering with differential operator representation," ibid. 198, 363-396 (1998).
33. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
34. P. Sharan, "Star-product representation of path integrals," Phys. Rev. D 20, 414-418 (1979).
35. J. Dito, "Star-product approach to quantum field theory: the free scalar field," Lett. Math. Phys. 20, 125-134 (1990).
36. H. Leschke, A. C. Hirshfeld, and T. Suzuki, "Canonical perturbation theory for nonlinear systems," Phys. Rev. D 18, 2834-2848 (1978).
37. A. S. Cattaneo and G. Felder, "A path integral approach to the Kontsevich quantization formula," Commun. Math. Phys. 212, 591-611 (2000).
38. P. Schaller and T. Strobl, "Poisson structure induced (topological) field theories in two dimensions," Mod. Phys. Lett. A 9, 3129-3136 (1994).
39. A: C. Hirshfeld and T. Schwarzweller, "Path integral quantization of the Poisson-Sigma model," Ann. Phys. (Leipzig) 9, 83-101 (2000).
40. J. Stasheff, "Deformation theory and the Batalin-Vilkovisky master equation," q-alg/9702012 (1997).
41. H. Garcia-Compean, J. F. Plebanski, M. Przanowski, and F. J. Turrubiates, "Deformation quantization of bosonic strings," J. Phys. A 33, 7935-7954 (2000).
42. F. Antonsen, "Deformation quantization of gravity," Bohr Institute preprint (1997).