Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash 91 tok/s
Gemini 2.5 Pro 46 tok/s Pro
GPT-5 Medium 33 tok/s
GPT-5 High 27 tok/s Pro
GPT-4o 102 tok/s
GPT OSS 120B 465 tok/s Pro
Kimi K2 205 tok/s Pro
2000 character limit reached

Classical Limits of Hilbert Bimodules as Symplectic Dual Pairs (2403.08060v1)

Published 12 Mar 2024 in math-ph, math.MP, and quant-ph

Abstract: Hilbert bimodules are morphisms between C*-algebraic models of quantum systems, while symplectic dual pairs are morphisms between Poisson geometric models of classical systems. Both of these morphisms preserve representation-theoretic structures of the relevant types of models. Previously, it has been shown that one can functorially associate certain symplectic dual pairs to Hilbert bimodules through strict deformation quantization. We show that, in the inverse direction, strict deformation quantization also allows one to functorially take the classical limit of a Hilbert bimodule to reconstruct a symplectic dual pair.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (45)
  1. Toeplitz Operators and Quantum Mechanics. Journal of Functional Analysis, 68:273–399.
  2. Deformation Quantization for Actions of Kählerian Lie Groups, volume 236 of Memoirs of the American Mathematical Society. American Mathematical Society, Providence, RI.
  3. Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization. Annales de l’Institut Henri Poincaré, 5:327–346.
  4. Picard Groups in Poisson Geometry. Moscow Mathematical Journal, 4(1):39–66.
  5. Coburn, L. A. (1992). Deformation estimates for the Berezin-Toeplitz quantization. Communications in Mathematical Physics, 149(2):415–424.
  6. Integrability of Poisson Brackets. Journal of Differential Geometry, 66(1).
  7. Dixmier, J. (1977). C*-Algebras. North Holland, New York.
  8. Feintzeig, B. (2024). Quantization as a Categorical Equivalence. Letters in Mathematical Physics, forthcoming.
  9. Feintzeig, B. H. (2023). The Classical–Quantum Correspondence. Cambridge University Press, Cambridge.
  10. Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles. Academic Press, Boston.
  11. Some Continuous Field Quantizations, Equivalent to the C*-Weyl Quantization. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 41(113-138).
  12. Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras. Symmetry, Integrability and Geometry: Methods and Applications, 4:047–084.
  13. Fundamentals of the Theory of Operator Algebras. American Mathematical Society, Providence, RI.
  14. Operations on continuous bundles of C*-algebras. Mathematische Annalen, 303:677–697.
  15. Lance, C. (1976). Tensor Products of Non-Unital C*-Algebras. Journal of the London Mathematical Society, 2(12):160–168.
  16. Lance, E. C. (1995). Hilbert C*-Modules. Cambridge University Press, Cambridge.
  17. Landsman, N. P. (1993). Strict deformation quantization of a particle in external gravitational and Yang-Mills fields. Journal of Geometry and Physics, 12:93–132.
  18. Landsman, N. P. (1995). Rieffel induction as generalized quantum Marsden-Weinstein reduction. Journal of Geometry and Physics, 15:285–319.
  19. Landsman, N. P. (1998a). Mathematical Topics Between Classical and Quantum Mechanics. Springer, New York.
  20. Landsman, N. P. (1998b). Twisted Lie Group C*-Algebras as Strict Quantizations. Letters in Mathematical Physics, 46:181–188.
  21. Landsman, N. P. (1999). Lie Groupoid C*-algebras and Weyl Quantization. Communications in Mathematical Physics, 206(2):367–381.
  22. Landsman, N. P. (2001a). Bicategories of operator algebras and Poisson manifolds. In Mathematical physics in mathematics and physics. Quantum and operator algebraic aspects. Proceedings of a conference, Siena, Italy, June 20–24, 2000. Dedicated to Sergio Doplicher and John E. Roberts on the occasion of their 60th birthday, pages 271–286. AMS, American Mathematical Society, Providence, RI.
  23. Landsman, N. P. (2001b). Operator Algebras and Poisson Manifolds Associated to Groupoids. Communications in Mathematical Physics, 222:97–116.
  24. Landsman, N. P. (2001c). Quantized reduction as a tensor product. In Quantization of Singular Symplectic Quotients, pages 137–180. Birkhäuser, Basel.
  25. Landsman, N. P. (2003). Quantization as a Functor. In Voronov, T., editor, Quantization, Poisson Brackets and beyond, pages 9–24. Contemp. Math., 315, AMS, Providence.
  26. Landsman, N. P. (2005). Functorial Quantization and the Guillemin–Sternberg Conjecture. In Twenty Years of Bialowieza: A Mathematical Anthology, pages 23–45. World Scientific, Singapore.
  27. Landsman, N. P. (2007). Between Classical and Quantum. In Butterfield, J. and Earman, J., editors, Handbook of the Philosophy of Physics, volume 1, pages 417–553. Elsevier, New York.
  28. Mackey, G. W. (1968). Induced Representations of Groups and Quantum Mechanics. W. A. Benjamin. Inc., New York.
  29. Reduction of symplectic manifolds with symmetry. Reports on Mathematical Physics, 5(1):121–30.
  30. Nestruev, J. (2003). Smooth Manifolds and Observables. Springer-Verlag, New York.
  31. Morita Equivalence and Continuous-Trace C*-Algebras, volume 60. American Mathematical Society, Providence.
  32. Rieffel, M. (1974a). Induced Representations of C*-Algebras. Advances in Mathematics, 13:176–257.
  33. Rieffel, M. (1974b). Morita Equivalence for C*-Algebras and W*-Algebras. Journal of Pure and Applied Algebra, 5:51–96.
  34. Rieffel, M. (1978). Induced Representations of C*-Algebras. Bulletin of the American Mathematical Society, 4:606–609.
  35. Rieffel, M. (1989). Deformation Quantization of Heisenberg manifolds. Communications in Mathematical Physics, 122:531–562.
  36. Rieffel, M. (1993). Deformation quantization for actions of ℝdsuperscriptℝ𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT. Memoirs of the American Mathematical Society. American Mathematical Society, Providence, RI.
  37. Extensions of bundles of C*-algebras. Reviews in Mathematical Physics, 33(8):2150025.
  38. Is the classical limit “singular”? Studies in History and Philosophy of Science Part A, 88:263–279.
  39. Sternberg, S. (1977). Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field. Proceedings of the National Academy of the Sciences, 74(12):5253–4.
  40. van Nuland, T. D. H. (2019). Quantization and the resolvent algebra. Journal of Functional Analysis, 277(8):2815–2838.
  41. Weinstein, A. (1983). The local structure of Poisson manifolds. Journal of Differential Geometry, 18(3).
  42. Weinstein, M. (1978). A universal phase space for particles in Yang-Mills fields. Letters in Mathematical Physics, 2:417–20.
  43. Xu, P. (1991). Morita Equivalence of Poisson Manifolds. Communications in Mathematical Physics, 142:493–509.
  44. Xu, P. (1992). Morita equivalence and symplectic realizations of Poisson manifolds. Annales scientifiques de l'École normale supérieure, 25(3):307–333.
  45. Xu, P. (1994). Classical Intertwiner Space and Quantization. Communications in Mathematical Physics, 164:473–488.
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Dice Question Streamline Icon: https://streamlinehq.com

Follow-up Questions

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets