Skip to main content
SpringerLink
Log in

Quantum Mechanics and Representation Theory: The New Synthesis

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

Quantum mechanics and representation theory, in the sense of unitary representations of groups on Hilbert spaces, were practically born together between 1925–1927, and have continued to enrich each other till the present day. Following a brief historical introduction, we focus on a relatively new aspect of the interaction between quantum mechanics and representation theory, based on the use of K-theory of C *-algebras. In particular, the study of the K-theory of the reduced C *-algebra of a locally compact group (which for a compact group is just its representation ring) has culminated in two fundamental conjectures, which are closely related to quantum theory and index theory, namely the Baum–Connes conjecture and the Guillemin–Sternberg conjecture. Although these conjectures were both formulated in 1982, and turn out to be closely related, so far there has been no interplay between them whatsoever, either mathematically or sociologically. This is presumably because the Baum–Connes conjecture is nontrivial only for noncompact groups, with current emphasis entirely on discrete groups, whereas the Guillemin–Sternberg conjecture has so far only been stated for compact Lie groups. As an elementary introduction to both conjectures in one go, indicating how the latter can be generalized to the noncompact case, this paper is a modest attempt to change this state of affairs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abraham, R. and Marsden, J. E.: Foundations of Mechanics, 2nd edn, revised and enlarged. With the assistance of Tudor Ratiu and Richard Cushman, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, MA, 1978.

    Google Scholar 

  2. Atiyah, M. F.: Elliptic operators, discrete groups and von Neumann algebras, In: Colloque Analyse et Topologie en l'Honneur de Henri Cartan (Orsay, 1974), Asterisque 32–33, Soc. Math. France, Paris, 1976, pp. 43–72.

    Google Scholar 

  3. Atiyah, M. F.: The Dirac equation and geometry, In: Paul Dirac, Cambridge Univ. Press, Cambridge, 1998, pp. 108–124.

    Google Scholar 

  4. Atiyah, M. and Schmid, W.: A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62.

    Google Scholar 

  5. Baum, P. and Connes, A.: Geometric K-theory for Lie groups and foliations, Enseign. Math. (2) 46 (2000), 3–42.

    Google Scholar 

  6. Baum, P., Connes, A. and Higson, N.: Classifying space for proper actions and K-theory of group C *-algebras, In: C*-Algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291.

    Google Scholar 

  7. Blackadar, B.: K-Theory for Operator Algebras, 2nd edn, Mathematical Sciences Research Institute Publications 5, Cambridge Univ. Press, Cambridge, 1998.

    Google Scholar 

  8. Braverman, M.: Vanishing theorems for the kernel of a Dirac operator, arXiv:math.DG/9805127.

  9. Carter, R., Segal, G. and Macdonald, I.: Lectures on Lie Groups and Lie Algebras, London Math. Soc. Student Texts 32, Cambridge Univ. Press, Cambridge, 1995.

    Google Scholar 

  10. Chabert, J., Echterhoff, S. and Nest, R.: The Connes-Kasparov conjecture for almost connected groups, arXiv:math.OA/0110130.

  11. Connes, A.: Noncommutative Geometry, Academic Press, Inc., San Diego, CA, 1994.

    Google Scholar 

  12. Connes, A. and Moscovici, H.: The L 2-index theorem for homogeneous spaces of Lie groups, Ann. of Math. (2) 115 (1982), 291–330.

    Google Scholar 

  13. Connes, A. and Moscovici, H.: Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), 345–388.

    Google Scholar 

  14. Dirac, P. A. M.: Lectures on Quantum Mechanics, Belfer School of Science, Yeshiva University, New York, 1964.

    Google Scholar 

  15. Duistermaat, J. J.: The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Progress in Nonlinear Differential Equations and their Applications 18, Birkhäuser, Boston, MA, 1996.

    Google Scholar 

  16. Gracía-Bondia, J. M., Várilly, J. C. and Figueroa, H.: Elements of Noncommutative Geometry, Birkhäuser, Boston, MA, 2001.

    Google Scholar 

  17. Guillemin, V., Ginzburg, V. and Karshon, Y.: Moment Maps, Cobordisms, and Hamiltonian Group Actions, Math. Surveys Monographs 98, Amer. Math. Soc., Providence, RI, 2002.

    Google Scholar 

  18. Guillemin, V. and Sternberg, S.: Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515–538.

    Google Scholar 

  19. Higson, N. and Roe, J.: Analytic K-Homology, Oxford Math. Monographs, Oxford Science Publications, Oxford Univ. Press, Oxford, 2000.

    Google Scholar 

  20. Kasparov, G. G.: The operator K-functor and extensions of C *-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 571–636, 719 (Russian).

    Google Scholar 

  21. Kasparov, G. G.: Index of invariant elliptic operators, K-theory and representations of Lie groups, Dokl. Akad. Nauk SSSR 268 (1983), 533–537 (Russian).

    Google Scholar 

  22. Kasparov, G. G.: Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988), 147–201.

    Google Scholar 

  23. Kirillov, A. A.: Unitary representations of nilpotent Lie groups, Uspekhi Mat. Nauk 17(106) (1962), 57–110 (Russian).

    Google Scholar 

  24. Kirillov, A. A.: Merits and demerits of the orbit method, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 433–488.

    Google Scholar 

  25. Kostant, B.: Orbits, symplectic structures and representation theory, In: 1966 Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965), p. 71.

  26. Landsman, N. P.: Mathematical Topics Between Classical and Quantum Mechanics, Springer Monographs in Mathematics, Springer-Verlag, New York, 1998.

    Google Scholar 

  27. Landsman, N. P.: Deformation quantization and the Baum-Connes conjecture, Comm. Math. Phys. 237 (2003), 87–103.

    Google Scholar 

  28. Lafforgue, V.: K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. Math. 149 (2002), 1–95.

    Google Scholar 

  29. Lafforgue, V.: Banach KK-theory and the Baum-Connes conjecture, In: Proceedings of the International Congress of Mathematicians (Beijing, 2002), Vol. II, Higher Ed. Press, Beijing, 2002, pp. 795–812.

    Google Scholar 

  30. Lück, W.: The relation between the Baum-Connes conjecture and the trace conjecture, Invent. Math. 149 (2002), 123–152.

    Google Scholar 

  31. Mackey, G. W.: Unitary Group Representations in Physics, Probability, and Number Theory, Mathematics Lecture Note Series 55, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1978.

    Google Scholar 

  32. Meinrenken, E.: Symplectic surgery and the Spin C-Dirac operator, Adv. Math. 134 (1998), 240–277.

    Google Scholar 

  33. Meinrenken, E. and Sjamaar, R.: Singular reduction and quantization, Topology 38 (1999), 699–762.

    Google Scholar 

  34. Muller, F. A.: The equivalence myth of quantum mechanics I, II, Stud. Hist. Phil. Mod. Phys. 28 (1997), 35–61, 219–247.

    Google Scholar 

  35. Paradan, P.-E.: Localization of the Riemann-Roch character, J. Funct. Anal. 187 (2001), 442–509.

    Google Scholar 

  36. Paradan, P.-E.: Spin c quantization and the K-multiplicities of the discrete series, arXiv:math.DG/0103222.

  37. Parthasarathy, R.: Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30.

    Google Scholar 

  38. Penington, M. G. and Plymen, R. J.: The Dirac operator and the principal series for complex semisimple Lie groups, J. Funct. Anal. 53 (1983), 269–286.

    Google Scholar 

  39. Raeburn, I. and Williams, D. P.: Morita Equivalence and Continuous-Trace C *-Algebras, Math. Surveys Monographs 60, Amer. Math. Soc., Providence, RI, 1998.

    Google Scholar 

  40. Rieffel, M. A.: Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 122 (1989), 531–562.

    Google Scholar 

  41. Rosenberg, J.: K-theory of group C *-algebras, foliation C *-algebras, and crossed products, In: Index Theory of Elliptic Operators, Foliations, and Operator Algebras (New Orleans, LA/Indianapolis, IN, 1986), Contemp. Math. 70, Amer. Math. Soc., Providence, RI, 1988, pp. 251–301.

    Google Scholar 

  42. Sjamaar, R. and Lerman, E.: Stratified symplectic spaces and reduction, Ann. Math. 134 (1991), 375–422.

    Google Scholar 

  43. Souriau, J.-M.: Quantification géométrique, Comm. Math. Phys. 1 (1966), 374–398.

    Google Scholar 

  44. Upmeier, H.: Toeplitz Operators and Index Theory in Several Complex Variables, Operator Theory: Advances and Applications 81, Birkhäuser, Basel, 1996.

    Google Scholar 

  45. Valette, A.: Introduction to the Baum-Connes Conjecture, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2002.

  46. Mislin, G. and Valette, A.: Proper Group Actions and the Baum-Connes Conjecture, Advanced Courses in Mathematics, Birkhäuser, Basel, 2003.

  47. von Neumann, J.: Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931), 570–578.

    Google Scholar 

  48. Wassermann, A.: Une démonstration de la conjecture de Connes-Kasparov pour les groupes de Lie linéaires connexes réductifs, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 559–562.

    Google Scholar 

  49. Weinstein, A.: The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), 523–557.

    Google Scholar 

  50. Wigner, E. P.: Unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40 (1939), 149–204.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Korteweg–de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, NL-1018 TV, Amsterdam, The Netherlands

    N. P. Landsman

Authors
  1. N. P. Landsman
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Landsman, N.P. Quantum Mechanics and Representation Theory: The New Synthesis. Acta Applicandae Mathematicae 81, 167–189 (2004). https://doi.org/10.1023/B:ACAP.0000024199.14167.44

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:ACAP.0000024199.14167.44

Search

Navigation