Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 047, 37 pages      arXiv:0805.4536      http://dx.doi.org/10.3842/SIGMA.2008.047
Contribution to the Special Issue on Deformation Quantization

Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras

Reinhard Honegger, Alfred Rieckers and Lothar Schlafer
Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany

Received December 20, 2007, in final form May 06, 2008; Published online May 29, 2008

Abstract
C*-algebraic Weyl quantization is extended by allowing also degenerate pre-symplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-*-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-*-algebras instead of C*-algebras. Fourier transformation and representation theory of the measure Banach-*-algebras are combined with the theory of continuous projective group representations to arrive at the genuine C*-algebraic strict deformation quantization in the sense of Rieffel and Landsman. Weyl quantization is recognized to depend in the first step functorially on the (in general) infinite dimensional, pre-symplectic test function space; but in the second step one has to select a family of representations, indexed by the deformation parameter h. The latter ambiguity is in the present investigation connected with the choice of a folium of states, a structure, which does not necessarily require a Hilbert space representation.

Key words: Weyl quantization for infinitely many degrees of freedom; strict deformation quantization; twisted convolution products on measure spaces; Banach-*- and C*-algebraic methods; partially universal representations.

pdf (519 kb)   ps (306 kb)   tex (51 kb)

References

  1. Emch G.G., Algebraic methods in statistical mechanics and quantum field theory, John Wiley and Sons, New York, 1972.
  2. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics, Vol. II, Springer, New York, 1981.
  3. Reed S., Simon B., Fourier analysis, self-adjointness, Vol. II, Academic Press, New York, 1975.
  4. Weyl H., Quantenmechanik und Gruppentheorie, Z. Phys. 46 (1928), 1-46.
  5. Weyl H., The theory of groups and quantum mechanics, Methuen, London, 1931 (reprinted by Dover Publ., New York, 1950).
  6. Putnam C.R., Commutation properties of Hilbert space operators and related topics, Springer, New York - Berlin - Heidelberg, 1967.
  7. Galindo A., Pascual P., Quantum mechanics, Vols. I, II, Springer, New York - Berlin, 1989, 1991.
  8. Honegger R., On Heisenberg's uncertainty principle and the CCR, Z. Naturforsch. A 48 (1993), 447-451.
  9. von Neumann J., Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931), 570-578.
  10. Rieffel M.A., Quantization and C*-algebras, in "C*-Algebras: 1943-1993", Editor R.S. Doran, Contemp. Math. 167 (1994), 67-97.
  11. Honegger R., Rieckers A., Some continuous field quantizations, equivalent to the C*-Weyl quantization, Publ. RIMS Kyoto Univ. 41 (2005), 113-138.
  12. Garcia-Bondía J.M., Varilly C., Algebras of distributions suitable for phase-space quantum mechanics I, J. Math. Phys. 29 (1988), 869-879.
  13. Kastler D., The C*-algebras of a free Boson field, Comm. Math. Phys. 1 (1965), 14-48.
  14. Binz E., Honegger R., Rieckers A., Field-theoretic Weyl quantization as a strict and continuous deformation quantization, Ann. Henri Poincaré 5 (2004), 327-346.
  15. Loudon R., The quantum theory of light, Clarendon Press, Oxford, 1979.
  16. Cohen-Tannoudji C., Dupont-Roc J., Grynberg G., Photons & atoms, introduction to QED, John Wiley & Sons, New York - Toronto - Singapore, 1989.
  17. Schwarz G., Hodge decompositions - a method for solving boundary value problems, Lecture Notes in Mathematics, Vol. 1607, Springer, Berlin - New York, 1995.
  18. Buchholz D., Mack G., Todorov I., The current algebra on the circle as a germ of the local field theories, Nuclear Phys. B 5 (1988), 20-56.
  19. Bayen F., Flato M., Fronsdal C., Lichnerovicz A., Sternheimer D., Deformation theory and quantization, Ann. Phys. 111 (1978), 61-151.
  20. Dito J., Star-products and nonstandard quantization for the Klein-Gordon equation, J. Math. Phys. 33 (1992), 791-801.
  21. Dito J., An example of cancellation of infinities in the star-product quantization of fields, Lett. Math. Phys. 27 (1993), 73-80.
  22. Dütsch M., Fredenhagen K., Algebraic quantum field theory, perturbation theory, and the loop expansion, Comm. Math. Phys. 219 (2001), 5-30, hep-th/0001129.
  23. Sternheimer D., Alcade C.A., Analytic vectors, anomalies and star representations, Lett. Math. Phys. 17 (1989), 117-127.
  24. Fronsdal C., Normal ordering and quantum groups, Lett. Math. Phys. 22 (1991), 225-228.
  25. Dito J., Star-product approach to quantum field theory: the free scalar field, Lett. Math. Phys. 20 (1990), 125-134.
  26. Rieffel M.A., Deformation quantization for actions of Rd, Mem. Amer. Math. Soc. 106 (1993), no. 506.
  27. Rieffel M.A., Questions on quantization, in Operator Algebras and Operator Theory (1997, Shanghai), Contemp. Math. 228 (1998), 315-326, quant-ph/9712009.
  28. Landsman N.P., Strict quantization of coadjoint orbits, J. Math. Phys. 39 (1998), 6372-6383, math-ph/9807027.
  29. Landsman N.P., Mathematical topics between classical and quantum mechanics, Springer, New York, 1998.
  30. Manuceau J., Sirugue M., Testard D., Verbeure A., The smallest C*-algebra for canonical commutation relations, Comm. Math. Phys. 32 (1973), 231-243.
  31. Binz E., Honegger R., Rieckers A., Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic form, J. Math. Phys. 45 (2004), 2885-2907.
  32. Hewitt E., Ross K.A., Abstract harmonic analysis, Vols. I, II, Springer, New York, 1963, 1970.
  33. Abraham R., Marsden J.E., Foundations of mechanics, 2nd ed., Benjamin/Cummings Publishing Company, London, Amsterdam, 1978.
  34. Arnold V.I., Mathematical methods of classical mechanics, Springer, New York, 1985.
  35. Libermann P., Marle C.-M., Symplectic geometry and analytical mechanics, D. Reidel Publ. Co, Dordrecht, 1987.
  36. Binz E., Sniatycki J., Fischer H., Geometry of classical fields, Mathematics Studies, Vol. 154, North Holland, Amsterdam, 1988.
  37. Marsden J.E., Ratiu T., Introduction to mechanics and symmetry, Springer, New York, Berlin, 1994.
  38. Schaefer H.H., Topological vector spaces, Macmillan Company, New York, 1966.
  39. Conway J.B., A course in functional analysis, Springer, New York, 1985.
  40. Cohn D.L., Measure theory, Birkhäuser, Boston, 1980.
  41. Edwards C.M., Lewis J.T., Twisted group algebras I, II, Comm. Math. Phys. 13 (1969), 119-141.
  42. Busby R.C., Smith H.A., Representations of twisted group algebras, Trans. Amer. Math. Soc. 149 (1970), 503-537.
  43. Dixmier J., C*-algebras, North-Holland, Amsterdam, 1977.
  44. Pedersen G.K., C*-algebras and their automorphism groups, Academic Press, London, 1979.
  45. Packer J.A., Raeburn I., Twisted crossed products of C*-algebras, Math. Proc. Camb. Phil. Soc. 106 (1989), 293-311.
  46. Grundling H., A group algebra for inductive limit groups, continuity problems of the canonical commutation relations, Acta Appl. Math. 46 (1997), 107-145.
  47. Haag R., Kadison R.V., Kastler D., Nets of C*-algebras and classification of states, Comm. Math. Phys. 16 (1970), 81-104.
  48. Sewell G.L., States and dynamics of infinitely extended physical systems, Comm. Math. Phys. 33 (1973), 43-51.
  49. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras, Vols. I, II, Academic Press, New York, 1983, 1986.
  50. Honegger R., Global cocycle dynamics for infinite mean field quantum systems interacting with the Boson gas, J. Math. Phys. 37 (1996), 263-282.
  51. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics, Vol. I, Springer, New York, 1987.
  52. Honegger R., Rieckers A., Partially classical states of a Boson field, Lett. Math. Phys. 64 (2003), 31-44.
  53. Robinson P.L., Symplectic pathology, Quart. J. Math. Oxford 44 (1993), 101-107.
  54. Honegger R., On the continuous extension of states on the CCR-algebra, Lett. Math. Phys. 42 (1997), 11-25.
  55. Honegger R., Enlarged testfunction spaces for the global free folia dynamics on the CCR-algebra, J. Math. Phys. 39 (1998), 1153-1169.
  56. Segal I.E., Distributions in Hilbert space and canonical systems of operators, Trans. Amer. Math. Soc. 88 (1958), 12-41.
  57. Hörmann G., Regular Weyl-systems and smooth structures on Heisenberg groups, Comm. Math. Phys. 184 (1997), 51-63.
  58. Segal I.E., Representations of the canonical commutation relations, in Cargese Lectures in Theoretical Physics, Gordon and Breach, 1967, 107-170.
  59. Folland G., Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122, Princeton University Press, Princeton, 1989.
  60. Fell J.M.G., Doran R.S. Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles, Pure and Applied Mathematics, Vol. 126, Academic Press, New York, London, 1988.
  61. Takesaki M., Theory of operator algebras, Vol. I, Springer, New York, 1979.

Previous article   Next article   Contents of Volume 4 (2008)