Fermionic Second Quantization and the Geometry of the Restricted Grassmannian

  • Tilmann Wurzbacher
Part of the DMV Seminar Band book series (OWS, volume 31)


We explain how fermionic second quantization leads to G res, the restricted Grassmannian of a polarized Hilbert space, and its homogeneous Kähler geometry, and how vice-versa G res encodes - via its holomorphic determinant bundle - the basic ingredients of fermionic second quantization, as, e.g., the fermionic Fock space and the “Schwinger term”. Using this approach we derive a new construction of the universal central extension of U res, the restricted unitary group. Furthermore, we develop the general theory of symplectic manifolds and symplectic actions in infinite dimensions and apply it notably to the U res -action on G res. Finally we construct the determinant bundle on G res functorially using “C*-algebro-geometric methods” naturally arising from the use of CAR-algebras in fermionic second quantization.


Line Bundle Frechet Space Determinant Bundle Symplectic Action Canonical Anticommutation Relation 
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  1. [Ar]
    H. Araki, Bogoliubov automorphisms and Fock representations of canonical anticommutation relations, in: Operator algebras and mathematical physics (Iowa City 1985), 23–141, Contemp. Math. 62, Amer. Math. Soc., Providence, R.I., 1987.CrossRefGoogle Scholar
  2. [AB]
    M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 no. 1505 (1983), 523–615.MathSciNetCrossRefGoogle Scholar
  3. [Au]
    M. Audin, Lectures on gauge theory and integrable systems, in: Gauge theory and symplectic geometry (Montreal, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 488, 1–48 Kluwer, Dordrecht, 1997.Google Scholar
  4. [Ba]
    V. Bargmann, On unitary ray representations of continuous groups, Ann. Of Math. 59 (1954), 1–46.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Beg]
    E.J. Beggs, The de Rham complex on infinite dimensional manifolds, Quart. J. Math. Oxford (2), 38 (1987), 131–154.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Ber]
    F.A. Berezin, The method of second quantization, Academic Press, Orlando, 1966.zbMATHGoogle Scholar
  7. [BW]
    B. Booß-Bavnbek and K.P. Wojciechowski, Elliptic boundary problems for Dirac operators, Birkhauser, Boston, 1993.CrossRefzbMATHGoogle Scholar
  8. [BSW]
    B. Booß-Bavnbek, S.G. Scott and K.P. Wojciechowski, Elliptic boundary problems for Dirac operators II - The heat kernel and determinants, Grassmannians, and Dirac operators on manifolds with boundary, Birkhauser, Boston, in preparation.Google Scholar
  9. [Bo]
    N. Bourbaki, Élements de mathématique. Fasc. XXX VII Groupes et algèbras de Lie. Chapitre III: Groupes de Lie, Actualites scientifiques et industrielles, No. 1349, Hermann, Paris, 1972.Google Scholar
  10. [BR]
    O. Bratteli and D.W. Robinson, Operator algebras and quantum statistical mechanics. II. Equilibrium states. Models in quantum statistical mechanics,Springer, Berlin, 1981.zbMATHGoogle Scholar
  11. [Br]
    R.L. Bryant, An introduction to Lie groups and symplectic geometry, in: Geometry and quantum field theory (Park City, 1991), Ed.D. Freed and K. Uhlenbeck, 5–181, IAS/Park City Math. Ser. 1, Amer. Math. Soc., Providence RI, 1995.Google Scholar
  12. [CH]
    A.L. Carey and C.A. Hurst, A note on the boson-fermion correspondence and infinite dimensional groups, Commun. Math. Phys. 98 (1985), 435–448.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [CHO’B]
    A.L. Carey, C.A. Hurst and D.M. O’Brien, Automorphisms of the canonical anticommutation relations and index theory, J. Funct. Anal. 48 (1982), 360–393.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [CMM]
    A.L. Carey, J. Mickelsson and M.K. Murray, Bundle gerbes applied to quantum field theory, Reviews in Mathematical Physics Vol. 12, No. 1 (2000), 65–90.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [CR]
    A.L. Carey and S. N. M. Ruijsenaars, On fermion gauge groups, current algebras and Kac-Moody algebras, Acta Appl. Math. 10 (1987), no. 1, 1–86.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Con]
    A. Connes, Noncommutative geometry,Academic Press, San Diego, 1994.zbMATHGoogle Scholar
  17. [Coo]
    J.M. Cook, The mathematics of second quantization, Trans. of the Amer. Math. Soc. 74 (1953), 222–245.CrossRefzbMATHGoogle Scholar
  18. [DD]
    J. Dixmier and A. Douady, Champs confirms d’espaces hilbertiens et de C*-algèbres, Bull. Soc. Math. France 91 (1963), 227–284.MathSciNetzbMATHGoogle Scholar
  19. [Don]
    S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3) 50, no. 1 (1985), 1–26.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [Dou]
    R.G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York London, 1972.zbMATHGoogle Scholar
  21. [E]
    J. Eells, On the geometry of function spaces, in: Symposium internacional de topología algebraica, 303–308, Universidad Nacional Autßnoma de México y UNESCO, Mexico City, 1958.Google Scholar
  22. [EE]
    J. Eells and K. D. Elworthy, Open embeddings of certain Banach manifolds, Ann. of Math., Sec. Ser. 91 (1970), 465–485.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Fo]
    V. Fock, Konfigurationsraum and zweite Quantelung, Zeitschrift für Physik 75 (1932), 622–647.CrossRefGoogle Scholar
  24. [Fr]
    K.O. Friedrichs, Mathematical aspects of the quantum theory of fields, New York, Interscience, 1953.zbMATHGoogle Scholar
  25. [FG]
    J. Fröhlich and K. Gawędzki, Conformal field theory and geometry of strings, in: Mathematical quantum field theory I: field theory and many-body theory (CRM proceedings & lecture notes), ed. J. Feldman, R. Froese, L. Rosen, 57–97, American. Math. Soc., Providence RI, 1994.Google Scholar
  26. [FGR]
    J. Fröhlich, O. Grandjean and A. Recknagel, Supersymmetric quantum theory, non-commutative geometry, and gravitation, in: Symetries quantiques (Les Houches, 1995), 221–385, North-Holland, Amsterdam, 1998.Google Scholar
  27. [Fuj]
    A. Fujiki, Moduli space of polarized algebraic manifolds and Kahler metrics, Sugaku expositions 5 (1992), n.2, 173–191.MathSciNetGoogle Scholar
  28. [Fuk]
    D.B. Fuks, Cohomology of infinite-dimensional Lie algebras, Consultants Bureau, New York, 1986.zbMATHGoogle Scholar
  29. [Ga]
    S.A. Gaal, Linear analysis and representation theory, Springer, Berlin, 1973.CrossRefzbMATHGoogle Scholar
  30. [Gl]
    H. Glockner, Infinite-dimensional complex groups and semigroups: representations of cones, tubes and conelike semigroups, Dissertation TU Darmstadt, 2000.Google Scholar
  31. [GHL]
    S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry, Sec. Ed., Springer, Berlin, 1990.CrossRefzbMATHGoogle Scholar
  32. [GM]
    H. Grosse, and W. Maderner, On the classical origin of the fermionic Schwinger term Lett. Math. Phys. 31 (1994), no. 1, 57–64.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [GS]
    V. Guillemin and S. Sternberg Symplectic techniques in physics, Cambridge University Press, Cambridge New York, 1984.zbMATHGoogle Scholar
  34. [Ha]
    R.S. Hamilton, The inverse function theorem of Nash and Moser, Bull. of the AMS (N.S.) Vol.7, No.1 (1982), 65–222.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [d1HJ]
    P. de la Harpe and V. Jones, An introduction to C*-algebras. Chapters 1–9, Publication de l’Université de Genève, 1995.Google Scholar
  36. [HeHu]
    P. Heinzner and A.T. Huckleberry, Kählerian potentials and convexity properties of the moment map, Invent. Math. 126 (1996), no. 1, 65–84.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [HeHe]
    G.F. Helminck and A. G. Helminck, The structure of Hilbert flag varieties, Publ. Res. Inst. Math. Sci. 30 (1994), no. 3, 401–441.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [He]
    M. Hervé, Analyticity in infinite-dimensional spaces,de Gruyter Studies in Mathematics 10, Walter de Gruyter, Berlin New York, 1989.CrossRefzbMATHGoogle Scholar
  39. [I]
    D. Iagolnitzer The S-matrix, North-Holland, Amsterdam New York, 1978.Google Scholar
  40. [J]
    R. Jackiw, Topological investigations of quantized gauge theories. Notes by R. Young, in: Relativity,groups and topology II, (Les Houches, 1983), Ed. B.S. DeWitt and R. Stora, 221–331, North-Holland, Amsterdam New York, 1984.Google Scholar
  41. [Ke]
    H.H. Keller, Differential calculus in locally convex spaces, Lect. Notes Math. 417, Springer, Berlin New York, 1974.zbMATHGoogle Scholar
  42. [Ki]
    A.A. Kirillov, Geometric quantization, in: Dynamical systems. IV. Symplectic geometry and its applications, Eds. V.I. Arnol’d and S.P. Novikov, 137–172, Encycl. Math. Sci. 4, Springer, Berlin, 1990.Google Scholar
  43. [Ko]
    B. Kostant, Quantization and unitary representations I: Prequantization, in: Lectures in Modern Analysis and Applications III,Lect. Notes Math. 170, 87–207, Springer, Berlin, 1970.CrossRefGoogle Scholar
  44. [KM]
    A. Kriegl and P. Michor, The convenient setting of global analysis, Mathematical Surveys and Monographs 53, Amer. Math. Soc., Providence RI, 1997.zbMATHGoogle Scholar
  45. [Ku]
    N.H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology 3 (1965), 19–30.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [La]
    E. Langmann, On anomalies and noncommutative geometry, in: Low-dimen-sional models in statistical physics and quantum field theory (Schladming, 1995), 291–297, Lect. Notes Phys. 469, Springer, Berlin, 1996.Google Scholar
  47. [LM1]
    E. Langmann and J. Mickelsson, (3+1)-dimensional Schwinger terms and noncommutative geometry, Phys. Lett. B 338 (1994), no. 2–3, 241–248.MathSciNetGoogle Scholar
  48. [LM2]
    E. Langmann and J. Mickelsson, Scattering matrix in external field problems, J. Math. Phys. 37 (1996), no. 8, 3933–3953.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [Lu]
    L.-E. Lundberg, Quasi-free “second quantization”, Commun. Math. Phys. 50 (1976), 103–112.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [Mic]
    J. Mickelsson, Current algebras and groups,Plenum Monographs in Nonlinear Physics, Plenum Press, New York London, 1989.zbMATHGoogle Scholar
  51. [MR]
    J. Mickelsson and S.G. Rajeev, Current algebras in d + 1-dimensions and determinant bundles over infinite-dimensional Grassmannians, Commun. Math. Phys. 116 (1988), no. 3, 365–400.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [Mil1]
    J. Milnor, Morse theory. Based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963.zbMATHGoogle Scholar
  53. [Mil2]
    J. Milnor, Remarks on infinite-dimensional Lie groups, in: Relativity,groups and topology II (Les Houches, 1983), Ed. B.S. DeWitt and R. Stora, 1007–1057, North-Holland, Amsterdam New York, 1984.Google Scholar
  54. [MFK]
    D. Mumford and J. Fogarty and F. Kirwan, Geometric invariant theory, Third edition, Ergebnisse der Mathematik and ihrer Grenzgebiete 34, Springer, Berlin, 1994.CrossRefGoogle Scholar
  55. [Muj]
    J. Mujica, Complex analysis in Banach spaces,North-Holland, Amsterdam, 1986.zbMATHGoogle Scholar
  56. [Mur]
    G.J. Murray, C*- algebras and operator theory, Academic Press, Boston, 1990.Google Scholar
  57. [N1]
    K.H. Neeb, Infinite dimensional groups and their representations, in: this volume. Google Scholar
  58. [N2]
    K.H. Neeb, Borel-Weil theory for loop groups, in: this volume. Google Scholar
  59. [N3]
    K.H. Neeb, Central extensions of infinite-dimensional Lie groups, Preprint 2084, TU Darmstadt, 2000.Google Scholar
  60. [N4]
    K.H. Neeb, Lecture Notes: Infinite-dimensional groups and their representations, Preprint, TU Darmstadt, 2000.Google Scholar
  61. [Ov]
    V.Y. Ovsienko, Coadjoint representation of Virasoro-type Lie algebras and differential operators on tensor-densities, in: this volume. Google Scholar
  62. [Ot]
    J. T. Ottesen, Infinite dimensional groups and algebras in quantum physics, Lect. Notes in Physics, m 27, Springer, Berlin, 1995.zbMATHGoogle Scholar
  63. [Pa]
    R. Palais, On the homotopy type of certain groups of operators, Topology 3 (1965), 271–279.MathSciNetCrossRefzbMATHGoogle Scholar
  64. [Pe]
    M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley, Reading, MA, 1995.Google Scholar
  65. [Pil]
    D. Pickrell, Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal. 70, no. 2 (1987), 323–356.MathSciNetCrossRefzbMATHGoogle Scholar
  66. [Pi2]
    D. Pickrell, On the support of quasi-invariant measures on infinite-dimensional Grassmann manifolds, Proc. Amer. Math. Soc. 100, no. 1 (1987), 111–116.MathSciNetCrossRefzbMATHGoogle Scholar
  67. [PR]
    R.J. Plymen and P.L. Robinson, Spinors in Hilbert space, Cambridge University Press, Cambridge, 1994.zbMATHGoogle Scholar
  68. [PoSt]
    R.T. Powers and E. Stormer, Free states of the canonical anticommutation relations, Commun. Math. Phys. 16 (1970), 1–33.MathSciNetCrossRefzbMATHGoogle Scholar
  69. [PrSe]
    A. Pressley and G. Segal, Loop groups, Oxford University Press, New York, 1986.zbMATHGoogle Scholar
  70. [RS1]
    M. Reed and B. Simon, Methods of modern mathematical physics. I: Functional analysis, Academic Press, New York, 1972.Google Scholar
  71. [RS2]
    M. Reed and B. Simon, Methods of modern mathematical physics. II: Fourier analysis,self-adjointness, Academic Press, New York, 1975.zbMATHGoogle Scholar
  72. [RS3]
    M. Reed and B. Simon, Methods of modern mathematical physics. III: Scattering theory, Academic Press, New York, 1979.zbMATHGoogle Scholar
  73. [RS4]
    M. Reed and B. Simon, Methods of modern mathematical physics. IV: Analysis of operators, Academic Press, New York, 1978.zbMATHGoogle Scholar
  74. [Re]
    A.G. Reznikov, Characteristic classes in symplectic topology. Appendix D by L. Katzarkov, Selecta Math. (N.S.) 3 (1997), no. 4, 601–642.MathSciNetCrossRefzbMATHGoogle Scholar
  75. [Rud]
    W. Rudin, Functional analysis,McGraw-Hill, New York, 1973.zbMATHGoogle Scholar
  76. [Rui]
    S. N. M. Ruijsenaars, Index theorems and anomalies: a common playground for mathematicians and physicists, CWI Quarterly 3 (1990) no. 1, 3–19.MathSciNetzbMATHGoogle Scholar
  77. [Sa]
    M. Sato, The KP hierarchy and infinite-dimensional Grassmann manifolds, in: Theta functions - Bowdoin 1987, Part 1 (Brunswick, 1987), 51–66, Proc. Sympos. Pure Math. 49, Part 1, Amer. Math. Soc., Providence, RI, 1989.Google Scholar
  78. [Sc]
    J. Schwinger, Field theoretic commutators, Physical reviews letters,Vol. 3, n.6 (1959), 296–297.CrossRefGoogle Scholar
  79. [Se]
    G. Segal, The definition of conformal field theory, unpublished manscript.Google Scholar
  80. [SeWi]
    G. Segal and G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. No. 61 (1985), 5–65.MathSciNetCrossRefzbMATHGoogle Scholar
  81. [Si]
    B. Simon, Trace ideals and their applications,Cambridge University Press, Cambridge, 1979.zbMATHGoogle Scholar
  82. [ShSt]
    D. Shale and W.F. Stinespring, Spinor representations of infinite orthogonal groups, J. Math. Mech. 14 (1965) 315–322.MathSciNetzbMATHGoogle Scholar
  83. [So]
    J.-M. Souriau, Quantification geometrique, Comm. Math. Phys. 1 (1966), 374–398.MathSciNetzbMATHGoogle Scholar
  84. [SV]
    M. Spera and G. Valli, Plucker embedding of the Hilbert space Grassmannian and the CAR algebra, Russ. Journal of Math. Physics 2 (1994), 383–392.MathSciNetzbMATHGoogle Scholar
  85. [SpWul]
    M. Spera and T. Wurzbacher, Determinants, Pfaffians, and quasi-free representations of the CAR algebra, Reviews in Mathematical Physics Vol. 10, No. 5 (1998), 705–721.MathSciNetCrossRefzbMATHGoogle Scholar
  86. [SpWu2]
    M. Spera and T. Wurzbacher, The differential geometry of Grassmannian embeddings of based loop groups, Diff. Geometry and its Appl. 13 (2000), 43–75.MathSciNetCrossRefzbMATHGoogle Scholar
  87. [Ta]
    M.E. Taylor, Partial differential equations. II. Qualitative studies of linear equations,Applied Math. Sciences 116, Springer, New York, 1996.Google Scholar
  88. [Tha]
    B. Thaller, The Dirac equation,Texts and Monographs in Physics, Springer, Berlin, 1992.Google Scholar
  89. [Tho]
    E. G. F. Thomas, Vector fields as derivations on nuclear manifolds, Math. Nachr. 176 (1995), 277–286.MathSciNetCrossRefzbMATHGoogle Scholar
  90. [To]
    V. Toledano Laredo, Integrating unitary representations of infinite-dimensional Lie groups, J. Funct. Anal. 161 (1999), no. 2, 478–508.MathSciNetCrossRefzbMATHGoogle Scholar
  91. [Tr]
    F. Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York London, 1967.zbMATHGoogle Scholar
  92. [Wa]
    A. Wassermann, Operator algebras and conformal field theory. III. Fusion of positive energy representations of LSU(N) using bounded operators, Invent. Math. 133 (1998), no. 3, 467–538.MathSciNetCrossRefzbMATHGoogle Scholar
  93. [Wu]
    T. Wurzbacher, Symplectic geometry of the loop space of a Riemannian manifold, J. Geom. Phys. 16 (1995), no. 4, 345–384.MathSciNetCrossRefzbMATHGoogle Scholar
  94. [Wu2]
    T. Wurzbacher, An “elementary” proof of the homotopy equivalence between the restricted general linear group and the space of Fredholm operators, Preprint, IRMA Strasbourg, in preparation.Google Scholar

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© Springer Basel AG 2001

Authors and Affiliations

  • Tilmann Wurzbacher
    • 1
  1. 1.Institut de Recherche Mathématique AvancéeUniversité Louis Pasteur et C.N.R.S.StrasbourgFrance

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