Non-Standard Analysis; Polymer Models, Quantum Fields

  • S. Albeverio
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 26/1984)


We give an elementary introduction to non-standard analysis and its applications to the theory of stochastic processes. This is based on a joint book with J. E. Fenstad, R. Høegh-Krohn and T. Lindstrøm. In particular we give a discussion of an hyperfinite theory of Dirichlet forms with applications to the study of the Hamiltonian for a quantum mechanical particle in the potential created by a polymer. We also discuss new results on the existence of attractive polymer measures in dimension d ≤ 5, with applications to the (φ 1 2 φ 2 2 )d-mode1 of interacting quantum fields.


Brownian Motion Standard Analysis Dirichlet Form Standard Part Brownian Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Robinson, Non-standard analysis, North-Holland, Amsterdam (1970).Google Scholar
  2. 2.
    D. Laugwitz, Inf initesimalkalkiil, Bibl. Institut, Mannheim (1978).Google Scholar
  3. 3.
    M. Davis, Applied non standard analysis, Wiley, New York (1977).MATHGoogle Scholar
  4. 4.
    K.D. Stroyan, W.A.J. Luxemburg, Introduction to the theory of infinitesimals. Academic Press, New York (1976).MATHGoogle Scholar
  5. 5.
    H.J. Keisler, Foundations of infinitesimal calculus, Prindle, Weber & Schmidt, Boston (1976).Google Scholar
  6. 6.
    N. Cutland, Non standard measure theory and its applications. Bull. London Math. Soc. 15 (1983) 525–589.CrossRefMathSciNetGoogle Scholar
  7. 7.
    S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, Non standard methods in stochastic analysis and mathematical physics, Acad. Press.Google Scholar
  8. 8.
    H.J. Keisler, Elementary Calculus, Prindle, Weber & Schmidt, Boston (1976).Google Scholar
  9. 9.
    J.M. Henle, E.M. Kleinberg, Infinitesimal Calculus, MIT, Cambridge (1979).Google Scholar
  10. 10.
    R. Lutz, M. Goze, Non-standard analysis: a practical guide with applications, Lect. Notes Maths. 881 (1981) Springer.Google Scholar
  11. 11.
    P. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans. Amer. Math. Soc. 211, (1975) 113–122.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    R.M. Anderson, A non standard representation for Brownian motion and Ito integration, Isr. J. Math. 25 (1976) 15–46.Google Scholar
  13. 13.
    J. Keisler, An infinitesimal approach to stochastic analysis, Mem. Am. Math. Soc.Google Scholar
  14. 14.
    T. Lindstr0m, Hyperfinite stochastic integration I-III, Math. Scand. 46 (1980).Google Scholar
  15. 15.
    E. Perkins, A global intrinsic characterization of Brownian local time, Ann. of Prob. 9 (1981) 800–817.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    A. Stoll, A self-repelling random walk and polymer measures in two dimension,in preparation.Google Scholar
  17. 17.
    M. Fukushima, Dirichlet forms and Markov processes, North-Holland (1980).Google Scholar
  18. 18.
    S. Albeverio, R. Høegh-Krohn, L. Streit, Energy forms, Hamiltonians, and distorted Brownian paths, J. Math. Phys. 18 (1977) 907–917.CrossRefMATHADSGoogle Scholar
  19. 19.
    S. Albeverio, R. Hj&egh-Krohn, Dirichlet forms and diffusion processes on rigged Hilbert spaces, Z. Wahrscheinlichk. verwe. Geb. 40 (1977) 1–57.CrossRefMATHGoogle Scholar
  20. 20.
    S. Kusuoka, Dirichlet forms and diffusion processes on Banach spaces, J. Fac. Sci. Univ. Tokyo JA 1A 29 (1982) 79–95.MathSciNetGoogle Scholar
  21. 21.
    Contribution by S. Albeverio, R. Høegh-Krohn, M. Fukushima, L. Streit in New Stochastic Methods in Physics, Ed. C. De Witt-Morette, K.D« Elworthy, Phys. Repts.77 (1982) 121–382.Google Scholar
  22. 22.
    S. Albeverio, R. Høegh-Krohn, Hunt processes and analytic potential theory on rigged Hilbert spaces, Ann. Inst. H. Poincarê B13 (1977) 269–291.Google Scholar
  23. 23.
    S. Albeverio, R. Hjøegh-Krohn, Diffusion fields, quantum fields, fields with values in Lie groups, to appear in Adv. in Prob., Stochastic analysis and application, Ed. M. Pinsky, Dekker (1984).Google Scholar
  24. 24.
    S. Albeverio, R. Høegh-Krohn, Schrödinger operators with point interactions and short range expansions, pp.11–27,in Proc. VII Int. Congress Math. Physics, Boulder, Eds. W.E. Brittin, K.E. Gustavson, W. Wyss, North Holland (1984)Google Scholar
  25. 25.
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Some exactly solvable models in quantum mechanics and the low energy expansions, to appear in Proc. Leipzig Conf. Operator Algebras, 1983, Teubner 1984,and book in preparation.Google Scholar
  26. 26.
    S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, Singular perturbations and non standard analysis. Trans. Am. Math. Soc. 252 (1979) 275–295.CrossRefMATHGoogle Scholar
  27. 27.
    S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, W. Karwowski, T. Lindstrç5m, Perturbation of the Laplacian supported by null sets, with applications to polymer measures and quantum fields, Bochum Preprint (1984).Google Scholar
  28. 28.
    S. Albeverio, Ph. Blanchard, R. Høegh-Krohn, Some applications of functional integration, Proc. Int. AMP. Conf., Berlin, 1981, Ed. R. Schräder, R. Seiler, D.A. Uhlenbrock, Lect. Notes Phys. 153 (1982) Springer,Berlin.Google Scholar
  29. 29.
    S. Albeverio, Ph. Blanchard, R. Høegh-Krohn, Newtonian diffusions and planets, with a remark on non-standard Dirichlet forms and polymers, to appear in Proc. LMS, Symposium Stoch. Analysis, Swansea, 1983, Ed. A. Truman, D. Williams, Lect. Notes, Maths..Google Scholar
  30. 30.
    E. Nelson, Internal set theory: a new approach to non standard analysis. Bull. Am. Math. Soc. 83 (1977) 1165–1198.CrossRefMATHGoogle Scholar
  31. 31.
    Alonso y Coria, Skrinking potentials in the Schrodinger equation. Ph. D. Thesis, Princeton Univ. (1978).Google Scholar
  32. 32.
    S. Edwards, Y.B. Gulyaev, Proc. Phys. Soc. 83 (1964) 495.CrossRefADSGoogle Scholar
  33. 33.
    Kirsch, Martinelli, On the spectrum of Schrodinger operators with a random potential. Comm. Math. Phys. 85 (1982) 329–350.CrossRefMATHADSMathSciNetGoogle Scholar
  34. 34.
    H. Holden, F. Martinelli, On absence of diffusion near the bottom of spectrum for a random Schrodinger operator on L2(Rυ), Bochum Preprint 19 83, to appear in Comm. Math. Phys.Google Scholar
  35. 35.
    S.F. Edwards, The theory of polymer solutions at intermediate concentration, Proc. Phys. Soc. 88 (1966) 265–280; Functional problems in the theory of polymers, pp. 53–59, in Functional Integration, Ed. A.M. Arthurs, Clarendon Press, Oxford (1975); The mathematics of a rubber band. New Scientist, Febr. 1981, pp. 480–482.CrossRefADSGoogle Scholar
  36. 36.
    S. Kusuoka, unpublished. See [24], [29].Google Scholar
  37. 37.
    J. Westwater, On Edward’s model for long polymer chains I, Comm. Math. Phys. 72 (1980) 131–174.CrossRefMATHADSMathSciNetGoogle Scholar
  38. 38.
    J. Rosen, A local time approach to the self-intersections of Brownian paths in space. Comm. Math. Phys. (1983) 327–338.Google Scholar
  39. 39.
    S. Kusuoka, Asymptotics of polymer measures in one dimension; and On the path property of Edward’s model for long polymer chains in three dimensions, to appear in Proc. Bielefeld Conf. Infinite dimensional analysis and stochastic processesEd. S. Albeverio.Google Scholar
  40. 40.
    K. Symanzik, Euclidean quantum field theory, in Local Quantum Theory, Ed. R. Jost, Academic Press, New York (1969).Google Scholar
  41. 41.
    G. Gallavotti, V. Rivasseau, (p field theory in dimension 4: a modern approach to its unsolved problems, to appear; A comment on 04 4 Euclidean field theory, Phys. Lett. 122B (1983) 268–270.ADSMathSciNetGoogle Scholar
  42. 42.
    P.M. Stevenson, On the physics of 04 in 3 + 1 dimensions, Madison Preprint, 1983.Google Scholar
  43. 43.
    S. Albeverio, G. Gallavotti, R. Høegh-Krohn, The exponential interaction in Rn, Phys. Letts. 83B (1979) 177; Some results for the exponential interaction in two or more dimensions. Comm. Math. Phys. 70 (1979) 187–192.ADSGoogle Scholar
  44. 44.
    M. Aizeman, Proof of the triviality of (f) field theory and some mean field features of Ising models for d4 dPhys. Rev. Letts. 4 7 (1981) 1–4; Geometric analysis of ø4 dfields and Ising models. Comm. Math. Phys. 86 (1982) 1–48.CrossRefADSGoogle Scholar
  45. 45.
    J. Fröhlich, On the triviality of λød4 theories and the approach to the critical point in d(≥)4 dimensions, Nucl. Phys. B 200 FS4 (1982) 281–296, and Lecture at this school.Google Scholar
  46. 46.
    T. Hattori, A generalization of the proof of triviality of scalar field theories, Tokyo Preprint (1983).Google Scholar
  47. 47.
    Y. Matsubara, T. Suzuki, I. Yotsuyanagi, On a possible inconsistency of the (p and the Yukawa theories in four dimensions, Kanazawa Prepr. (1983)Google Scholar
  48. 48.
    K.R. Ito, Trajectories of the λø44-model by the block spin transformations, ZiF-Preprint (1984).Google Scholar
  49. 49.
    A. Bover, G. Felder, J. Fröhlich, On the critical properties of the Edwards and the self-avoiding walk model of polymer chains, ETH Preprint (1983).Google Scholar
  50. 50.
    A. Stoll, A non standard construction of Levy Brownian motion, Ruhr-University Bochum, Preprint (1984).Google Scholar
  51. 51.
    C. Kessler, Hyperfinite representation of generalized random fields, Bochum Preprint (1984).Google Scholar
  52. 52.
    J. Westwater, On Edward’s model for polymer chains, to appear in Proc. Bielefeld Encounters Math, and Phys. IV, Edts. S. Albeverio, Ph. Blanchard, World Sclent. Publ. (1984).Google Scholar
  53. 53.
    G.F. Lawler, A self-avoiding random walk, Duke Math. J. 47 (1980) 655–692.CrossRefMATHMathSciNetGoogle Scholar
  54. 54.
    A.D. Sokal, An alternate constructive approach to the λø34 quantum field theory, and a possible destructive approach to <mml:math>, Ann. I.H. Poincaré A 37 (1982) 317.MathSciNetGoogle Scholar
  55. 55.
    R.A. Brandt, Asymptotically free λ4 theory, Phys. Rev. D14 (1976) 3381–3394.ADSGoogle Scholar
  56. 56.
    D. Brydges, T. Spencer, Self-avoiding walk in 5 or more dimensions, Viriginia Courant Preprint (1984).Google Scholar
  57. 57.
    G. Felder, J. Fröhlich, Intersection properties of simple random walks: a renormalization group approach, ETH-Preprint (1984).Google Scholar
  58. 58.
    Ph. Blanchard, J. Tarski, Renormalizable interactions in two dimensions and sharprtime fields. Acta Phys. Austr. 49 (1978) 129–152.MathSciNetGoogle Scholar
  59. 59.
    L. Arkeryd, Asymptotic behaviour of the Boltzmann equation with infinite range forces. Comm. Math. Phys. (1982) 475–484.Google Scholar
  60. 60.
    J.E. Fenstad, Is non standard analysis relevant for the philosophy of mathematics? Synthese (1984).Google Scholar
  61. 61.
    Ch. Kessler, Non standard treatment of the global Markov property of lattice fields, in preparation.Google Scholar
  62. 62.
    D.N. Hoover, H.J. Perkins, Non standard construction of the stochastic integral and applications to stochastic differential equations, I, II, Trans. Amer. Math. Soc. 275 (1983) 1–58.CrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    M.M. Richter, Ideale Punkte, Monaden und Nichtstandard- Methoden, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden (1982).Google Scholar
  64. 64.
    J. Rosen, A local time approach to the self-intersections of Brownian paths in space, Commun. Math. Phys. 88 (1983) 327–338.CrossRefMATHADSGoogle Scholar
  65. 65.
    K. Gawedzki, A. Kupiainen, Triviality of λ44 and all that in a hierarchical model approximation, IHES Preprint (1982)Google Scholar
  66. 66.
    J.E. Fenstad, Non standard methods in stochastic methods in stochastic analysis and mathematical physics, jber. d. Dt. Math.-Verein (1980) 167–180.Google Scholar
  67. 67.
    K. Symanzik, Lett. Nuovo Cim. 6 (1973) 77CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • S. Albeverio
    • 1
  1. 1.Mathematisches InstitutRuhr-UniversitätBochum 1Fed. Rep. Germany

Personalised recommendations