Communications in Mathematical Physics

, Volume 132, Issue 1, pp 39–71

Construction of convergent simplicial approximations of quantum fields on Riemannian manifolds

  • Sergio Albeverio
  • Boguslav Zegarlinski
Article
  • 71 Downloads

Abstract

We construct simplicial approximations of random fields on Riemannian manifolds of dimensiond. We prove convergence of the fields to the continuum limit, for arbitraryd in the Gaussian case and ford=2 in the non-Gaussian case. In particular we obtain convergence of the simplicial approximation to the continuum limit for quantum fields on Riemannian manifolds with exponential interaction.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A] Albeverio, S.: Some new developments concerning Dirichlet forms, Markov fields and quantum fields, pp. 250–259. In: Mathematical Physics. Swansea '88. Simon, B., Truman, A., Davies, I. M. (eds.). Bristol: Adam Hilger 1988Google Scholar
  2. [AFHKL] Albeverio, S., Fenstad, J. E.: Høegh-Krohn, R., Lindstrøm, T.: Non-standard methods in stochastic analysis and mathematical physics. Orlando: Academic Press 1986Google Scholar
  3. [AHK] Albeverio, S., Høegh-Krohn, R.: The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space time. J. Funct. Anal.16, 39–82 (1974)Google Scholar
  4. [AHKH] Albeverio, S., Høegh-Krohn, R., Holden, H.: Markov cosurfaces and gauge fields. Acta Phys. Austr. [Suppl.]XXVI, 211–231 (1984)Google Scholar
  5. [AHKHK1] Albeverio, S., Høegh-Krohn, R., Holden, H., Kolsrud, T.: Stochastic multiplicative measures, generalized Markov semigroups and group valued stochastic processes and fields. J. Funct. Anal.78, 154–184 (1988)Google Scholar
  6. [AHKHK2] Albeverio, S., Høegh-Krohn, R., Holden, H., Kolsrud, T.: Representation and construction of multiplicative noise, J. Funct. Anal.87, 250–272Google Scholar
  7. [AHHHK3] Albeverio, S., Høegh-Krohn, R., Holden, H., Kolsrud, T.: Construction of quantized Higgs-like fields in two dimensions. Physics LettersB 222, 263–268 (1989)Google Scholar
  8. [AHKI] Albeverio, S., Høegh-Krohn, R., Iwata, K.: Covariant Markovian random fields in four space-time dimensions with nonlinear electromagnetic interaction, pp. 69–83. Exner, P., Seba, P. (eds.), Lecture Notes Physics, vol.324. Berlin, Heidelberg, New York: Springer 1989Google Scholar
  9. [AHKMTT] Albeverio, S., Høegh-Krohn, R., Marion, J., Testard, D., Torresani, B.: Non-commutative distribution theory, book in preparationGoogle Scholar
  10. [AHKPS1] Albeverio, S., Høegh-Krohn, R., Paycha, S., Scarlatti, S.: A probability measures for random surfaces of arbitrary genus and bosonic strings in 4 dimensions. Nucl. Phys. B (Proc. Suppl.)6, 180–182 (1989)Google Scholar
  11. [AHKPS2] Albeverio, S., Høegh-Krohn, R., Paycha, S., Scarlatti, S.: A global and stochastic analysis approach to bosonic strings and associated quantum fields. Bochum Preprint (1989)Google Scholar
  12. [AHPRS] Albeverio, S., Hida, T., Potthoff, J. Röckner, M., Streit, L.: Dirichlet forms in terms of white noise analysis. I. Construction of QFT examples; II. Construction of infinite dimensional diffusions. BiBoS Preprint (1989), to appear in Rev. Math. Phys. (1990)Google Scholar
  13. [AIK] Albeverio, S., Iwata, K., Kolsrud, T.: Random fields as solutions of the inhomogeneous quaternionic Cauchy-Riemann equation. I. Invariance and analytic continuation. Bochum Preprint (1989), to appear in Commun. Math. Phys. (1990)Google Scholar
  14. [AK] Albeverio, S., Kusuoka, S.: Maximality of infinite dimensional Dirichlet forms and R. Høegh-Krohn's model of quantum fields. Memorial Volume for Raphael Høegh-Krohn (to appear)Google Scholar
  15. [AR1] Albeverio, S., Röckner, M.: Classical Dirichlet forms on topological vector spaces—The construction of the associated diffusion process. Prob. Th. Rel. Fields83, 405–434 (1989)Google Scholar
  16. [AR2] Albeverio, S., Röckner, M.: New developments in theory and applications of Dirichlet forms, to appear in Stochastic Processes—Geometry and Physics, Proc. Ascona-Locarno-Como, Singapore: World Scientific 1989Google Scholar
  17. [Ad] Adler, R. J.: The geometry of random fields. Chichester: J. Wiley 1981Google Scholar
  18. [AmD] Ambjørn, J., Durhuus, B.: Regularized bosonic strings need extrinsic curvature. Phys. Letts.188B, 253–257 (1987)Google Scholar
  19. [As] Ashtekar, A.: Recent developments in Hamiltonian gravity, pp. 268–271. In: Mathematical Physics. Simon, B., Truman, A., Davies, I. M. (eds.). Bristol: Adam Hilger 1989Google Scholar
  20. [BaJ] Balaban, T., Jaffe, A.: Constructive gauge theory, pp. 207–263. In: Velo, G., Wightman, A. S. (eds.). Fundamental Problems of Gauge Field Theory. Amsterdam: D. Reidel 1986Google Scholar
  21. [BraLie] Brascamp, H. J., Lieb: E. H.: On Extensions of the Brunn-Minkowski and Prékopa-Leindler Theorems, including Inequalities for Log Concave Functions, and with an Applications to the Diffusion Equation. J. Funct. Anal.22, 366–389 (1976)Google Scholar
  22. [BriFLS] Bricmont, J., Fontaine, J. R., Lebowitz, J. L., Spencer, T.: Lattice systems with a continuous symmetry; Part I. Commun. Math. Phys.78, 281–302 (1980); Part II. Commun. Math. Phys.78, 263-372 (1981); Part III with Lieb, E.: Commun. Math. Phys.78, 545–566 (1981)Google Scholar
  23. [CDeA] Dell' Antonio, G. F., Cotta-Ramusino, P.: Self duality and topologial-like properties of lattice gauge field theories. A proposal. Commun. Math. Phys.70, 75–95 (1979)Google Scholar
  24. [CMS] Cheeger, J., Müller, W., Schrader, R.: On the curvature of piecewise flat spaces. Commun. Math. Phys.92, 405–454 (1984)Google Scholar
  25. [Che] Cheeger, J.: Analytic torsion and the heat equation. Ann. Math.109, 259–322 (1979)Google Scholar
  26. [Cia] Ciarlet, Ph.: Lectures on The Finite Element Method. Tata Institute of Fundamental Research, Bombay, India 1975Google Scholar
  27. [DeADFG] De Angelis, G. E., De Falco, D., Guerra, F.: Scalar quantum electrodynamics on the lattice as classical statistical mechanics. Commun. Math. Phys.57, 201–212 (1977)Google Scholar
  28. [Do] Dodziuk, J.: Finite-Difference approach to the Hodge Theory of Harmonic Forms. Am. J. Math.98, 79–104 (1976)Google Scholar
  29. [DoP] Dodziuk, J., Patodi, V. K.: Riemannian structures and triangulations of manifolds. J. Indian Math. Soc.40, 1–52 (1976)Google Scholar
  30. [Dr] Driver, B. K.: Convergence of theU(1)4 lattice gauge theory to its continuum limit. Commun. Math. Phys.110, 479–501 (1987)Google Scholar
  31. [DrM] Drouffe, J. M., Moriarty, K. J. M.: Gauge theories on a simplicial lattice. Nucl. Phys. B220, 253–268 (1983)Google Scholar
  32. [Eck] Eckman, B.: Harmonische Funktionen and Randwertaugfaben in einem Komplex. Commentari Math. Helv.17, 240–245 (1944/45)Google Scholar
  33. [El] Elworthy, D.: Lectures on stochastic differential equations, St. Flour (1988)Google Scholar
  34. [EIR] Elworthy, K. D., Rosenberg, S.: Spectral bounds and the shape of manifolds near infinity, pp. 369–373. In: Mathematical Physics. Swansea '88. (eds.). Simon, B., Truman, A., Davies, I. M.. Bristol: Adam Hilger 1989Google Scholar
  35. [FKG] Fortuin, C. M., Kastelyn, P. W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89–103 (1971)Google Scholar
  36. [FreLM] Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the monster. Boston: Academic Press 1988Google Scholar
  37. [Fo] Fontaine, J. R.: Bounds on the Decay of Correlations for λ(∇Φ)4 Models. Commun. Maths.87, 835–394 (1982)Google Scholar
  38. [Frö] Fröhlich, J.: Statistics of fields, the Yang-Baxter equation and the theory of knots and links. t' Hooft, G. et al. (eds.). New York: Plenum Press 1988Google Scholar
  39. [FröZe] Fröhlich, J., Zegarlinski, B.: Spin Glasses and other Lattice Systems with Long Range Interactions. Commun. Math. Phys.120, 665–668 (1989)Google Scholar
  40. [Ga] Gawedzki, K.: Conformal field theory. Sém. Bourbaki 41e année, 1988–89, No. 704Google Scholar
  41. [GlJ] Glimm, J., Jaffe, A.: Quantum physics: A functional integral point of view. Second ed, Berlin, Heidelberg New York: Springer 1987Google Scholar
  42. [GRS] Guerra, F., Rosen, L., Simon, B.: TheP(ϕ)2 Euclidean quantum field theory as classical statistical Mechanics. Ann. Math.101, 111–259 (1975)Google Scholar
  43. [GSW] Green, M. B., Schwartz, J. H., Witten, E.: Superstring theory. Cambridge University Press 1982Google Scholar
  44. [Gi] Ginibre, J.: General formulation of Griffith's inequalities. Commun. Math. Phys.16, 310 (1970)Google Scholar
  45. [Gr1] Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math.97, 1061–1083 (1976)Google Scholar
  46. [Gr2] Gross, L.: Convergence ofU(1)3 lattice gauge theory to its continuum limit. Commun. Math. Phys.92, 137–162 (1983)Google Scholar
  47. [GrKS] Gross, L., King, Ch., Sengupta, A.: Two dimensional Yang-Mills theory via stochastic differential equations. Cornell University Preprint (1989)Google Scholar
  48. [Gri] Griffiths, R.: Correlations in Ising Ferromagnets I, II, III. J. Math. Phys.B (1967) 478–483, 484–489; Commun. Math. Phys.6, 121–127 (1967)Google Scholar
  49. [HaI] Hawking, S. W., Israel, W. eds.: Three hundred years of gravitation (1987)Google Scholar
  50. [I] Ito, K.: Isotropic random currents, pp. 125–132 in Proc. 3 d Berkeley Sympos. Math. Sta. Prob. (1956)Google Scholar
  51. [ISZ] Itzykson, C., Saleur, H., Zuber, J. B. eds.: Conformal invariance and applications to statistical mechanics. Singapore: World Scientific 1988Google Scholar
  52. [KS] Kelly, D., Sherman, S.: General Griffith's inequalities on Correlations on Ising Ferromagnets. J. Math. Phys.9, 466–488 (1968)Google Scholar
  53. [Kac] Kac, V.: Infinite dimensional Lie algebras. Boston: Birkhäuser 1983Google Scholar
  54. [Kau] Kaufmann, L. H.: Knot polynomial and Yang-Baxter models, pp. 438–441. In: Mathematical Physics. Swansea 88. Simon, B., Trumann, A., Davies, I. M. (eds.) Briston: Adam Hilger 1989Google Scholar
  55. [Kha] Khatsymovsky, V. M.: Vector fields and gravity on the lattice. Novosibirsk Preprint (1988)Google Scholar
  56. [Kr] Krylov, A. L.: Difference approximations to differential operators of mathematical physics. Sov. Math. Dokl.9, 138–141 (1968)Google Scholar
  57. [Ku] Kusuoka, S.: Høegh-Krohn's model of quantum fields and the absolute continuity of measures. To appear in Memorial Volume for R. Høegh-KrohnGoogle Scholar
  58. [Le] Leff, H. S.: Correlation Inequalities for Coupled Oscillators. J. Math. Phys.1, 569–578 (1971)Google Scholar
  59. [Ma] Mandelbrot, B. B.: The fractal geometry of nature. New York: W. H. Freeman 1983Google Scholar
  60. [MBF] Markov, M. A., Berezin, V. A., Frolov, V. P.: Quantum Gravity, Moscow 1987, Singapore: World Scientific 1988Google Scholar
  61. [Mü] Müller, W.: Analytic Torsion and R-Torsion of Riemannian Manifolds. Adv. Math.28, 233–305 (1978)Google Scholar
  62. [Ne] Nelson, E.: The free Markoff field. J. Funct. Anal.12, 211–227 (1973)Google Scholar
  63. [Nit] Nitsche, J.:L -convergence of finite element approximations, pp. 261–274. In: Mathematical Aspects of Finite Element Methods. Galligani, I., Magenes, E. (eds.) Lecture Notes in Mathematics. vol.606. Berlin, Heidelberg, New York: Springer 1977, see also ref. [12] thereGoogle Scholar
  64. [PrS] Pressley, A., Segal, G.: Loop groups. Oxford: Oxford University Press 1986Google Scholar
  65. [RayS] Ray, D. B., Singer, L. M.: R-Torsion and the Laplacian on Riemannian Manifolds. Adv. Math.7, 145–210 (1971)Google Scholar
  66. [Re] Regge, T.: General relativity without coordinates. Nuovo Cim.10, 558–571 (1961)Google Scholar
  67. [RehS] Rehren, K. H., Schroer, B.: Einstein causality and Artin braids, FU Preprint (1988)Google Scholar
  68. [Sa] Santaló, L. A.: Integral geometry and geometric probability. Enc. Maths. Appl., Vol. 1, Reading, MA: Addison-Wesley 1986Google Scholar
  69. [Sc] Scarlatti, S.: PhD Thesis, Roma, 1989Google Scholar
  70. [Si] Simon, B.: TheP(ϕ)2 Euclidean (Quantum) Field Theory. Princeton, NJ: Princeton University Press 1974Google Scholar
  71. [SiTh] Singer, I. M., Thorpe, J. A.: Lecture notes on elementary topology and geometry. Berlin, Heidelberg, New York: Springer 1967Google Scholar
  72. [StrF] Strang, G., Fix, G. J.: An analysis of the finite element method. Englewood Cliffs, NJ: Prentice-Hall 1973Google Scholar
  73. [Wa] Warner, F. W.: Foundations of differentiable manifolds and Lie groups. Gelnview: Scott, Foresman and Company 1971Google Scholar
  74. [Whi] Whitney, H.: Geometric Integration Theory. Princeton NJ: Princeton University Press 1957Google Scholar
  75. [Wi] Williams, D.: Diffusions, Markov Processes and Martingales. Chichester: Wiley 1979Google Scholar
  76. [WoZ] Wong, E., Zakai, M.: Isotropic Gauss-Markov currents. Prob. Theory Rel. Fields82, 137–154 (1989)Google Scholar
  77. [Ze] Zegarlinski, B.: Uniqueness and the Global Markov Property for Euclidean fields: The case of general exponential interaction. Commun. Math. Phys.96, 195–221 (1984)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Sergio Albeverio
    • 1
    • 2
  • Boguslav Zegarlinski
    • 1
    • 2
  1. 1.Fakultät für MathematikRuhr-UniversitätBochumFederal, Republic of Germany
  2. 2.SFBBochum-Essen-Düsseldorf
  3. 3.CERFIMLocarno

Personalised recommendations