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Some stochastic techniques in quantization, new developments in Markov fields and quantum fields

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Probabilistic Methods in Quantum Field Theory and Quantum Gravity

Part of the book series: NATO ASI Series ((NSSB,volume 224))

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Abstract

In these lectures we intend to discuss a few recent developments in the area of interactions between quantum fields and Markov fields in which we have been involved. We stress particularly developments involving techniques of stochastic analysis and where mathematical results have been obtained. By this we do not want at all to detract importance to more heuristic, physical approaches, on the contrary, we have been ourselves often inspired by such approaches. We hope however our exposition can provide some complement to other lectures. We also would like to stress that we are convinced that new techniques in stochastic analysis should become very useful also for the entire area of stochastic quantisation, and perhaps some of the remarks and References we give can be useful to a reader who would like to go deeper in this direction. There are many topics we unfortunately do not discuss. We refer to the talks by G.F. Dell’ Antonio and G. Jona-Lasinio for other approaches to stochastic quantization. Also we would like to refer to other exposés and excellent reviews, like e.g. [1], for background and possible areas of further applications of the techniques we discuss. We are also forced, by lack of space and time, to refer to some papers of ours which have a large bibliography for additional References, especially to the work of other authors.

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References

  1. P.H. Damgaard, H. Hüffel, Phys. Rep. 152, 227 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  2. M. Fukushima, Dirichlet forms and Markov processes, North-Holland, Amsterdam (1980)

    MATH  Google Scholar 

  3. S. Albeverio, J. Brasche, M. Röckner, Dirichlet forms and generalized Schrödinger operators, to appear in Proc. Sønderborg Conf. “Schrödinger operators”, Edts. H. Holden, A. Jensen, Lect. Notes Phys., Springer, Berlin (1989)

    Google Scholar 

  4. L. Arnold, Stochastic differential equations: theory and applications, J. Wiley, New York (1974)

    Google Scholar 

  5. B. Øksendahl, Stochastic differential equations, Springer, Berlin (1985)

    Book  Google Scholar 

  6. Th. C. Gard, Introduction to stochastic differential equations, M. Dekker (1988)

    Google Scholar 

  7. L. C. G. Rogers, D. Williams, Diffusions, Markov processes, and martingales, J. Wiley, Cichester (1987)

    Google Scholar 

  8. S. Albeverio, R. Høegh-Krohn, L. Streit, Energy forms, Hamiltonians and distorted Brownian paths, Journal of Mathematical Physics 18, 907–917 (1977)

    Google Scholar 

  9. S. Albeverio, M. Fukushima, W. Karwowski, L. Streit, Capacity and quantum mechanical tunneling, Commun. Math. Phys. 80, 301–342 (1981)

    Google Scholar 

  10. M. Fukushima, Energy forms and diffusion processes, pp. 65–98 in “Mathematics and Physics”, Vol. 1, Ed. L. Streit, World Scient., Singapore (1985)

    Google Scholar 

  11. S. Albeverio, M. Röckner, Inßnite dimensional stochastic equations: solutions via Dirichlet forms, Bochum Preprint (1989)

    Google Scholar 

  12. S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, Non standard methods in stochastic analysis and mathematical physics, Academic Press (1986)

    Google Scholar 

  13. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable models in quantum mechanics, Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  14. D. Pantic, Stochastic calculus on distorted Brownian motion, J. Math. Phys. 39, 207–210 (1988)

    Google Scholar 

  15. S. Albeverio, Zhiming Ma, Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms, in preparation

    Google Scholar 

  16. S. Albeverio, Zhiming Ma, Perturbation of Dirichlet forms — lower semiboundedness, closability and form cores, to appear in J.Funct.Anal.

    Google Scholar 

  17. S. Albeverio, M. Röckner, Classical Dirichlet forms on topological vector spaces — closability and a Cameron-Martin formula, to appear in J. Funct. Anal. (1990)

    Google Scholar 

  18. S. Albeverio, S. Kusuoka, Maximality of inßnite dimensional Dirichlet forms and R. Høegh-Krohn’s model of quantum fields, to appear in Memorial Volume for Raphael Høegh-Krohn.

    Google Scholar 

  19. S. Albeverio, S. Kusuoka, M. Röckner, On partial integration in infinite dimensional space and applications to Dirichlet forms (with S. Kusuoka, M. Röckner), SFB 237 — Preprint to appear in J. London Math. Soc. (1989).

    Google Scholar 

  20. S. Albeverio. Röckner, New developments in theory and applications of Dirichlet forms, to appear in “Stochastic Processes, Physics and Geometry”, Proc. 2nd Int. Conf. Ascona — Locarno — Como 1988, Ed. S. Albeverio, G. Casati, U. Cattaneo, D. Merlini, R. Moresi, World Scient. (1989)

    Google Scholar 

  21. S. Albeverio, R. Høegh-Krohn, A remark on the connection between stochastic mechanics and the heat equation, Journal of Mathematical Physics, 15, 1745–1747 (1974)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Ph. Blanchard, Ph. Combe, W. Zheng, Physical and mathematical aspects of stochastic mechanics, Lect. Notes Phys. 281, Springer, Berlin (1987)

    Google Scholar 

  23. S. Albeverio, L. Streit, S. Kusuoka, Convergence of Dirichlet forms and associated Schrödinger operators, J. Funct. Anal. 68, 130–148 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  24. B. Zegarlinski, Log-Sobolev inequalities for inßnite one-dimensional lattice systems, to appear in Commun. Math. Phys. (1990).

    Google Scholar 

  25. L. Schwartz, Radon measures, Oxford University Press (1973)

    Google Scholar 

  26. S. Albeverio, M. Röckner, Dirichlet forms, quantum ßelds and stochastic quantisation, pp. 1–21 in “Stochastic Analysis, path integration and dynamics”, Emanations from “Summer Stochastics”, Warwick 1987, Edts. D. Elworthy, J. C. Zambrini, Pitman Res. Notes, Longman, Harlow (1989)

    Google Scholar 

  27. S. Albeverio, M. Röckner, Classical Dirichlet forms on topological vector spaces — the construction of the associated diffusion process, Prob.Th.Rel. Fields 83, 405–434 (1989)

    Article  MATH  Google Scholar 

  28. S. Albeverio, M. Röckner, On Dirichlet forms on topological vector spaces: Existence and maximality, SFB 237-Preprint, to appear in Proc. Bad Honnef ′88 Conf., Edts. N. Christopeit et al., Lect. Notes Inform. Control, Springer Verlag (1989)

    Google Scholar 

  29. S. Albeverio, R. Høegh-Krohn Quasi invariant measures, symmetric diffusion processes and quantum fields, pp. 11–59 in “Proceedings of the International Colloquium on Mathematical Methods of Quantum Field Theory”, Editions du CNRS. 1976, ( Colloques Internationaux du Centre National de la Recherche Scientifique, No. 248 ).

    Google Scholar 

  30. S. Albeverio, R. Høegh-Krohn, Dirichlet forms and diffusion processes on rigged Hilbert spaces, Z.für Wahr.und verw.Gebiete 40, 1–57 (1977).

    Article  MATH  Google Scholar 

  31. S. Albeverio, T. Hida, J. Potthoff, M. Röckner, L. Streit, Dirichlet forms in terms of white noise analysis I — Construction and QFT examples, Rev.Math.Phys.l (1990).

    Google Scholar 

  32. S. Albeverio, T. Hida, J. Potthoff, M. Röckner, L. Streit, Dirichlet forms in terms of white noise analysis II — Closability and Diffusion Processes, Rev.Math.Phys.l (1990).

    Google Scholar 

  33. S. Albeverio, R. Høegh-Krohn, Diffusion Fields, Quantum Fields and Fields with Values in Groups, pp. 1–98, “Stochastic Analysis and Applications”, Adv. in Probability, Edt. M. Pinsky, M. Dekker, New York (1984)

    Google Scholar 

  34. G. Jona-Lasinio, P.K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys. 101, 409–436 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. G. Jona-Lasinio, P.K. Mitter, Large deviation estimates in the stochastic quantization of, Roma Preprint (1989)

    Google Scholar 

  36. V.S. Borkar, R.T. Chari, S.K. Mitter, Stochastic quantization of field theory in finite and infinite volume, J. Funct. Anal. 81, 184–206 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  37. C.R. Doering, Nonlinear parabolic stochastic differential equations with additive colored noise on: IR d × IR + a regulated stochastic quantization, Commun. Math. Phys. 109, 537 (1987)

    Article  ADS  Google Scholar 

  38. S. Albeverio, R. Høegh-Krohn, B. Zegarlinski, Uniqueness and global Markov property for Euclidean fields: The case of general polynomial interactions, Commun. Math. Phys. 123, 377–424 (1989)

    Article  ADS  MATH  Google Scholar 

  39. S. Albeverio, R. Høegh-Krohn, B. Zegarlinski, Uniqueness of Gibbs states for general P (φ)2 — weak coupling models by cluster expansion, Commun. Math. Phys. 121, 683–697 (1989)

    Article  ADS  MATH  Google Scholar 

  40. R. Gielerak, On the DLR equation for the Euclidean quantum Geld theory: the uniqueness theorem, Ann. of Phys. 189., 1–28 (1989)

    Google Scholar 

  41. S. Albeverio, B. Zegarlinski, Construction of convergent simplicial approximations of quantum ßelds on Riemannian manifolds, to appear in Commun. Math. Phys.

    Google Scholar 

  42. S. Albeverio, R. Høegh-Krohn, L. Streit, Regularization of processes and Hamiltonians, J. Math. Phys. 21, 1636–1642 (1980).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. M. Takeda, On the uniqueness of the Markovian self-adjoint extension, pp. 319–325 in “Stochastic Processes — Mathematics and Physics II”, Edts. S. Albeverio, Ph. Blanchard, L. Streit, Lect. Notes Maths. 1250. Springer, Berlin (1987)

    Google Scholar 

  44. S. Albeverio, T. Arede, The relation between quantum mechanics and classical mechanics: a survey of some mathematical aspects, pp. 37–76 in “Chaotic behavior in Quantum Systems, Theory and Applications”, (Proc. Como 1983) Ed. G. Casati, Plenum Press, New York (1985)

    Google Scholar 

  45. F. Guerra, Quantum field theory and probability theory. Outlook on new possible developments, pp. 214–243 in “Trends and Developments in the Eighties”, Edts. S. Albeverio, Ph. Blanchard, World Scient., Singapore (1985)

    Google Scholar 

  46. G.F. Dell’Antonio, D. Zwanziger, these Proceedings

    Google Scholar 

  47. Z. Brzezniak, Some remarks on Feynman-Kac formulae, SFB 237-Preprint, Bochum (1989)

    Google Scholar 

  48. S. Albeverio, R. Høegh-Krohn, H. Holden, Markov cosurfaces and gauge fields, pp. 211–231 in Proc. Schladming 84, 23. Intern. Universitätswochen für Kernphysik, Acta Phys. Austr. Suppl. XXVI (1984): “Stochastic Methods and Computer Techniques in Quantum Dynamics, Edts. H. Mitter, L. Pittner.

    Google Scholar 

  49. S. Albeverio, R. Høegh-Krohn, H. Holden, Markov processes on infinite dimensional spaces, Markov fields and Markov cosurfaces, pp. 11–40 in “Stochastic Space-Time Models and Limit Theorems”, Ed. L. Arnold, P. Kotelenez, D. Reidel, Dordrecht (1985).

    Google Scholar 

  50. S. Albeverio, R. Høegh-Krohn, Brownian motion, Markov cosurfaces, Higgs Gelds, pp. 95–104 in “Fundamental Aspects of Quantum Theory”, Proc. Como. Conf. 1985 Ed. V. Gorini, A. Frigerio, Plenum Publ, New York (1986).

    Google Scholar 

  51. S. Albeverio, R. Høegh-Krohn, H. Holden, Random fields with values in Lie groups and Higg fields, pp. 1–13 in “Stochastic Processes in Classical and Quantum Systems”, Procedings, Ascona 1985, S. Albeverio, G. Casati, D. Merlini, Lect. Notes in Physics 262, Springer, Berlin (1986)

    Google Scholar 

  52. S. Albeverio, R. Høegh-Krohn, H. Holden, Stochastic Lie group-valued measures and their relation to stochastic curve integrals, gauge fields and Markov cosurfaces, pp. 1–24 in “Stochastic Processes in Mathematics and Physics”, Proc. BiBoS I, Edts. S. Albeverio, Ph. Blanchard, L. Streit, Lect. Notes Maths. 1158, Springer (1986).

    Google Scholar 

  53. S. Albeverio, R. Høegh-Krohn, Euclidean Markov fields and relativistic quantum fields from stochastic partial differential equations in four dimensions, Phys. Letts. B177. 175–179 (1986).

    Google Scholar 

  54. S. Albeverio, R. Høegh-Krohn, Quaternionic non abelian relativistic quantum fields in four space-time dimensions, Phys. Letts. B189, 329–336 (1987)

    Google Scholar 

  55. S. Albeverio, R. Høegh-Krohn, H. Holden, T. Kolsrud, Representation and construction of multiplicative noise, to appear in J. Funct. Anal.

    Google Scholar 

  56. S. Albeverio, R. Høegh-Krohn, Some recent interactions between mathematics and physics in connection with generalized random fields, pp. 455–472 in Proc. 1st World Congress of the Bernouilli Society, Edts. Yu. Prokhorov, V.V. Sazonov, Vol. 1 Probability Theory and Applications, Tashkent 1986, VNU-Press, Utrecht (1987).

    Google Scholar 

  57. S. Albeverio, R. Høegh-Krohn, H. H.lden, Stochastic multiplicative measures, generalized Markov semigroups and group valued stochastic processes and fields, J. Funct. Anal. 78, 154–184 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  58. S. Albeverio, R. Høegh-Krohn, Construction of interacting local relativistic quantum fields in four space-time dimensions, Phys. Letts. B200, 108–114 (1988), Err. Phys. Letts, (1988). B 202 621

    Google Scholar 

  59. S. Albeverio, R. Høegh-Krohn, H. Holden, T. Kolsrud, A covariant Feynman-Kac formula for unitary bundles over Euclidean space, Stockholm Preprint 1988, to appear in Proc. Trento Conference “Stochastic Partial Differential Equations”, Edts. G. Da Prato, L. Tubaro, Lect. Notes Maths., Springer (1988).

    Google Scholar 

  60. S. Albeverio, K. Iwata, T. Kolsrud, Random fields as solutions of the inhomogeneous quaternionic Cauchy-Riemann equation I. Invariance and Analytic Continuation, SFB 237 — Preprint, to appear in Commun. Math. Phys. (1989)

    Google Scholar 

  61. S. Albeverio, R. Høegh-Krohn, K. Iwata, Covariant Markovian random fields in four space-time dimensions with nonlinear electromagnetic interaction, pp. 69–83 in “Applications of Self-Adjoint Extensions in Quantum Physics”, in Proc. Dubna Conf. 1987, Edts. P. Exner, P. Seba, Lec. Notes in Physics 324, Springer, Berlin (1989)

    Google Scholar 

  62. S. Albeverio, R. Høegh-Krohn, H. Holden, T. Kolsrud, Construction of quantised Higgs-like fields in two dimensions, Phys. Letts. B 222. 263–268 (1989).

    Article  Google Scholar 

  63. S. Albeverio, Some new developments concerning Dirichlet forms, Markov fields and quantum fields, pp. 250–259 in “IXth International Congress on Mathematical Physics”, July 1988, Swansea ’88, Edts., B. Simon, A. Truman, I.M. Davis, Adam Hilger, Bristol and New York (1989)

    Google Scholar 

  64. E.P. Osipov, Euclidean Markov fields from stochastic partial differential equation in eight-dimensional space, Novosibirsk Preprint (1987)

    Google Scholar 

  65. E.P. Osipov, Octavic Markov fields from stochastic partial differential equations in eight-dimensional space, Novosibirsk Preprint (1987)

    Google Scholar 

  66. E.P. Osipov, Two-dimensional random fields as solutions of stochastic differential equations, Bochum Preprint (1989)

    Google Scholar 

  67. E.P. Osipov, Markov properties of solutions of stochastic partial differential equations in a finite volume, Novosibirsk Preprint (1989)

    Google Scholar 

  68. Z. Haba, Stochastic equations for some Euclidean fields, pp. 315–328 in “Stochastic Processes in classical and quantum systems”, Eds. S. Albeverio, G. Casati, D. Merlini, Lect. Notes Phys. 262, Springer, Berlin (1986)

    Google Scholar 

  69. S. Klimek, W. Kondracki, A construction of two-dimensional quantum chromodynamics, Commun. Math. Phys. 113, 389–402 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  70. L. Gross, Ch. King, A. Sengupta, Two dimensional Yang-Mills theory via stochastic differential equations, Cornell Univ. Preprint (1989)

    Google Scholar 

  71. B. Driver, YM2: continuum expectations, lattice convergence and lassos, Commun. Math. Phys. 123, 575–616 (1989)

    Article  ADS  Google Scholar 

  72. H.G. Dosch, V.F. Müller, Lattice gauge theory in two space-time dimensions, Fortschr. d. Phys. 27, 547–559 (1979)

    MathSciNet  Google Scholar 

  73. E. Seiler, Gauge theories as a problem of constructive quantum field theory and statistical mechanics, Lect. Notes Phys. 159, Springer, Berlin (1982)

    Google Scholar 

  74. H. Tamura, Nonlinear electromagnetic fields conßne charges, Kanazawa University Preprint (1989)

    Google Scholar 

  75. S. Albeverio, S. Paycha, S. Scarlatti, A short overview of mathematical approaches to functional integration, Bochum Preprint 1989, to appear in Proc. Karpacz 1989, Ed. Z. Haba, Birkhäuser, Basel (1990)

    Google Scholar 

  76. S. Albeverio, R. Høegh-Krohn, S. Paycha, S. Scarlatti, A probability measure for random surfaces of arbitrary genus and bosonic strings in 4 dimensions, Nucl. Phys. B (Proc. Suppl.), 6, 180–182 (1989) (Proc. Eugene Wigner Symposium on Space-Time Symmetries, Washington 1988 ).

    Google Scholar 

  77. S. Albeverio, R. Høegh-Krohn, S. Paycha, S. Scarlatti, A global and stochastic analysis approach to bosonic strings and associated quantum fields, SFB 237-Preprint (1989)

    Google Scholar 

  78. B. Eckman, Harmonische Funktionen und Randwertaufgaben in einem Komplex, Commentari Math. Helv. 17 (1944/45) 240–245

    Google Scholar 

  79. J. Dodziuk, Finite -Difference approach to the Hodge Theory of Harmonic Forms, Ann. J. Math. 98 (1976) 79–104

    Google Scholar 

  80. H. Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton N.J. 1957

    Google Scholar 

  81. I.M. Singer, J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer-Verlag 1967

    Google Scholar 

  82. J. Dodziuk, V.K. Patodi, Riemannian Structures and triangulations of Manifolds, J. Indian Math. Soc. 40 (1976) 1–52

    MathSciNet  MATH  Google Scholar 

  83. W. Müller, Analytic Torsion and R-Torsion of Riemannian Manifolds, Adv. Math. 28 (1978) 233–305

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics Department, Ruhr-University, D-4630, Bochum 1, Germany

    Sergio Albeverio

  2. BiBoS, Bielefeld, Germany

    Sergio Albeverio & Boguslav Zegarlinski

  3. CERFIM, Locarno, Switzerland

    Sergio Albeverio

Authors
  1. Sergio Albeverio
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  2. Boguslav Zegarlinski
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Editors and Affiliations

  1. The Niels Bohr Institute, Copenhagen, Denmark

    P. H. Damgaard

  2. University of Vienna, Vienna, Austria

    H. Hüffel

  3. Utah State University, Logan, Utah, USA

    A. Rosenblum

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Albeverio, S., Zegarlinski, B. (1990). Some stochastic techniques in quantization, new developments in Markov fields and quantum fields. In: Damgaard, P.H., Hüffel, H., Rosenblum, A. (eds) Probabilistic Methods in Quantum Field Theory and Quantum Gravity. NATO ASI Series, vol 224. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3784-7_1

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