Skip to main content

Along Paths Inspired by Ludwig Streit: Stochastic Equations for Quantum Fields and Related Systems

  • Conference paper
  • First Online:
Stochastic and Infinite Dimensional Analysis

Part of the book series: Trends in Mathematics ((TM))

Abstract

The interaction between quantum mechanics, quantum field theory, stochastic partial differential equations and infinite dimensional analysis is briefly surveyed, referring in particular to models and techniques to which L. Streit has given outstanding contributions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Albeverio, S.: Loop groups, random gauge fields, Chern-Simons models, strings: some recent mathematical developments. In: Léandre, R., Paycha, S., Wurzbacher, T. (eds.) Espace des lacets, Proceedings of the Conference Loop Spaces ’94, Strasbourg, pp. 5–34 (1996) (Publ. Inst. R. Descartes)

    Google Scholar 

  2. Albeverio, S.: Wiener and Feynman-Path Integrals and Their Applications. Proceedings of Symposia in Applied Mathematics, vol. 52, pp. 163–194. AMS (1997)

    Google Scholar 

  3. Albeverio, S.: Some themes of the scientific work of Ludwig Streit. In: Albeverio, S., Blanchard, Ph., Ferreira, L., Hida, T., Kondratiev, Y., Vilela Mendes, R. (eds.) Mathematical Physics and Stochastic Analysis, Essays in Honour of Ludwig Streit, pp. 1–7. World Scientific, Singapore (2000).

    Chapter  Google Scholar 

  4. Albeverio, S.: Theory of Dirichlet forms and applications. In: Bernard, P. (ed.) Lectures on Probability Theory and Statistics (Saint-Flour, 2000). Lecture Notes in Mathematics, vol. 1816, pp. 1–106. Springer, Berlin (2003)

    Google Scholar 

  5. Albeverio, S., di Persio, L., Mastrogiacomo, Z.: Small noise asymptotic expansions for stochastic PDE’s I. The case of a dissipative polynomially bounded nonlinearity. Tohôku Math. J. 63, 877–898 (2011)

    MATH  Google Scholar 

  6. Albeverio, S., di Persio, L., Mastrogiacomo, E., Smii, B.: Invariant measures for infinite dimensional SDEs driven by Lévy noise with dissipative nonlinear drift (2014, Preprint); and paper in preparation.

    Google Scholar 

  7. Albeverio, S., Fan, R., Herzberg, F.S.: Hyperfinite Dirichlet Forms and Stochastic Processes. Lecture Notes of the UMI. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  8. Albeverio, S., Ferrario, B.: Some methods of infinite dimensional analysis in hydrodynamics: an introduction. In: SPDE in Hydrodynamic: Recent Progress and Prospects. Lecture Notes in Mathematics, vol. 1942, pp. 1–50. Springer, Berlin/London (2008)

    Google Scholar 

  9. Albeverio, S., Fenstad, J.E., Høegh-Krohn, R., Lindstrøm, T.: Nonstandard Methods in Stochastic Analysis and Mathematical Physics. Pure and Applied Mathematics, vol. 122. Academic, Orlando (1986). Reprint: Dover Publications, Mineola (2009)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Albeverio, S., Gallavotti, G., Høegh-Krohn, R.: Some results for the exponential interaction in two or more dimensions. Commun. Math. Phys. 70, 187–192 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Albeverio, S., Gottschalk, H., Wu, J.-L.: Models of local relativistic quantum fields with indefinite metric (in all dimensions). Commun. Math. Phys. 184(3), 509–531 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Albeverio, S., Gottschalk, H., Yoshida, M.W.: Systems of classical particles in the grand canonical ensemble, scaling limits and quantum field theory. Rev. Math. Phys. 17(2), 175–226 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Albeverio, S., Haba, Z., Russo, F.: A two-space dimensional semilinear heat equation perturbed by (Gaussian) white space. PTRF 121, 319–366 (2001)

    MathSciNet  MATH  Google Scholar 

  15. Albeverio, S., Hahn, A., Sengupta, A.: Chern-Simons theory, Hida distributions and state models. IDAQP 6, 65–81 (2003). (Special Issues on Diff. Geom. Stoch. Anal. I).

    Google Scholar 

  16. Albeverio, S., Hida, T., Potthoff, J., Röckner, M., Streit, L.: Dirichlet forms in terms of white noise analysis I – Construction and QFT examples. Rev. Math. Phys. 1, 291–312 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Albeverio, S., Hida, T., Potthoff, J., Röckner, M., Streit, L.: Dirichlet forms in terms of white noise analysis II – Closability and Diffusion Processes. Rev. Math. Phys. 1, 313–323 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Albeverio, S., Hida, T., Potthoff, J., Streit, L.: The vacuum of the Høegh-Krohn model as a generalized white noise functional. Phys. Lett. B 217, 511–514 (1989)

    Google Scholar 

  19. Albeverio, S., Høegh-Krohn, R.: Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non-polynomial interactions. Commun. Math. Phys. 30, 171–200 (1973)

    Article  MathSciNet  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Albeverio, S., Høegh-Krohn, R.: Canonical relativistic quantum fields in two space-time dimensions. Preprint Series, Institute of Mathematics, University of Oslo, No. 23 (1975)

    Google Scholar 

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

    Google Scholar 

  23. Albeverio, S., Høegh-Krohn, R.: Dirichlet forms and diffusion processes on rigged Hilbert spaces. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 40, 1–57 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  24. Albeverio, S., Høegh-Krohn, R.: Diffusions, Quantum Fields and Groups of Mappings. Functional Analysis in Markov Processes. Lecture Notes in Mathematics, vol. 923, pp. 133–145 (1982)

    Article  MATH  Google Scholar 

  25. Albeverio, S., Høegh-Krohn, R., Holden, H.: Stochastic multiplicative measures, generalized Markov semigroups and group valued stochastic processes and fields. J. Funct. Anal. 78(1), 154–184 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  26. Albeverio, S., Høegh-Krohn, R., Mazzucchi, S.: Mathematical Theory of Feynman Path Integrals, An Introduction, 2nd edn. Springer (2008)

    Google Scholar 

  27. Albeverio, S., Høegh-Krohn, R., Streit, L.: Energy forms, Hamiltonians, and distorted Brownian paths. J. Math. Phys. 18, 907–917 (1977)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Albeverio, S., Kawabi, H., Mihalache, S.: Stochastic quantization of exponential interactions (in preparation)

    Google Scholar 

  31. Albeverio, S., Kawabi, H., Mihalache, S., Röckner, M.: Dirichlet form approach to stochastic quantization under exponential interaction in finite volume (in preparation)

    Google Scholar 

  32. Albeverio, S., Kawabi, H., Röckner, M.: Strong uniqueness for both Dirichlet operators and stochastic dynamics for Gibbs measures on a path space with exponential interactions. J. Funct. Anal. 262(2), 602–638 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Albeverio, S., Kondratiev, Y., Kozitsky, Y., Röckner, M.: Statistical Mechanics of Quantum Lattice Systems: A Path Integral Approach. EMS Tracts in Mathematics, vol. 8. European Mathematical Society (2009)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Albeverio, S., Liang, S.: A remark on the nonequivalence of the time-zero \(\phi _{3}^{4}\)-measure with the free field measure. Markov Process. Rel. Fields 14, 159–164 (2008)

    MathSciNet  MATH  Google Scholar 

  36. Albeverio, S., Liang, S., Zegarliński, B.: Remark on the integration by parts formula for the \(\phi _{3}^{4}\)-quantum field model. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9, 149–154 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Albeverio, S., Ma, Z.M., Röckner, M.: Quasi-regular Dirichlet forms and the stochastic quantization problem, Festschrift Masatoshi Fukushima. In: Chen, Z.-Q., Jacob, N., Takeda, M., Uemura, T. (eds.) Honor of Masatoshi Fukushima’s Sanju, pp. 27–58. World Scientific, Singapore (2015)

    Chapter  Google Scholar 

  38. Albeverio, S., Mandrekar, V.: A remark on measure indexed Gaussian random fields (2015, Preprint).

    Google Scholar 

  39. Albeverio, S., Mazzucchi, S.: Path integrals, mathematical aspects, tiblo-Scholarpedia (2015)

    Google Scholar 

  40. Albeverio, S., Oliveira, M.J., Streit, L.: Intersection local times of independent Brownian motions as generalized white noise functionals. Acta Appl. Math. 69(3), 211–241 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Albeverio, S., Röckner, M.: Dirichlet forms, quantum fields and stochastic quantization. In: Elworthy, K.D., Zambrini, J.C. (eds.) Stochastic analysis, path integration and dynamics. Research Notes in Mathematics, vol. 200, pp. 1–21. Longman, Harlow (1989)

    Google Scholar 

  42. Albeverio, S., Röckner, M.: Classical Dirichlet forms on topological vector spaces – the construction of the associated diffusion process. Probab. Theory Relat. Fields 83(3), 405–434 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  43. Albeverio, S., Röckner, M.: Stochastic differential equations in infinite dimensions: solution via Dirichlet forms. Probab. Theory Relat. Fields 89(3), 347–386 (1991)

    Article  MATH  Google Scholar 

  44. Albeverio, S., Rüdiger, B.: Infinite dimensional Stochastic Differential Equations obtained by subordination and related Dirichlet forms. J. Funct. Anal. 204(1), 122–156 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. Albeverio, S., Rüdiger, B.: Subordination of symmetric quasi-regular Dirichlet forms. Random Oper. Stoch. Equ. 13(1), 17–38 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  46. Albeverio, S., Rüdiger, B., Wu, J.-L.: Analytic and probabilistic aspects of Lévy processes and fields in quantum theory. In: Barndorff-Nielsen, O., Mikoschi, T., Resnick, S. (eds.) Lévy Processes, Theory and Application, pp. 187–224. Birkhäuser, Basel (2001)

    Chapter  Google Scholar 

  47. Albeverio, S., Sengupta, A.: A mathematical construction of the non-abelian Chern-Simons functional integral. Commun. Math. Phys. 186, 563–579 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  48. Albeverio, S., Sengupta, A.: From classical to quantum fields – a mathematical approach (in preparation)

    Google Scholar 

  49. Albeverio, S., Ugolini, S.: Complex Dirichlet forms: non symmetric diffusion processes and Schrödinger operators. Potential Anal. 12, 403–417 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  50. Andrisani, A., Cufaro-Petroni, N.: Markov processes and generalized Schrödinger operators. J. Math. Phys. 52, 113509, 22pp (2011)

    Google Scholar 

  51. Benth, F., Streit, L.: The Burgers equation with a non-Gaussian random force. In: Decreusefond, L., Gjerde, J., Øksendal, B., Üstünel, A.S. (eds.) Stochastic Analysis and Related Topics. Stochastic Monographs vol. 6, pp. 187–210. Birkhäuser, Basel (1998)

    Chapter  Google Scholar 

  52. Berezansky, Y.M., Kondratiev, Y.G.: Spectral Methods in Infinite-Dimensional Analysis, vol. 1, 2. Kluwer (1995) (first issued in URSS 1988)

    Google Scholar 

  53. Bock, W., Oliveira, M.J., da Silva, J.L., Streit, L.: Polymer measure: Varadhan’s renormalization revisited. Rev. Math. Phys. 27(3), 1550009 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  54. Bonaccorsi, S., Mastrogiacomo, E.: Analysis of the stochastic Fitz-Hugh-Nagumo System. IDAQP 11, 427–446 (2008).

    MathSciNet  MATH  Google Scholar 

  55. Borasi, L.: Complex scaled time oscillatory infinite dimensional integrals and the Gell-Mann Low formula. Master Thesis, University of Pisa (2014)

    Google Scholar 

  56. Borkar, V.S., Chari, R.T., Mitter, S.K.: Stochastic quantization of field theory in finite and infinite volume. J. Funct. Anal. 81(1), 184–206 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  57. Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space. Studies in Mathematics, vol. 14. deGruyter (1991)

    Google Scholar 

  58. Bovier, A.: Metastability. In: Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics 1970, pp. 177–221. Springer, Berlin (2009)

    Google Scholar 

  59. Chen, Z.Q., Fukushima, M.: Symmetric Markov Processes, Time change, and Boundary Theory. Princeton University Press (2012)

    Google Scholar 

  60. Da Prato, G., Debussche, A.: Strong solutions to the stochastic quantization equations. Ann. Probab. 31(4), 1900–1916 (2003)

    Google Scholar 

  61. Da Prato, G., Tubaro, L.: Self-adjointness of some infinite dimensional elliptic operators and applications to stochastic quantization. Probab. Theory Relat. Fields 118(1), 131–145 (2000)

    Article  MATH  Google Scholar 

  62. de Faria, M., Hida, T., Watanabe, H., Streit, L.: Intersection local times as generalized white noise functionals. Acta Appl. Math. 46, 351–362 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  63. de Faria, M., Potthoff, J., Streit, L.: The Feynman integrand as a Hida distribution. J. Math. Phys. 32, 2123–2127 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  64. Doering, C.R.: Nonlinear parabolic stochastic differential equations with additive colored noise on \(\mathbb{R}^{d} \times \mathbb{R}_{+}\): a regulated stochastic quantization. Commun. Math. Phys. 109, 537–561 (1987)

    Article  MathSciNet  Google Scholar 

  65. Eachus, W.J., Streit, L.: Exact solution of the quadratic interaction Hamiltonian. Rep. Math. Phys. 4, 161–182 (1973)

    Article  MathSciNet  Google Scholar 

  66. Eberle, A.: Uniqueness and Non-Uniqueness of Semigroups generated by Singular Diffusion Operators. Lecture Notes in Mathematics, vol. 1718. Springer, Berlin (1999)

    Google Scholar 

  67. Fröhlich, J., Park, T.M.: Remarks on exponential interactions and the quantum sine-Gordon equation in two space-time dimensions. Helv. Phys. Acta 50, 315–329 (1977)

    MathSciNet  Google Scholar 

  68. Fröhlich, J., Seiler, E.: The massive Thirring-Schwinger model (QED2): convergence of perturbation theory and particle structure. Helv. Phys. Acta 49, 889–924 (1976)

    MathSciNet  Google Scholar 

  69. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. 2nd revised and extended edn. de Gruyter Studies in Mathematics, vol. 19. Walter de Gruyter & Co., Berlin (2011)

    Google Scholar 

  70. Gatarek, D., Goldys, B.: Existence, uniqueness and ergodicity for the stochastic quantization equation. Studia Math. 119, 179–193 (1996)

    MathSciNet  MATH  Google Scholar 

  71. Glimm, J., Jaffe, A.: Quantum Physics. Springer, New York (1981). 2nd edn., (1986)

    Google Scholar 

  72. Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  73. Grothaus, M., Streit, L.: Construction of relativistic quantum fields in the framework of white noise analysis. J. Math. Phys. 40, 5387–5405 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  74. Grothaus, M., Streit, L., Vogel, A.: Feynman integrals as Hida distributions: the case of non-perturbative potentials. Astrisque 327, 55–68 (2010)

    MathSciNet  MATH  Google Scholar 

  75. Grothaus, M., Streit, L., Vogel, A.: The complex scaled Feynman-Kac formula for singular initial distributions. Stochastics 84(2–3), 347–366 (2012)

    MathSciNet  MATH  Google Scholar 

  76. Guerra, F., Rosen, J., Simon, B.: The P(Φ)2-Euclidean quantum field theory as classical statistical mechanics. Ann. Math. 101, 111–259 (1975)

    Article  MathSciNet  Google Scholar 

  77. Hairer, M.: A theory of regularity structures. Invent. Math. 198, 269–504 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  78. Hida, T., Kuo, H.-H., Potthoff, J., Streit, L.: White Noise: An Infinite Dimensional Calculus. Mathematics and Its Applications, vol. 253. Kluwer Academic Publishers Group, Dordrecht (1993)

    Google Scholar 

  79. Hida, T., Potthoff, J., Streit, L.: Dirichlet forms and white noise analysis. CMP 116, 235–245 (1988).

    MathSciNet  MATH  Google Scholar 

  80. Hida, T., Streit, L.: Generalized Brownian functionals and the Feynman integral. Stoch. Process. Appl. 16, 55–69 (1983)

    MathSciNet  MATH  Google Scholar 

  81. Hida, T., Streit, L.: On quantum theory in terms of white noise. Nagrega Math. J. 68, 21–34 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  82. Høegh-Krohn, R.: A general class of quantum fields without cut-offs in two space-time dimensions. Commun. Math. Phys. 21, 244–255 (1971)

    Article  MathSciNet  Google Scholar 

  83. Hu, Y., Kallianpur, G.: Exponential integrability and application to Stochastic quantization. Appl. Math. Optim. 37, 295–353 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  84. Iwata, K.: Reversible Measures of a P(ϕ)1-Time Evolution. Probabilistic Methods in Mathematical Physics, pp. 195–209 (1985)

    Google Scholar 

  85. Iwata, K.: An infinite dimensional stochastic differential equation with state space \(C(\mathbb{R})\). Probab. Theory Relat. Fields 74, 141–159 (1987)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  87. Kawabi, H., Röckner, M.: Essential self-adjointness of Dirichlet operators on a path space with Gibbs measures via an SPDE approach. J. Funct. Anal. 242, 486–518 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  88. Kondratiev, Y.G., Leukert, P., Potthoff, J., Streit, L., Westerkamp, W.: Generalized functionals in Gaussian spaces: the characterization theorem revisited. J. Funct. Anal. 141(2), 301–318 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  89. Kusuoka, S.: Høegh-Krohn’s model of quantum fields and the absolute continuity of measures In: Albeverio, S., Fenstad, J.E., Holden, H., Lindstrøm, T. (eds.) Ideas and Methods in Mathematical Analysis, stochastics, and Applications, pp. 405–424. Cambridge University Press (1992)

    Google Scholar 

  90. Liskevich, V., Röckner, M.: Strong uniqueness for certain infinite dimensional Dirichlet operators and applications to stochastic quantization. Ann. Scuola Norm. Pisa 27(1), 69–91 (1998)

    MathSciNet  MATH  Google Scholar 

  91. Leukert, S., Schäfer, J.: A rigorous construction of Abelian Chern-Simons path integrals using white noise analysis. Rev. Math. Phys. 8, 445–456 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  92. Ma, Z.M., Röckner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  93. Mazzucchi, S.: Mathematical Feynman Path Integrals and Applications. World Scientific, Singapore (2009)

    Book  MATH  Google Scholar 

  94. Mazzucchi, S.: Functional-integral solution for the Schrödinger equation with polynomial potential: a white noise approach. DAQP 14, 675–688 (2011)

    MathSciNet  MATH  Google Scholar 

  95. Mihalache, S.: Stochastische Quantisierung bei exponentieller Wechselwirkung. Diploma Thesis, Bonn (2006)

    Google Scholar 

  96. Mikulevicius, R., Rozovskii, B.: Martingale problems for stochastic PDE’s. In: Carmona, R., Rozovskii, B. (eds.) Stochastic Partial Differential Equations: Six Perspectives, pp. 243–325. AMS (1999)

    Google Scholar 

  97. Mitter, S.: Markov random fields, stochastic quantization and image analysis. Math. Appl. 56, 101–109 (1989)

    MathSciNet  MATH  Google Scholar 

  98. Nelson, E.: The free Markov field. J. Funct. Anal. 12, 211–227 (1973)

    Article  MATH  Google Scholar 

  99. Otto, F., Weber, H., Westdickenberg, M.G.: Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size. Electron. J. Probab. 19, 76 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  100. Parisi, G., Wu, Y.S.: Perturbation theory without gauge fixing. Sci. Sin. 24, 483–496 (1981)

    MathSciNet  Google Scholar 

  101. Potthoff, J., Streit, L.: A characterization of Hida distributions. J. Funct. Anal. 10, 212–229 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  102. Potthoff, J., Streit, L.: Invariant states on random and quantum fields: ϕ bounds and white noise analysis. J. Funct. Anal. 111, 295–311 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  103. Pitt, L.: A Markov property for Gaussian processes with a multidimensional time. Arch. Ration. Mech. Anal. 43, 367–391 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  104. Röckner, M.: Traces of harmonic functions and a new path space for the free quantum field. J. Funct. Anal. 79, 211–249 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  105. Röckner, M., Zhang, T.S.: Uniqueness of generalized Schrödinger operators and applications. J. Funct. Anal. 105, 187–231 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  106. Simon, B.: The P(Φ)2-Euclidean (Quantum) Field Theory. Princeton University Press, Princeton NJ, (1974)

    Google Scholar 

  107. Streit, L.: An introduction to theories of integration over function spaces. Acta Phys. Austriaca, Suppl. II, 2–21 (1965)

    Google Scholar 

  108. Streit, L.: Stochastic differential equations. A pedagogical random walk. In: Nonlinear Klein-Gordon and Schrödinger Systems: Theory and Applications (Madrid, 1995), pp. 87–109. World Scientific, River Edge (1996)

    Google Scholar 

  109. Streit, L.: Beyond Fock Space. An Introduction to Infinite Dimensional Analysis for Physicists. Methods and Applications of Infinite Dimensional Analysis, vol. 3–34. Central Visayan Institute Foundation, Jagna (2006)

    Google Scholar 

  110. Streit, L.: Feynman Integrals as Generalized Functions on Path Space: Things Done and Open Problems. Path Integrals, vol. 78–85. World Scientific, Hackensack (2008)

    Google Scholar 

  111. Streater, R.: Euclidean quantum mechanics and stochastic integrals. In: Stochastic Integrals. Lecture Notes in Mathematics, vol. 851, pp. 371–393. Springer (1981)

    Google Scholar 

  112. Symanzik, K.: Euclidean quantum field theory. In: Jost, R. (ed.) Local Quantum Theory. Academic, New York (1969)

    Google Scholar 

  113. Varadhan, S.R.S.: Appendix to “Euclidean quantum field theory” by K. Symanzik. In: Jost, R. (ed.) Local Quantum Theory. Academic Press, New York (1969)

    Google Scholar 

Download references

Acknowledgements

I am very grateful to the organizers for the honour and pleasure they gave me by their invitation to present this little “homage” to my very dear friend Ludwig. I thank Luigi Borasi, Tobias Kuna, anonymous referees and the editors for helpful discussions respectively suggestions. I am also grateful to Nadine Kunze and Luigi for their help with the setting of the manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sergio Albeverio .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Albeverio, S. (2016). Along Paths Inspired by Ludwig Streit: Stochastic Equations for Quantum Fields and Related Systems. In: Bernido, C., Carpio-Bernido, M., Grothaus, M., Kuna, T., Oliveira, M., da Silva, J. (eds) Stochastic and Infinite Dimensional Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-07245-6_1

Download citation

Publish with us

Policies and ethics