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Martingale Convergence and the Exponential Interaction in ℝn

  • Sergio Albeverio
  • Raphael Høegh-Krohn

Abstract

In this lecture we discuss the exponential interaction in ℝn. We give a proof that the ultraviolet cut-off exponential interaction \( {m_{k}}\left( \xi \right) = \int {_{\Lambda }} :{e^{{\alpha {\xi _{k}}\left( x \right)}}} \): dx, where Λ is a fixed bounded region in Rn and k the ultraviolet cut-off parameter, is a martingale in k with respect to the free Euclidean measure µO. Moreover \( {{\text{e}}^{{{\text{ - }}\lambda {{\text{m}}_{{\text{k}}}}\left( \xi \right)}}} \) is a positive bounded submartingale and we prove that \( {{\text{e}}^{{{\text{ - }}\lambda {{\text{m}}_{{\text{k}}}}\left( \xi \right)}}} \) converges pointwise µO-almost everywhere and strongly in L1(dµO) as the ultraviolet cut-off k tends to infinity. Especially \( {{\text{e}}^{{{\text{ - }}\lambda {{\text{m}}_{{\text{k}}}}\left( \xi \right)}}}{\text{d}}{\mu _{{\text{o}}}}\left( \xi \right) \) converges weakly as k → ∞ which implies that the ultraviolet cut-off Schwinger functions for the exponential interaction in ℝn converge as k → ∞. For n > 4 this limit is the free Euclidean field. Further results concerning the cases n < 3 are also mentioned, as well as applications to the study of the energy representations of groups of mappings from Riemann manifolds into compact Lie groups.

Keywords

Gaussian Measure Borel Subset Energy Representation Borel Probability Measure Real Separable Hilbert Space 
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.

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Copyright information

© Springer-Verlag/Wien 1980

Authors and Affiliations

  • Sergio Albeverio
    • 1
    • 2
    • 3
    • 4
    • 5
  • Raphael Høegh-Krohn
    • 1
    • 2
    • 3
    • 4
    • 5
  1. 1.Fakultät für MathematikUniversität BielefeldFederal Republic of Germany
  2. 2.Fakultät für MathematikRuhr-Universität BochumFederal Republic of Germany
  3. 3.Matematisk InstituttUniversitetet i OsloNorway
  4. 4.Centre de Physique ThéoriqueCNRSMarseilleFrance
  5. 5.Université d’Aix-Marseille IIUER de LuminyFrance

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