Brownian Motion, Markov Cosurfaces, Higgs Fields

  • Sergio Albeverio
  • Raphael Høegh-Krohn
Part of the NATO ASI Series book series (NSSB, volume 144)


Brownian motion enters quantum theory in many ways, e.g. in the stochastic mechanical formulation of quantum theory, see e.g. ([1],[2],°) in the formulation of quantum field theory as Euclidean field theory, with covariances given by potential operators belonging to Brownian motions ([29], [3],[4]), in the expression of certain field theoretical models as “gases of local times of Brownian motion” ([5] – [7],[26],[33],[36]). In this lecture we shall discuss yet other uses of Brownian motion, and related processes, in the description of the quantum world, more precisely in connection with random group-valued hypersurfaces and gauge fields, leading to a (noncommutative) stochastic analysis with “higher dimensional time”. Let us start with gauge fields. Formally a pure Yang-Mills Euclidean measure gives a “white noise” type of distribution to the curvature 2-form F. Finding the corresponding connection a s.t. F = Da (D covariant derivative) implies then solving a stochastic partial differential equation for Lie algebra-valued one forms. The holonomyoperator is a stochastic one. This is one motivation for developing a suitable theory of stochastic mapping from curves into Lie groups, and extending the study of multiplicative stochastic differential equations to the case of multiplicative stochastic partial differential equations.


Brownian Motion Continuum Limit Gauge Field Stochastic Partial Differential Equation Higgs Field 
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Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • Sergio Albeverio
    • 1
  • Raphael Høegh-Krohn
    • 2
  1. 1.Mathematisches Institut, Bielefeld-Bochum Research Centre Stochastics (BiBoS)Ruhr-UniversitätBochum 1Deutschland
  2. 2.Matematisk InstituttUniversitetet i OsloOslo 3Norway

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