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Some applications of functional integration

  • Sergio Alheverio
  • Philippe Blanchard
  • Raphael Høegh-Krohn
Functional Integration General Plenary Session
Part of the Lecture Notes in Physics book series (LNP, volume 153)

Abstract

We discuss some new developments in the theory of functional integration and some applications in nonrelativistic quantum theory, quantum field theory, statistical mechanics, solid state physics and hydrodynamics.

Keywords

Gauge Group Functional Integration Trace Formula Feynman Path Feynman Path Integral 
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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References

Ch. I.

  1. 1).
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  2. 2).
    See [3], [4], [8]. The derivation of asymptotic expansions for the solution of Schrödinger equation has been discussed also by methods of p.d.e. by Maslov, Duistermaat, Fujiwara (e.g. [22]) and others. There is a rich literature concerning compact manifolds (see e.g. [8]). The basic ideas for writing in the WKBJ approach the leading term in the trace formula for Schrödinger operators goes back to M. Gutzwiller (for discussions:and references, also in connection with quantum fields see e.g. [9]), The approach to the leading term as h → O is also discussed from a mathematical definition of Feynman path integrals in [10], [11]. The trace formula has been proven recently by methods of p.d.e. by Chazarain [12]. Recent very interesting work on the structure of classical orbits with prescribed period has been given by C. Conley and E. Zehnder [13].Google Scholar
  3. 3).
    A note on probabilitstic methods related to the Feynman path integral methods. We shall mention some recent related work which uses probabilistic path integrals rather than Feynman path integrals. 1. Asymptotics-of-function-space-integrals-with-respect-to-Gaussian measures There is a tradition (Donsker, Schilder, Pinkus, Varadhan) in such studies, see e.g. [14]. Recent results related to the one in 12) have been obtained by R. Ellis, J. Rosen concerning the case where the formal complex Gaussian measure of (2), is replaced by a Gaussian probability measure. Here instead of the stationary points of the phase one has to look at minimum points. The expansion corresponds to the one given in our case. The results have applications to the convergence of associated random variables and by this to some problems in classical statistical mechanics (for a very recent result see [15]). Related methods with other applications have been used recently by I. Davies and A. Truman [16] (see also [11] for manifolds and e.g. [17], for quantum fields). 2. Methods of stochastic-equations. The methods have been used in connection with solutions of the Schrödinger equations [18] and their eigenvalues and eigenfunctions [12], [19]. Recently an interesting relation between heuristic Feynman path integrals for the Coulomb problem on ℝ3 and Feynman path integrals for the harmonic oscillator on ℝ4 has been discovered by I.H. Duru and H. Kleinert [20]. Ph. Blanchard and M. Sirugue [21] have proven the validity of this relation replacing Feynman path integrals by Wiener ones.Google Scholar
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Ch. II. The results have connections with many other domains, e.g.: — Affine Lie algebras and number theory (see e.g. [11]) — Current groups (see e.g. [11], [12]) — The energy representation can be looked upon as a representation of gauge groups. There are also some connections with recent work on constructing approximate Euclidean measures for Yang-Mills fields [13]. — More generally new developments in stochastic geometry (see e.g. contributions by Bismut, Pinsky, Stroock in the volume [4]).

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Ch. III

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Ch. IV

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    A basic reference for early work in the statistical approach is A.S. Manin, A.M. Yaglom, Statistical Fluid Mechanics, M.I.T. Press: (1975).Google Scholar
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Ch. V

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    G.G. Emch, S. Albeverio, J.P. Eckmann, Rept. Math. Phys. 13, 73–85 (1978).Google Scholar
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    E.B. Davies, Quantum theory of open systems, Acad. Press, N.Y. (1976).Google Scholar
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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Sergio Alheverio
    • 1
    • 2
    • 3
  • Philippe Blanchard
    • 1
    • 2
    • 3
  • Raphael Høegh-Krohn
    • 1
    • 2
    • 3
  1. 1.Mathematisches InstitutRuhr-UniversitätBochum
  2. 2.Fakultät für PhysikUniversität BielefeldBielegeld
  3. 3.Matematisk InstituttUniversitetet i OsloOslo

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