Functional Integration

1. Front Matter

   Pages i-x

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2. A Rigorous Mathematical Foundation of Functional Integration

   1. A Rigorous Mathematical Foundation of Functional Integration

      • P. Cartier, C. DeWitt-Morette

      Pages 1-50

3. Physics on and near Caustics

   1. Physics on and near Caustics

      • C. DeWitt-Morette, P. Cartier

      Pages 51-66

4. Quantum Equivalence Principle

   1. Quantum Equivalence Principle

      • H. Kleinert

      Pages 67-92

5. Variational Perturbation Theory for Path Integrals

   1. Variational Perturbation Theory for Path Integrals

      • H. Kleinert

      Pages 93-95

6. Functional Integration and Wave Propagation

   1. Functional Integration and Wave Propagation

      • S. K. Foong

      Pages 97-130

7. Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms

   1. Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms

      • A. Sokal

      Pages 131-192

8. Path Integral Simulation of Long-Time Dynamics in Quantum Dissipative Systems

   1. Path Integral Simulation of Long-Time Dynamics in Quantum Dissipative Systems

      • Nancy Makri

      Pages 193-211

9. Développements Récents sur les Groupes de Tresses Applications à la Topologie et à l’Algèbre

   1. Développements Récents sur les Groupes de Tresses Applications à la Topologie et à l’Algèbre

      • P. Cartier

      Pages 213-246

10. An Introduction to Knot Theory and Functional Integrals

    1. An Introduction to Knot Theory and Functional Integrals

       • L. Kauffman

       Pages 247-308

11. Locally Self-Avoiding Walks

    1. Locally Self-Avoiding Walks

       • D. Iagolnitzer, J. Magnen

       Pages 309-325

12. Gauge Theory without Ghosts

    1. Gauge Theory without Ghosts

       • B. DeWitt, C. Molina-París

       Pages 327-361

13. Localization and Diagonalization: A Review of Functional Integral Techniques for Low-Dimensional Gauge Theories and Topological Field Theories

    1. Localization and Diagonalization: A Review of Functional Integral Techniques for Low-Dimensional Gauge Theories and Topological Field Theories

       • M. Blau, G. Thompson

       Pages 363-410

14. Participants Contributions by Title and Abstract

    1. Front Matter

       Pages 411-411

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    2. A Constructive Analog of Zimmermann’s Formula

       • Abdelmalek Abdesselam

       Pages 413-413

    3. Heat Kernel as the Basis for the Functional Integration in Quantum Field Theory

       • Ivan G. Avramidi

       Pages 413-413

    4. Wiener Integration for Quantum Systems: A Unified Approach to the Feynman-Kac Formula

       • Bernhard Bodmann, Simone Warzel

       Pages 414-414

    5. Anomalies in Brst Anti Brst Approach as a Functional Integration Problem

       • Violeta Calian

       Pages 414-414

    6. Semiclassical Approximation Using Wiener Measure

       • Fabienne Castell

       Pages 415-415

    7. Open Problem: The Measure for the Functional Integral

       • Bryce DeWitt

       Pages 415-415

The program of the Institute covered several aspects of functional integration -from a robust mathematical foundation to many applications, heuristic and rigorous, in mathematics, physics, and chemistry. It included analytic and numerical computational techniques. One of the goals was to encourage cross-fertilization between these various aspects and disciplines. The first week was focused on quantum and classical systems with a finite number of degrees of freedom; the second week on field theories. During the first week the basic course, given by P. Cartier, was a presentation of a recent rigorous approach to functional integration which does not resort to discretization, nor to analytic continuation. It provides a definition of functional integrals simpler and more powerful than the original ones. Could this approach accommodate the works presented by the other lecturers? Although much remains to be done before answering "Yes," there seems to be no major obstacle along the road. The other courses taught during the first week presented: a) a solid introduction to functional numerical techniques (A. Sokal) and their applications to functional integrals encountered in chemistry (N. Makri). b) integrals based on Poisson processes and their applications to wave propagation (S. K. Foong), in particular a wave-restorer or wave-designer algorithm yielding the initial wave profile when one can only observe its distortion through a dissipative medium. c) the formulation of a quantum equivalence principle (H. Kleinert) which. given the flat space theory, yields a well-defined quantum theory in spaces with curvature and torsion.

• Monte Carlo method
• algorithms
• dynamics
• gauge theory
• mathematics
• mechanics
• physics
• quantum theory
• topology
• wave

• The University of Texas at Austin, Austin, USA

  Cecile DeWitt-Morette

• Ecole Normale Supérieure, Paris, France

  Pierre Cartier

• Université de Corse, Corte, France

  Antoine Folacci

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