CONTENTS -- Preface -- Rigorous Feynman Path Integrals, with Applications to Quantum Theory, Gauge Fields, and Topological Invariants S. Albeverio, A . Hahn and A. N. Sengupta -- 1. Introduction -- 2. Rigorous Feynman path integrals and probabilistic functional integrals. Some applications to quantum theory -- 2.1. A short reminder of Wiener integrals and some of their applications to quantum theory -- 2.2. Rigorous Feynman Path integrals -- 2.2.1. Definition and properties of mathematical Feynman path integrals -- 2.2.2. An infinite dimensional version of the stationary phase method applied to a mathematical realization of Feynman's path integrals -- 2.2.3. Applications to the Schrodinger equation -- 2.3. The White Noise Analysis framework -- 3. The problem of gauge fields. Yang-Mills theory and Chern- Simons theory -- 3.1. Principal Bundles -- 3.2. Connection -- 3.3. Parallel Transport -- 3.4. Holonomy -- 3.5. The spaces K'(P -- V ) -- 3.6. Curvature and Covariant Derivatives -- 3.7. Gauge Transformations -- 3.8. The action of B on A -- 3.9. The Yang-Mills action -- 3.10. Yang-Mills as a dynamical system -- 3.11. Lagmngian to Hamiltonian -- 3.12. Comparison with Yang-Mills equations -- 3.13. Moment Map for the Gauge Group Action -- 3.14. The Chern-Simons Action -- 4. Quantization of Yang-Mills and Chern-Simons models -- 4.1. Quantization of Yang-Mills models -- 4.2. Quantization of Chern-Simons models -- 4.2.1. Axial Gauge fixing for Chern-Simons models on R3 -- 4.2.2. The Chern-Simons path integral on R3 in axial gauge as a generalized distribution -- 4.2.3. Loop-smearing -- 4.2.4. Framing -- 4.2.5. Admissible Links and admissible Fkamings -- 4.2.6. Computation of the WLOs -- 4.2.7. Examples -- Acknowledgements: -- References -- Analytic Yeh-Feynman Integral and Generalized Conditional Yeh-Wiener Integral J. S. Chang -- 1. Introduction.

2. Preliminiaries -- 3. Banach algebra S(L2(Q)) -- 4. Analytic Yeh-Feynman integrals -- 5. Generalized conditional Yeh-Wiener integral -- References -- The Selfadjoint Lie-Trotter-Kato Product Formula in Operator Norm and Time-Sliced Approximation to Imaginary-time Path Integral T. Ichinose -- 1. Introduction -- 2. Outline of Proof -- 3. The imaginary-time path integral for the heat equation -- 4. Speculative Comment -- Acknowledgements. -- References -- Vassiliev Invariants and Functional Integration without Integration L. H. Kauflman -- 1. Introduction -- 2. Vassiliev Invariants and Invariants of Rigid Vertex - Graphs -- 3. Integration without integration -- 3.1. Functional Derivatives -- 4. Vassiliev Invariants and Witten's Functional Integral -- References -- Wiener Analysis and Cyclic Homology R. Le'andre -- 1. Introduction -- 2. The first calculus -- 3. The second calculus -- References -- On the Affine Metaplectic Group O. Rusk -- 1. Introduction -- 2. Preliminaries -- 2.1. The Boson Fock algebra -- 2.2. The symplectic group -- 3. The complexification (H) -- 3.1. The affine symplectic group -- 4. The Fock space representations -- 4.1. Exponentiability -- 4.2. Examples -- References -- Open Quantum Systems and Classical Trajectories R. Rebolledo -- 1. Introduction -- 2. Open Quantum Systems -- 2.1. The approach through Feynman integrals -- 2.2. Master Equations and Quantum Markov semigroups -- 3. Classical reduction of a quantum Markov semigroup defined on a von Neumann algebra -- 3.1. The reduction by mmimal abelian subalgebras -- 3.2. Reduction by a bounded normal operator -- 3.3. Reduction by an unbounded self-adjoint operator -- 3.4. Construction of the canonical Markov process associated to the reduced semigroup -- 3.5. Master equations and form generators -- 4. Examples -- 4.1. Quantum Brownian Motion -- 4.2. Quantum exclusion model.

4.2.1. The initial space -- 4.2.2. The algebra (h0) -- 4.2.3. Configurations -- 4.2.4. The quantum Markov semigroup -- Acknowledgements -- References -- Stochastic Oscillatory Integrals: Asymptotics and Exact Expressions for Quadratic Phase Function S. Taniguchi -- Introduction -- 1. Analyticity -- 2. Asymptotics and localization -- 3. Explicit expression and exponential decay -- 4. Cameron-Martin transform and Jacobi equation -- References -- Fourier-Feynman Transforms on Wiener Spaces I. Yoo, K. S. Chang, D. H. Cho, B. S. Kim and T. S. Song -- 1. Introduction -- 2. Fourier-Feynman transform on classical Wiener space -- 3. Fourier-Feynman transform on abstract Wiener space -- 4. Fourier-Feynman transform on the space of abstract Wiener space valued continuous functions -- References -- Infrared Bounds and Bose-Einstein Condensation: Study of a Class of Diagonalizable Perturbations of the Free Boson Gas M. Corgini and H. Torres -- 1. Introduction -- 2. Basic Mathematical Notions -- 3. Bose Einstein Condensation -- 4. Previous Results -- 5. Huang Yang Luttinger Model System -- 6. A Class of Diagonalizable Perturbations of the F'ree Boson Gas -- References -- Quadratic Wiener Functionals and Dynamics on Grassmannians: The Framework and the Simplest Example K. Hara -- 1. Introduction -- 2. Framework -- 3. An example - LQvy's stochastic area -- References -- Feynman's Operational Calculi Blending Instantaneous and Continuous Phenomena in Feynman's Operational Calculi G. W. Johnson and L. Nielsen -- 1. Introduction -- 2. Time-Ordering within the Disentangling Algebra -- 3. Definition of the Disentangling Map -- the General Case -- 4. Calculation in Some Special Cases - Especially Discrete Case -- References -- On Quantum Stochastic Dynamics. Some Recent Developments A . W. Majewski -- 1. Introduction -- 2. Classical systems -- 3. Quantum spin systems.

4. Non-commutative Lp-spaces. -- 5. Quantum stochastic dynamical semigroups -- 6. Quantum diffusions for spin systems. -- 7. Discussion -- 7.1. Acknowledgements -- References -- Non Adapted Transformations of the Wiener Measure A . B. Cruzeiro -- 1. Introduction -- 2. Non linear transformations of the Wiener measure -- 3. Differentiation on the Wiener space. -- 3.1. The divergence operator -- 3.2. Non linear transformations -- 4. Tangent processes -- 5. Integration by parts on the path space of a Riemannian manifold -- 6. An asymptotic estimate for the vertical derivatives of the horizontal Laplacian -- References -- Feynman Integrals and White Noise Analysis J. L. Silva and L. Streit -- 1. Introduction -- 2. Review of white noise analysis -- 3. The free Feynman integrand -- 4. The perturbed Feynman integrand -- 5. The Feynman integrand for the harmonic oscillator -- 6. The Feynman integrand for a class of unbounded potentials -- Acknowledgments -- Bibliography.