on Feynman Path Integral and Microlocal Analysis |

This introductory workshop is open to everyne. We have no official registration procedures. Speaker must give introductory talks about his important result without respect to the past and the present. Especially, aged professor must give an introductory talk about his important result for young researchers, even if he talked many times decades ago. Because young researchers have never heard the talk of aged professor directly, and young researchers want to know what the aged professor was thinking and how the aged professor solved it.

Program Program(Japanese) Access to RIMS The previous workshop (2010)

Speakers (Alphabet's order) and Introductory Lectures

Dong Hyun Cho (Kyonggi University, Korea), A survey of an analogue of conditional analytic Feynman integrals on a function space

Daisuke Fujiwara (Gakushuin University, Japan), Stationay phase method for oscillatory integrals over a space of large dimension

Takeyuki Hida (Nagoya University, Japan), White noise approach to Feynman path integrals

Takashi Ichinose (Kanazawa University, Japan), Imaginary-time path integrals for three magnetic relativistic Schr\"odinger operators

Bong Jin Kim (Daejin University, Korea), A note on the integral transforms on function spaces

Kun Sik Ryu (Hannam University, Korea), Introduction to the analogue of Wiener measure space and its applications.

Byoung Soo Kim (Seoul National University of Technology, Korea), Introduction to Feynman's operational calculi for noncommuting operators

Naoto Kumano-go (Kogakuin University, Japan), Phase space Feynman path integrals by time slicing approximation

Organizers:Naoto Kumano-go (Kogakuin University, Japan) ft24343kogakuin.ac.jp,

Byoung Soo Kim (Seoul National University of Science and Technology, Korea),

Susumu Yamazaki (Nihon University),

Yasuo Chiba (Tokyo Univsersity of Technology)

Abstracts:

Kun Sik Ryu, "Introduction to the analogue of Wiener measure space and its applications"

In this lecture, we will introduce the definition of analogue of Wiener measure space and the motivation of it. We investigate various properties of it - integration formulae for some functionals on analogue of Wiener space,the relationship among the Bartle integral and the measure-valued measure on analogue of Wiener space,the relationship among the Dovrakov integral and the operator-valued measure on analogue of Wiener space,the simple formula for coditional expactation with respect to analogue of Wiener measure,the measure-valued Feynman-Kac formula and Volterra integral equation, including the basic calculation for it. Contents;1) Historical Background and Preliminies 2)The complex-valued analogue of Wiener measure 3) Fernique's Theorem and integration formula for some analogue of Wiener functionals4) Translation Theorem and Paley-Wiener -Zygmund integral for analogue of Wiener measure 5) the relationship among the Bartle integral and the measure-valued measure on analogue of Wiener space 6)the relationship among the Dovrakov integral and the operator-valued measure on analogue of Wiener space7) measure-valued Feynman-Kac formula 8) Volterra integral equation for the measure-valued Feynman-Kac formula.

Byoung Soo Kim,"Introduction to Feynman's operational calculi for noncommuting operators"

Feynman's 1951 paper on the operational calculus for noncommuting operators arouse out of his ingenious work on quantum electrodynamics and was inspired in part by his earlier work on the Feynman path integral.Indeed, Feynman thought of his operational calculus as a kind of generalized path integral.Much surprisingly varied work on the subject has been done since by mathematicians and physicists.Recently Jefferies and Johnson developed mathematical rigorous approach to Feynman's operational calculi.In this talk we give definitions and properties of Feynman's operational calculi initiated by Jefferies and Johnson. In particular, extraction of a linear factor and measure permutation formula for Feynman's operational calculi will be given.

Dong Hyun Cho,"A survey of an analogue of conditional analytic Feynman integrals on a function space"

In this lecture, we introduce two kinds of simple formulas for the conditional expectations on the analogue of Wiener space which is introduced by Ryu and Im. Using the simple formulas, we evaluate the conditional analytic Feynman integrals of various kinds of functions which are useful in Feynman integration theories itself and quantum mechanics. We then find a solution of an integral equation which is formally equivalent to the Schr\"odinger differential equation. We also provide a change of scale transformation using the simple formulas and prove that the operator-valued Feynman integral can expressed through the analogues of conditional analytic Feynman Feynman integrals on that space.