ISSN 1880-2818

数理解析研究所講究録 1797

RIMS 共同研究

Introductory Workshop on

Feynman Path Integral and Microlocal Analysis

京都大学数理解析研究所
2012 年 6 月

RIMS K\^okyuroku 1797

In troductory Workshop on
Feynman Path In tegral and Microlocal Analysis
June 2124, 2011$\sim$

edited by Naoto Kumano-go, Susumu Yamazaki,

Byoung $SooKim$ and Yasuo Chiba

June, 2012

Research Institute for Mathematical Sciences
Kyoto University, Kyoto, Japan
This is a report of research done at the Research Institute for Mathematical
Sciences, Kyoto University. The papers contained herein are in final formand will not be submitted for publication elsewhere.

PREFACE
This volume contains a set of introductory articles which were contributed by the invited
speakers of the RIMS joint research workshop: “Introductory Workshop on Feynman Path Integral and Microlocal Analysis“, which was held at Research Institute for Mathematical Sciences,
Kyoto University in June 21-24, 2011. The aims of this workshop were to overview the present
stage of Japanese and Korean research in Feynman path integral and microlocal analysis and toobtain prospects for the future research. In order to do so, all the speakers gave the introductorylectures about their research fields, which contributed to capturing a whole picture of the current
progress in the field. The editors would like to thank all the speakers who submitted their papers
to this volume and also all those participants who attended the workshop.
Naoto KUMANO-GO

Susumu YAMAZAKI
Byoung Soo KIM
Yasuo CHIBA

Thisjoint research is supported by RIMS and JSPS KAKENHI(C)21540196 and $(C)20540191$ .
We would like to express our sincere gratitude to RIMS and, in particular, to the secretary staffsof RIMS.

RIMS Joint Research

Introductory Workshop onFeynman Path Integral and Microlocal Analysis
Organizers: Naoto Kumano-go (Kogakuin University)
Byoung Soo Kim (Seoul National University of Science and Technology)

Susumu Yamazaki (Nihon University)
Yasuo Chiba (Tokyo University of Technology)

Date: from June 21 (Tuesday) to June 24 (Friday), 2011
Venue: RIMS, Kyoto University, Room No. 111
The following program is tentative. We can get the present version of the program $hom$
the URL: $htt\mathfrak{v}://www.ns.kogakuin.ac.in/\sim R24343/worksho\mathfrak{v}2011-6$

June 21 (Tuesday)
10:00 11:00 Takashi Ichinose (Kanazawa University)$\sim$

Imaginary-time path integrals for three magnetic relativistic Schr\"odinger operators,Part 1
11:20 12:20 Bong Jin Kim (Daejin University)
A note on the integral transforms on function spaces, Part 1
14:00 15:00 Takeyuki Hida (Nagoya University)
White noise approach to Feynman path integrals, Part 1$\sim$$\sim$

16:20 Kun Sik Ryu (Hannam University)
Introduction to the analogue of Wiener measure space and its applications, Part 1

15:20$\sim$

June 22 (Wednesday)
10:00 11:00 Daisuke Fujiwara (Gakushuin University)
Stationary phase method for $oSCm_{atory}$ integrals over a space of large dimension,
Part 1
11:20 12:20 Bong Jin Kim (Daejin University)
A note on the integral transforms on function spaces, Part 2$\sim$$\sim$

14:00

15:00 Takeyuki Hida (Nagoya University)$\sim$White noise approach to Feynman path integrals, Part 2

15:20$\sim$

16:20 Kun Sik Ryu (Hannam University)

Introduction to the analogue of Wiener measure space and its applications, Part 2
June 23 (Thursday)
10:00 11:00 Daisuke Fujiwara (Gakushuin University)$\sim$

Stationary phase method for oscillatory integrals over a space of large dimension,
Part 2
11:20 12:20 Dong Hyun Cho (Kyonggi University)$\sim$

A survey of an analogue of conditional analytic Feynman integrals on a function space,Part 1

14:00$\sim$

15:00 Byoung Soo Kim (Seoul National University of Science and Technology)

Introduction to $Feynman^{1}s$ operational calculi for noncommuting operators, Part 1
15:20 16:20 Naoto $Kumano\cdot go$ (Kogakuin University)$\sim$

Phase space Feynman path integrals by time slicing approximation, Part 1

June 24 (Friday)
10:00 11:00 Takashi Ichinose (Kanazawa University)$\sim$$Imaginary\cdot time$ path integrals

for three magnetic relativistic Schr\"odinger operators,Part 2
11:10 12: 10 Dong Hyun Cho (Kyonggi University)$\sim$

A survey of an analogue of conditional analytic Feynman integrals on a function space,Part 2
14:00 15:00 Byoung Soo Kim (Seoul National University of Science and Technology)
Introduction to Feynman’s operational calculi for noncommuting operators, Part 2
15:20 16:20 Naoto $Kumano\cdot go$ (Kogakuin University)
Phase space Feynman path integrals by time slicing approximation, Part 2$\sim$$\sim$

This joint research is supported by RIMS and JSPS KAKENHI(C)21540196 and$(C)20540191$ .

Introductory Workshop on Feymnan Path Integral and Microlocal Analysis

RIMS 共同研究報告集
2011 年 6 月 21 日
研究代表者

熊ノ郷

副代表者

山崎

$\sim 6$

月 24 日直人 (Naoto Kumano-go)
晋 (Susumu Yamazaki)’/

Byoung Soo Kim〃

千葉

康生 (Yasuo Chiba)
目次

1. Imagin$a1\gamma$-Time Path Integrals for Three Magnetic Relativistic Schr\"odinger Operators— 1

金沢大 (Kanazawa U.)

一瀬孝 (Takashi Ichinose)

2. A Note on the Integral Transforms on a Function Space I, I $————————–20$
Daejin U.

Bong Jin Kim

3. White Noise Approach to Feynman Path Integrals and Some of the Related ToPics
名大 (Nagoya U.)

飛田

武幸 (Takeyuki Hida)

4. Introduction to Analogue of Wiener Measure Space and Its Applications
Han Nam U.

$–42$$————51$

Kun Sik Ryu
Man Kyu’/${\rm Im}$

Kon Yang U.

Ki Seong Choi

5. Stationary Phase Method, Feynman Path Integrals and Integration by Parts Formula –86
学習院大理 (Gakushuin U.)

藤原大輔 (Daisuke Fujiwara)

6. A Survey of Conditional Expectations on an Analogue of Wiener Space
Kyonggi U.

115

Dong Hyun Cho

7. Feynman’s Operational Calculi for Noncommutin Operators$g$Seoul Nat. U. Sci. Tech.

$————————144$
Byoung Soo Kim

8. Phase Space Feynman Path Integrals –Calculation Examples via Piecewise
Bicharacteristic Paths$———————————167$

工学院大工 (Kogakuin U.)

熊ノ郷直人 (Naoto Kumano-go)

9. Phase Space Feynman Path Integrals –as Analysis on Path Space via Piecewise
Constant Paths$—————————————————187$

工学院大・工 (Kogakuin U.)

熊ノ郷直人 (Naoto Kumano-go)