Room 201, Astronomy-Mathematics Building, NTU

(台灣大學天文數學館 201室)

Free Fields and Analysis of the Wiener Space

Narn-Rueih Shieh (National Taiwan University)

Abstract

By the Wiener space

, it means the space

endowed with the

norm and with a unique p.m.

so that the

is the Brownian motion under

( a Gaussian stochastic process with independent stationary increments ). Thus the Wiener integral

is defined for suitable class of functionals

. This is a mathmatical version of the Feynman path-intergral. On the other hand, a free-field is assumed to be the building block of QFT (cf: J. Glimm and A. Jeff, Quantum Physics, a functional integral point of view, Springer 1981 and 1987) and is assumed to be a pure-flunciated Gaussian field one the space of paths (cf: S. Janson, Gaussian Hilbert Spaces, Cambridge 1997).

In this contributed talk, we discuss a math construction which may lead to a non-Gaussian free field (cf: Shieh, J. Pseudo-Diff. Op's 2012). This gives a puzzle questioned by Remi Rhodes ( Paris-Est, private communications). Maybe the solution depends on deeper understanding of Analysis on the Wiener Space (cf: N. Bouleau and F. Hirsch, Dirichlet forms and Analysis on Wiener Space, De Gruyler 1991)