On the Representations of the Local Current Algebra and the Group of Diffeomorphisms (I)

Abstract

Recently, several physicists and mathematicians^[1–8] have investigated the representations of the local current algebra and the group of diffeomorphisms, that are motivated by the theory of quantum physics and statistical physics. These representations are closely related to quasi-invariant measures ^[1–3],[6]. But the investigation on the measure, which is quasi-invariant with respect to the group of diffeomorphisms, began only a few years ago. In this paper, firstly we give the analytic expression of the Radon-Nikodym derivative of measure on the space of generalized functions, which is quasi-invariant with respect to the group of diffeomorphisms. Secondly, by means of this expression we give a method of finding out a class of representations of local current algebra in the theory of quantum physics.

The author thanks Professor Karl Gustafson for his invitation to attend the meeting and kind hospitality. The author is also acknowledging Professor Mityagin and the Department of Mathematics, The Ohio State University for their help to complete this paper during his visit in Columbus, Ohio.