Diffeomorphism Groups And Local Symmetries: Some Applications In Quantum Physics

Geometric hydrodynamics and infinite-dimensional Newton’s equations

“…Those of perhaps most interest from our point of view were proposed by Smolentsev [72,73], who used diffeomorphism groups to describe the motions of a barotropic fluid, and by Ebin [19], who used a similar framework to study, among others, the incompressible limit of slightly compressible fluids. In the early 1980s, Doebner, Goldin, and Sharp [17,27] began to develop links between representations of diffeomorphism groups, ideal fluids, and nonlinear quantum systems, revisiting in the process the classical transform of Madelung [48,49]. More recently, motivated by the problems of optimal transport, von Renesse [79] used it to relate the Schrödinger equations with a variant of Newton's equations defined on the space of probability measures (see Section 9 below for details).…”

Factor representations of diffeomorphism groups

“…[8,9,10,14,15], Kirillov [13], and Vershik-Gel'fand-Graev [24] in the early 1970's. The explicit link with the unitary group was mentioned by Okomoto and Sakurai [17].…”

“…These infinite volume limits can be naturally described in our framework. For a discussion of these limits see [8,10,14]. We first observe that if L is a real Hilbert space with a non-zero vector ω, then the subgroup K of O(L ) of orthogonal transformations W such that W ω = ω can be identified with the orthogonal group of L ω .…”