Nonrelativistic current algebra in the N / V limit

Abstract

The case of a noninteracting infinite Bose gas at zero temperature is studied in the formalism of local current algebras, using the representation theory of nuclear Lie groups. The class of representations describing such a system is obtained by taking an ``N / V limit'' of the finite case. These representations can also be determined uniquely from the solutions of a functional differential equation, which follows in turn from a condition on the ground state vector. Finally a system of functional differential equations is formulated for a theory with interactions, using a proposed definition of indefinite functional integration.

Factor representations of diffeomorphism groups

“…which is the functional for the free boson gas given in [8] and is called the Poisson function in [24]. These representations have also been systematically studied in [11] and [16].…”

“…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 .…”

“…[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].…”

“…We also describe the connections between the N/V (thermodynamic) limits [8,14] and the asymptotic character theory of the infinite unitary group [3,25].…”

Poisson configuration spaces, von Neumann algebras, and harmonic forms

“…A crucial role here is played by a probability measure on Γ X (in particular, a Poisson or Gibbs measure), which is quasiinvariant with respect to the action of a group of diffeomorphisms of X. This philosophy, inspired by the pioneering works [1] and [2], has been initiated and developed in [3], [4] and has lead to many interesting and important results in the field of stochastic analysis on configuration spaces and its applications, see also references in [5], [6].…”