Algebras of observables of nearly canonical physical theories. II (Q1071288)

This paper is the second in a series on algebras of observables [for part I see the review above]. In this paper an action of a Lie group G on a manifold M is the starting point. A vector bundle E over M is studied and \(\Gamma(M)\) is the space of sections. Associative algebras \(\{\Gamma(M),\pi\}\) are studied. Special cases for M,G were studied earlier. An observable algebra is a triple \(\{A,\pi,\sigma\}\) satisfying \[ \pi (AB)\sigma (C)=\sigma (AC)\pi (B)+\pi (\sigma(ABC))\quad and\quad \sigma (AB)\sigma (C)=\sigma (AC)\sigma (B)+\sigma (\sigma(ABC))) \] when \(\pi\), resp. \(\sigma\), correspond to operator multiplication, resp. commutator (Poisson) bracket. The operations \(\pi\), resp. \(\sigma\), are given by kernels of the form \[ \pi(AB)(x)=\int e^{i(k+p)x} \alpha(k,p)\tilde A(k)\tilde B(p)dk dp \] for the classical case when the Fourier transform \(\tilde{\;}\) is defined. The functions \(\alpha(\cdot,\cdot)\) are found as solution to functional equations. A class of binary Lie algebras is introduced (type \(\Delta)\) and it is proved that these are actually Lie algebras.