Quantum Mechanics and Operator algebras on the Hilbert ball

Abstract

Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C${}^{\ast }$-algebra is represented as a function algebra on the set of pure states with a noncommutative $\ast$-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space $\mathcal{P}(\mathcal{H})$ of a Hilbert space $\mathcal{H}$. The space $\mathcal{P}(\mathcal{H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$. The Hilbert ball $B_{\mathcal{H}}$ is defined by the open unit ball in $\mathcal{H}$ and it is a Kähler manifold with $c<0$. We introduce the quantum mechanics on $B_{\mathcal{H}}$. As an application, we show the structure of the noncommutative function algebra on $B_{\mathcal{H}}$.

31 pages, LaTeX

References in corpus (1)

• Deformation quantization of Poisson manifolds, I

Cited by in corpus (1)

• Exponential families, Kahler geometry and quantum mechanics