Year: 1999
Type: article
Abstract: We explore the relation between (spectral) noncommutative geometry and quantum mechanics. We consider a dynamical model based on a triple $(A,H,D)$, which mimics aspects of the spectral formulation of general relativity. Its phase space is the space of on-shell Dirac operators. We construct the corresponding quantum theory using a covariant canonical quantization. The Connes distance between two states over $A$ (``spacetime points'') turns out discrete and we compute its spectrum. The quantum states of the geometry form a Hilbert space $K$, and $D$ is promoted to an operator $\stackrel{^}{D}$ on $H\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}H\ensuremath{\bigotimes}K$. The triple $(A,H,\stackrel{^}{D})$ can be viewed as the quantization of the triples $(A,H,D)$.
Source: Physical Review Letters
Author: Carlo Rovelli
Institutions: University of Pittsburgh, Centre de Physique Théorique
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2245265755