Conceptual Issues for Noncommutative Gravity on Algebras and Finite Sets

Abstract

We discuss some of the issues to be addressed in arriving at a definitive noncommutative Riemannian geometry that generalises conventional geometry both to the quantum domain and to the discrete domain. This also provides an introduction to our 1997 formulation based on quantum group frame bundles. We outline now the local formulae with general differential calculus both on the base `quantum manifold' and on the structure group Gauge transforms with nonuniversal calculi, Dirac operator, Levi-Civita condition, Ricci tensor and other topics are also covered. As an application we outline an intrinsic or relative theory of quantum measurement and propose it as a possible framework to explore the link between gravity in quantum systems and entropy.

17 pages, to appear Proc. Euroconference on Noncommutative Geometry and Hopf Algebras in Field Theory and Particle Physics, Torino, 1999 -- this intro for theoretical physicists (mathematicians, see long paper)

References in corpus (1)

• Riemannian geometry of quantum groups and finite groups with nonuniversal differentials

Cited by in corpus (7)

• Riemannian geometry of quantum groups and finite groups with nonuniversal differentials
• Noncommutative BTZ Black Hole and Discrete Time
• Electromagnetism and Gauge Theory on the Permutation Group $S_3$
• Noncommutative Riemannian Geometry of the Alternating Group A_4
• Noncommutative Unification of General Relativity and Quantum Mechanics
• $Z_2\times Z_2$ Lattice as a Connes-Lott-Quantum Group Model
• Finite group discretization of Yang-Mills and Einstein actions