Quantized Riemannian Manifolds and Geometry

An interesting, but perhaps limiting approach to quantum gravity (QG), involves defining a quantum Riemannian geometry [2] in place of the classical Riemannian manifold that is employed in the well-known, Einstein's classical approach to general relativity (GR). Whereas a classical Riemannian manifold has a metric defined by a special, Riemannian tensor, the quantum Riemannian geometry may be defined in different theoretical approaches to QG by either quantum loops (or perhaps `strings'), or spin networks and spin foams (in locally covariant GR quantized space-times). The latter two concepts are related to the `standard' quantum spin observables and thus have the advantage of precise mathematical definitions. As spin foams can be defined as functors of spin network categories, quantized space-times (QST)s can be represented by, or defined in terms of, natural transformations of `spin foam' functors. The latter definition is not however the usual one adopted for quantum Riemannian geometry, and other (for example, noncommutative geometry) approaches attempt to define a QST metric not by a Riemannian tensor –as in the classical GR case– but in relation to a generalized, quantum `Dirac' operator in a spectral triplet.

Remarks. Other approaches to Quantum Gravity include: loop quantum gravity (LQG), AQFT approaches, topological quantum field theory (TQFT)/ homotopy Quantum Field Theories (HQFT; Tureaev and Porter, 2005), quantum theories on a lattice (QTL), string theories and spin network models.

A Result for quantum spin foam representations of quantum space-times (QST)s: There exists an -connected CW model for the pair such that: , is an isomorphism for , and it is a monomorphism for . The -connected CW model is unique up to homotopy equivalence. (The complex, , considered here is a homotopic `hybrid' between QSF and QST).

Bibliography

1

A. Connes. 1994. Noncommutative Geometry. Academic Press: New York and London.

2

Abhay Ashtekar and Jerzy Lewandowski.2005. Quantum Geometry and Its Applications. PDF file download.