quantized Riemann spaces

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.

Definition 1.1   Quantum Geometry is defined as a field of Mathematical or Theoretical Physics based on geometrical and Algebraic Topology approaches to Quantum Gravity- one such approach is based on Noncommutative Geometry and SUSY (the `Standard' Model in current Physics).

A Result for Quantum Spin Foam Representations of Quantum Space-Times (QST)s: There exists an $ n$-connected CW model $ (Z,QSF)$ for the pair $ (QST,QSF)$ such that: $ f_*: \pi_i(Z) \rightarrow \pi_i (QST)$, is an isomorphism for $ i>n$, and it is a monomorphism for $ i=n$. The $ n$-connected CW model is unique up to homotopy equivalence. (The $ CW$ complex, $ Z$, 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.



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