PlanetPhysics/Quantized Riemann Spaces

Quantized Riemannian Manifolds and Geometry

An interesting, but perhaps limiting approach to [[../LQG2|quantum gravity]] ([[../SUSY2|QG]]), involves defining a [[../Nonabelian Algebraic Topology 3|quantum Riemannian geometry]] in place of the classical Riemannian [[../Noncommutative Geometry 4|manifold]] that is employed in the well-known, [[../Albert Einstein|Einstein's]] classical approach to [[../SR|general relativity]] ([[../SR|GR]]). Whereas a classical Riemannian manifold has a [[../Metric Tensor|metric]] defined by a special, Riemannian [[../Tensor|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|spin networks and spin foams]] (in locally covariant GR quantized space-times). The latter two [[../Precise Idea|concepts]] are related to the 'standard' quantum [[../Quark Antiquark Pair|spin]] [[../Quantum Spin Network Functor 2|observables]] and thus have the advantage of precise mathematical definitions. As [[../Simplicial C W Complex|spin foams]] can be defined as [[../Functor|functors]] of [[../Simplicial C W Complex|spin network]] [[../Cod|categories]], quantized space-times (QST [[../SUSY2|SUSY2]])s can be represented by, or defined in terms of, natural transformations [[../Variable Category 2|Variable Category 2]] of '[[../Simplicial C W Complex|spin foam]]' functors. The latter definition is not however the usual one adopted for quantum Riemannian geometry, and other (for example, [[../Noncommutative Geometry 4|noncommutative geometry]]) approaches attempt to define a QST metric not by a Riemannian tensor --as in the classical GR case-- but in [[../Bijective|relation]] to a generalized, quantum 'Dirac' [[../Quantum Spin Network Functor 2|operator]] in a spectral triplet.

Remarks. Other approaches to Quantum Gravity include: [[../Triangulation Methods For Quantized Spacetimes 2|loop quantum gravity]] ([[../LQG2|LQG]]), [[../SUSY2|AQFT]] approaches, [[../Co Intersections|topological]] [[../Space Time Quantization In Quantum Gravity Theories|quantum field theory]] ([[../SUSY2|TQFT]])/ [[../Thin Equivalence|homotopy]] Quantum Field Theories ([[../QAT|HQFT]]; Tureaev and Porter, 2005), [[../FTNIR|quantum theories on a lattice]] ([[../FTNIR|QTL]]), [[../10D Brane|string theories]] and spin network models.

[[../NAQAT2|quantum geometry]] is defined as a field [[../Cosmological Constant 2|Cosmological Constant 2]] of Mathematical or [[../Physical Mathematics 2|theoretical physics]] based on geometrical and [[../Cubical Higher Homotopy Groupoid|algebraic topology]] approaches to Quantum Gravity - one such approach is based on Noncommutative Geometry and [[../Quark Antiquark Pair|SUSY]] (the 'Standard' Model in current Physics).

A Result for quantum spin foam [[../Triangulation Methods For Quantized Spacetimes 2|Triangulation Methods For Quantized Spacetimes 2]] [[../Categorical Group Representation|representations]] of [[../SUSY2|quantum space-times]] (QST)s: There exists an $n$-connected CW model $(Z,QSF)$ for the pair $(QST,QSF)$ such that: $f_{*}:{\pi }_i(Z)\to {\pi }_i(QST)$, is an [[../Isomorphic Objects Under AnIsomorphism|isomorphism]] for $i>n$, and it is a [[../Injective Map|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).