quantum spacetimes
This is a contributed topic on quantum spacetimes^{} viewed from general relativistic and quantum gravity standpoints, and includes, for example, quantum geometry fundamental notions.
1 Quantum Spacetimes (QST)
The concept of quantum spacetimes (QST) is fundamental to the development of relativistic quantum theories^{} and at this point it can only be broadly defined as a class of mathematical spaces that allow the construction of quantum physical theories in a manner consistent with both relativistic principles and quantum gravity. There is no universal agreement amongst either theoretical physicists or mathematicians who work on Physical Mathematics about either a specific definition of such quantum spacetimes or how to develop a valid classification theory of quantum spacetimes. However, several specific definitions or models were proposed and a list of such examples is presented next.
1.1 Specific Definitions for Models of Quantum SpaceTimes (QSTs) and Quantum Geometry

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QSTs represented by posets or causal sets

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QSTs represented by socalled quantum topoi (QTs) with Heyting logic algebra as a subobject classifier

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QSTs represented by Topological Quantum Field Theories (TQFTs) or Homotopy^{} QFTs

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QSTs represented as spin foams of spin networks (http://planetmath.org/SpinNetworksAndSpinFoams)

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QSTs represented as a noncommutative, algebraic– and/or “geometrical”–quantized space as in noncommutative geometry^{} models for SUSY

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QSTs represented as generalized Riemannian manifolds with quantum tangent spaces^{}

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QSTs represented as presheaves of local nets of quantum operators in Algebraic QFT

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QSTs represented as quantum fields (QF) on a (physical) lattice of geometric points

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QSTs represented as consisting of quantum loops

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QSTs represented as fractal dimension spaces

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QST represented as a quantum category^{} of locally compact grpoupoids

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QSTs represented as categories^{} or spaces of quantized strings as in string theories

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Twistor representations in quantum gravity (QG) (introduced by Sir Roger Penrose).
Title  quantum spacetimes 
Canonical name  QuantumSpacetimes 
Date of creation  20130322 18:15:56 
Last modified on  20130322 18:15:56 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  28 
Author  bci1 (20947) 
Entry type  Topic 
Classification  msc 81R60 
Classification  msc 81R50 
Classification  msc 81P10 
Classification  msc 83C47 
Classification  msc 83C75 
Classification  msc 83C45 
Classification  msc 81P05 
Synonym  QST 
Synonym  quantum state spaces 
Synonym  quantum spaces 
Related topic  QuantumFundamentalGroupoids 
Related topic  QuantumGroupoids2 
Related topic  QuantumGroups 
Related topic  QuantumSystem 
Related topic  QEDInTheoreticalAndMathematicalPhysics 
Related topic  HamiltonianOperatorOfAQuantumSystem 
Related topic  QuantumChromodynamicsQCD 
Related topic  QuantumAutomataAndQuantumComputation2 
Related topic  Quantization 
Related topic  QEDInTheoreticalAndMathema 
Defines  quantum spacetime structure 
Defines  QST 