This is a fundamental topic on quantum space-times viewed from general relativistic and quantum gravity (QG) standpoints, and includes, for example, quantum geometry fundamental notions.

Quantum Space-Times (QST)

The concept of quantum space-times (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 space-times or how to develop a valid classification theory of quantum space-times. However, several specific definitions or models were proposed and a list of such examples is presented next.

Specific Definitions for Models of Quantum Space-Times (QSTs) and Quantum Geometry

• QSTs represented by posets or causal sets
• QSTs represented by so-called quantum topoi with Heyting logic algebra as classifier
• QSTs represented by topological Quantum Field Theories (TQFTs) or homotopy QFTs
• QSTs represented as spin foams of spin networks
• QSTs represented as a noncommutative, algebraic– and/or “geometrical”–quantized space as in noncommutative geometry models for SUSY
• QSTs represented as generalized Riemannian manifolds with quantum tangent spaces
• QSTs represented as presheaves of local nets of quantum operators in algebraic QFT (AQFT)
• QSTs represented as quantum fields on a (physical) Lattice of geometric points
• QSTs represented as consisting of quantum loops
• QSTs represented as fractal dimension spaces
• QSTs represented as categories or spaces of quantized strings as in string theories
• Twistor representations in Quantum Gravity (QG) (introduced by Sir Roger Penrose).