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.
1 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.
1.1 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).
