Twisted geometries are discrete geometries that play a role in
loop quantum gravity
Loop quantum gravity (LQG) is a theory of quantum gravity that incorporates matter of the Standard Model into the framework established for the intrinsic quantum gravity case. It is an attempt to develop a quantum theory of gravity based direc ...
and spin foam models,
where they appear in the semiclassical limit of
spin networks. A twisted geometry can be visualized as collections of polyhedra dual to the nodes of the spin network's graph.
Intrinsic and extrinsic curvatures are defined in a manner similar to
Regge calculus
In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in 1961. Availabl ...
, but with the generalisation of including a certain type of metric discontinuities: the face shared by two adjacent polyhedra has a unique area, but its shape can be different.
This is a consequence of the quantum geometry of spin networks: ordinary Regge calculus is "too rigid" to account for all the geometric degrees of freedom described by the semiclassical limit of a spin network.
The name twisted geometry captures the relation between these additional degrees of freedom and the off-shell presence of torsion in the theory, but also the fact that this classical description can be derived from
twistor theory
In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics. Penrose's idea was that twistor space should ...
, by assigning a pair of twistors to each link of the graph, and suitably constraining their helicities and incidence relations.
References
Loop quantum gravity
Physics beyond the Standard Model
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