Quantum geometry

In quantum gravity, quantum geometry is the set of mathematical concepts that generalize geometry to describe physical phenomena at distance scales comparable to the Planck length. Each theory of quantum gravity uses the term "quantum geometry" in a slightly different fashion.

String theory uses it to describe exotic phenomena such as T-duality and other geometric dualities, mirror symmetry, topology-changing transitions^{[clarification needed]}, minimal possible distance scale, and other effects that challenge intuition. More technically, quantum geometry refers to the shape of a spacetime manifold as experienced by D-branes, which includes quantum corrections to the metric tensor, such as the worldsheet instantons. For example, the quantum volume of a cycle is computed from the mass of a brane wrapped on this cycle.^{[citation needed]}

In an alternative approach to quantum gravity called loop quantum gravity (LQG), the phrase "quantum geometry" usually refers to the formalism within LQG where the observables that capture the information about the geometry are well-defined operators on a Hilbert space. In particular, certain physical observables, such as the area, have a discrete spectrum. LQG is non-commutative.^[1] It is possible (but considered unlikely) that this strictly quantized understanding of geometry is consistent with the quantum picture of geometry arising from string theory.^{[citation needed]}

Another approach, which tries to reconstruct the geometry of space-time from "first principles" is Discrete Lorentzian quantum gravity.

See also

• Noncommutative geometry
• Quantum spacetime