Emergence of wave equations from quantum geometry
Abstract
We argue that classical geometry should be viewed as a special limit of noncommutative geometry in which aspects which are interconstrained decouple and appear arbitrary in the classical limit. In particular, the wave equation is really a partial derivative in a unified extradimensional noncommutative geometry and arises out of the greater rigidity of the noncommutative world not visible in the classical limit. We provide an introduction to this 'wave operator' approach to noncommutative geometry as recently used[27] to quantize any static spacetime metric admitting a spatial conformal Killing vector field, and in particular to construct the quantum Schwarzschild black hole. We also give an introduction to our related result that every classical Riemannian manifold is a shadow of a slightly noncommutative one wherein the meaning of the classical Ricci tensor becomes very natural as the square of a generalised braiding.
 Publication:

The Sixth International School on Field Theory and Gravitation2012
 Pub Date:
 October 2012
 DOI:
 10.1063/1.4756969
 Bibcode:
 2012AIPC.1483..169M
 Keywords:

 black holes;
 quantum gravity;
 spacetime configurations;
 wave equations;
 03.65.Ge;
 04.60.Pp;
 04.62.+v;
 04.70.Dy;
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 Loop quantum gravity quantum geometry spin foams;
 Quantum field theory in curved spacetime;
 Quantum aspects of black holes evaporation thermodynamics