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Title: 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 inter-constrained decouple and appear arbitrary in the classical limit. In particular, the wave equation is really a partial derivative in a unified extra-dimensional 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.

Authors:
 [1]
  1. School of Mathematical Sciences, Queen Mary University of London, 327 Mile End Rd, London E1 4NS (United Kingdom)
Publication Date:
OSTI Identifier:
22075569
Resource Type:
Journal Article
Journal Name:
AIP Conference Proceedings
Additional Journal Information:
Journal Volume: 1483; Journal Issue: 1; Conference: 6. international school on field theory and gravitation 2012, Petropolis, RJ (Brazil), 23-27 Apr 2012; Other Information: (c) 2012 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0094-243X
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; BLACK HOLES; COMMUTATION RELATIONS; GEOMETRY; QUANTUM GRAVITY; RICCI TENSOR; SCHWARZSCHILD METRIC; SPACE-TIME; VECTOR FIELDS; WAVE EQUATIONS

Citation Formats

Majid, Shahn. Emergence of wave equations from quantum geometry. United States: N. p., 2012. Web. doi:10.1063/1.4756969.
Majid, Shahn. Emergence of wave equations from quantum geometry. United States. https://doi.org/10.1063/1.4756969
Majid, Shahn. 2012. "Emergence of wave equations from quantum geometry". United States. https://doi.org/10.1063/1.4756969.
@article{osti_22075569,
title = {Emergence of wave equations from quantum geometry},
author = {Majid, Shahn},
abstractNote = {We argue that classical geometry should be viewed as a special limit of noncommutative geometry in which aspects which are inter-constrained decouple and appear arbitrary in the classical limit. In particular, the wave equation is really a partial derivative in a unified extra-dimensional 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.},
doi = {10.1063/1.4756969},
url = {https://www.osti.gov/biblio/22075569}, journal = {AIP Conference Proceedings},
issn = {0094-243X},
number = 1,
volume = 1483,
place = {United States},
year = {Mon Sep 24 00:00:00 EDT 2012},
month = {Mon Sep 24 00:00:00 EDT 2012}
}