Semiclassical quantum gravity: statistics of combinatorial Riemannian geometries

Authors

  • L. Bombelli,

    1. Department of Physics and Astronomy, University of Mississippi, University, MS 38677, USA
    2. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5
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  • A. Corichi,

    1. Department of Physics and Astronomy, University of Mississippi, University, MS 38677, USA
    2. Department of Gravitation and Field Theory, Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D.F. 04510, México
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    • Phone: +52 55 5622 4690, Fax: +52 55 5622 4693

  • O. Winkler

    1. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5
    2. Current address: Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada E3B 5A3
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Abstract

This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at “quantum scales” and continuum, classical geometries at large scales. Such a correspondence can be meaningfully established when one has a “semiclassical” state in the underlying quantum gravity theory, and the uncertainties in the correspondence arise both from quantum fluctuations in this state and from the kinematical procedure of matching a smooth geometry to a discrete one. We focus on the latter type of uncertainty, and suggest the use of statistical geometry as a way to quantify it. With a cell complex as an example of discrete structure, we discuss how to construct quantities that define a smooth geometry, and how to estimate the associated uncertainties. We also comment briefly on how to combine our results with uncertainties in the underlying quantum state, and on their use when considering phenomenological aspects of quantum gravity.

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