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A Discrete Analog of General Covariance -- Part 2: Despite what you've heard, a perfectly Lorentzian lattice theory

Grimmer, Daniel (2022) A Discrete Analog of General Covariance -- Part 2: Despite what you've heard, a perfectly Lorentzian lattice theory. [Preprint]

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Abstract

A crucial step in the history of General Relativity was Einstein's adoption of the principle of general covariance which demands a coordinate independent formulation for our spacetime theories. General covariance helps us to disentangle a theory's substantive content from its merely representational artifacts. It is an indispensable tool for a modern understanding of spacetime theories, especially regarding their background structures and symmetry. Motivated by quantum gravity, one may wish to extend these notions to quantum spacetime theories (whatever those are). Relatedly, one might want to extend these notions to discrete spacetime theories (i.e., lattice theories). In Part 1, I developed two discrete analogs of general covariance for non-Lorentzian lattice theories. This paper extends these results to a Lorentzian setting.

In either setting these discrete analogs of general covariance reveal that lattice structure is rather less like a fixed background structure and rather more like a coordinate system, i.e., merely a representational artifact. These discrete analogs are built upon a rich analogy between the lattice structures appearing in our discrete spacetime theories and the coordinate systems appearing in our continuum spacetime theories. From this, in Part 1, I argued that properly understood there are no such things as lattice-fundamental theories, rather there are only lattice-representable theories. It is well-noted by the causal set theory community that no theory on a fixed spacetime lattice is Lorentz invariant, however as I will discuss this is ultimately a problem of representation, not of physics. There is no need for the symmetries of our representational tools to latch onto the symmetries of the thing being represented. Nothing prevents us from using Cartesian coordinates to describe rotationally invariant states/dynamics. As this paper shows, the same is true of lattices in a Lorentzian setting: nothing prevents us from defining a perfectly Lorentzian lattice(-representable) theory.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Grimmer, Danieldaniel.grimmer@philosophy.ox.ac.uk0000-0002-8449-3775
Additional Information: 45 pages, 10 figures. Video abstract: https://youtu.be/dc58WyWX-z4 Part 1: https://arxiv.org/abs/2204.02276
Keywords: General Covariance, Background Independence, Quantum Spacetime, Discrete Spacetime, Lattice, Coordinates, Symmetry, Representation, Quantum Gravity
Subjects: Specific Sciences > Physics > Quantum Gravity
Specific Sciences > Physics > Relativity Theory
Specific Sciences > Physics > Symmetries/Invariances
Depositing User: Dr. Daniel Grimmer
Date Deposited: 21 May 2022 03:50
Last Modified: 21 May 2022 03:50
Item ID: 20638
Subjects: Specific Sciences > Physics > Quantum Gravity
Specific Sciences > Physics > Relativity Theory
Specific Sciences > Physics > Symmetries/Invariances
Date: 17 May 2022
URI: http://philsci-archive.pitt.edu/id/eprint/20638

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