Grimmer, Daniel (2022) A Discrete Analog of General Covariance  Part 1: Could the world be fundamentally set on a lattice? [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). This paper delivers such an extension with surprising consequences.
One's first intuition regarding discrete spacetime theories may be that they introduce a great deal of fixed background structure (i.e., a lattice) and thereby limit our theory's possible symmetries down to those which preserve this fixed structure (i.e., only certain discrete symmetries). By so restricting symmetries, lattice structures appear to be both theorydistinguishing and fundamentally ``bakedinto'' our discrete spacetime theories. However, as I will discuss, all of these intuitions are doubly wrong and overhasty. Discrete spacetime theories can and do have continuous translation and rotation symmetries. Moreover, the exact same lattice theory can be given a wide variety of lattice structures and can even be described making reference to no lattice whatsoever. As my discrete analog of general covariance will reveal: lattice structure is rather less like a fixed background structure and rather more like a coordinate system, i.e., merely a representational artifact. Ultimately, I show that the lattice structure supposedly underlying any discrete ``lattice'' theory has the same level of physical import as coordinates do, i.e., none at all. Thus, the world cannot be ``fundamentally set on a square lattice'' (or any other lattice) any more than it could be ``fundamentally set in a certain coordinate system''. Like coordinate systems, lattice structures are just not the sort of thing that can be fundamental; they are both thoroughly merely representational. Spacetime cannot be a lattice (even when it might be representable as such).
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Item Type:  Preprint  

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Additional Information:  50 pages, 10 figures. Video abstract: https://youtu.be/dc58WyWXz4 Part 2: https://arxiv.org/abs/2205.07701  
Keywords:  General Covariance, Background Independence, Quantum Spacetime, Discrete Spacetime, Lattice, Coordinates, Symmetry, Representation, Quantum Gravity  
Subjects:  Specific Sciences > Physics > Classical Physics Specific Sciences > Physics > Quantum Gravity Specific Sciences > Physics > Symmetries/Invariances 

Depositing User:  Dr. Daniel Grimmer  
Date Deposited:  18 May 2022 13:19  
Last Modified:  18 May 2022 13:19  
Item ID:  20471  
Subjects:  Specific Sciences > Physics > Classical Physics Specific Sciences > Physics > Quantum Gravity Specific Sciences > Physics > Symmetries/Invariances 

Date:  17 May 2022  
URI:  https://philsciarchive.pitt.edu/id/eprint/20471 
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