Abstract
To answer foundational questions in physics, physicists turn more and more to abstract advanced mathematics, even though its physical significance may not be immediately clear. What if we started to borrow ideas and approaches, with appropriate modifications, from the foundations of mathematics? In this paper we explore this route. In reverse mathematics one starts from theorems and finds the minimum set of axioms required for their derivation. In reverse physics we want to start from laws or more specific results, and find the physical concepts and starting points that recover them. We want to understand what physical results are implied by which physical assumptions. As an example of the technique, we will see six different characterizations of classical mechanics, show that the uncertainty principle depends only on the entropy bound on pure states and recast the third law of thermodynamics in terms of the entropy of an empty system. We believe the approach can provide greater insights into both current and new physical theories, put the physical concepts at the forefront of the discussion and provide a more unified view of physics by highlighting common patterns and ideas across different physical theories.
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Notes
This is essentially a short proof for Liouville’s theorem that can work in both directions.
The careful reader will note the units do not quite work. The issue here is the \(\log \rho\) in the Shannon/Gibbs entropy, as \(\rho\) is not a pure number. Introducing a dimensionful constant \(\log \rho {\hat{h}}\) would fix the expression, which would then fix the uncertainty relationship as well.
This formulation is given by Ref. [15].
If we consider systems as a monoid under composition, the empty system is the identity element, much like the number zero, the empty set or the identity map in their respective structures.
Philosophers may call these “constitutive” conditions.
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Acknowledgements
We acknowledge funding from the MCubed program of the University of Michigan. This work is part of a larger project, Assumptions of Physics [16], which aims to identify a handful of physical principles from which the basic laws can be rigorously derived.
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Communicated by Carlo Rovelli.
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Carcassi, G., Aidala, C.A. Reverse Physics: From Laws to Physical Assumptions. Found Phys 52, 40 (2022). https://doi.org/10.1007/s10701-022-00555-z
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DOI: https://doi.org/10.1007/s10701-022-00555-z