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This paper bridges algorithmic randomness theory and Bohmian mechanics to rigorously define physical typicality, offering a precise mathematical formulation for statistical laws in quantum foundations. By demonstrating how Martin-Löf randomness constrains the distribution postulate, the work provides a novel theoretical framework that could influence interpretations of quantum probability and experimental protocols.
Algorithmic Randomness and Physical Typicality
arXiv:2609.06268v1 Announce Type: new Abstract: Appeals to typicality are common in physics, but it is often unclear what it means for a physical state to be typical relative to a probability measure, and correspondingly unclear what a law that appeals to typicality asserts. Here we consider how one might characterize physical typicality using ideas from the theory of algorithmic randomness. As a concrete example, we show how taking a physical state to be typical relative to a computable measure when it is Martin-L\"of random allows one to formulate the distribution postulate in Bohmian mechanics as a statistical constraining law of the theory. Using a toy model, we show how this constraint guarantees the standard Born statistics for computable experimental protocols. Algorithmic Bohmian mechanics (aBM) thus illustrates how algorithmic randomness may be used to provide precise content to a statistical law.
Abstract:Appeals to typicality are common in physics, but it is often unclear what it means for a physical state to be typical relative to a probability measure, and correspondingly unclear what a law that appeals to typicality asserts. Here we consider how one might characterize physical typicality using ideas from the theory of algorithmic randomness. As a concrete example, we show how taking a physical state to be typical relative to a computable measure when it is Martin-Löf random allows one to formulate the distribution postulate in Bohmian mechanics as a statistical constraining law of the theory. Using a toy model, we show how this constraint guarantees the standard Born statistics for computable experimental protocols. Algorithmic Bohmian mechanics (aBM) thus illustrates how algorithmic randomness may be used to provide precise content to a statistical law.
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); General Relativity and Quantum Cosmology (gr-qc); Statistics Theory (math.ST); History and Philosophy of Physics (physics.hist-ph)
arXiv:2609.06268 [quant-ph]
(or arXiv:2609.06268v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.06268 arXiv-issued DOI via DataCite (pending registration)
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