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arXiv:2411.00972 · Physica Scripta

Classical mechanics as the high-entropy limit of quantum mechanics

Gabriele Carcassi, Manuele Landini, C. Aidala

Published Jan 19, 2026• 0 citations
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Abstract

Abstract We show that classical mechanics can be recovered as the high-entropy limit of quantum mechanics. That is, the high entropy masks quantum effects, and mixed states of high enough entropy can be approximated with classical distributions. The mathematical limit ℏ → 0 can be reinterpreted as setting the zero entropy of pure states to −∞, in the same way that non-relativistic mechanics can be recovered mathematically with c → ∞. Physically, these limits are more appropriately defined as S ≫ 0 and v ≪ c . Both limits can then be understood as approximations independently of what circumstances allow those approximations to be valid. Consequently, the limit presented is independent of possible underlying mechanisms and of what interpretation is chosen for both quantum states and entropy.

DOI
10.1088/1402-4896/ae3a20

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Abstract We show that classical mechanics can be recovered as the high-entropy limit of quantum mechanics. That is, the high entropy masks quantum effects, and mixed states of high enough entropy can be approximated with classical distributions. The mathematical limit ℏ → 0 can be reinterpreted as setting the zero entropy of pure states to −∞, in the same way that non-relativistic mechanics can be recovered mathematically with c → ∞. Physically, these limits are more appropriately defined as S ≫ 0 and v ≪ c . Both limits can then be understood as approximations independently of what circumstances allow those approximations to be valid. Consequently, the limit presented is independent of possible underlying mechanisms and of what interpretation is chosen for both quantum states and entropy.
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Reverse Physics is a methodology that breaks physical theories into separate mathematical and physical conditions to establish their logical relationships. To showcase the power of the methodology, we present several results for quantum mechanics and their related insights. The standard Hilbert-space formulation conflicts with basic physical requirements, while a minimal topological modification can solve these problems. The ensemble space, rather than the pure-state space, distinguishes classical from quantum systems. The Born rule is an additional assumption linking orthogonality, mutual exclusivity and information entropy. Under explicit background conditions, unitary evolution is equivalent to deterministic and reversible evolution. Nonselective projective measurements can be characterized as Lindblad equilibration processes, while unitary evolution can be characterized as a limit of infinitesimal projective processes. Classical mechanics is recovered as the high-entropy limit of quantum mechanics, and every quantum state, pure or mixed, is a dynamic, spectral and thermodynamic equilibrium. These results are self-contained, use the standard vector-space representation and can thus be used as common tools and constraints for teaching, interpretations, reconstructions and future theories.
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I argue that it is possible to give an interpretation of the classical ℏ→0 limit of quantum mechanics that results in a partial explanation of the success of classical mechanics. The interpretation is novel in that it allows one to explain the success of the theoretical structure of classical mechanics. This interpretation clarifies the relationship between physical quantities and propositions in quantum theories and provides a precise notion of a quantum theory holding “approximately on certain scales.”