Delimiting Limits: Quantum-Classical Relations, hbar, and Decoherence

Abstract

That there’s a sense in which classical physics reduces to and emerges from quantum physics is relatively uncontroversial but this talk will introduce much needed clarity on the role of the $\hbar$ limit for such inter-theoretic relations. This talk divides into two parts: I'll consider how classical physics emerges from quantum physics via decoherence, and I'll articulate the role for the $\hbar\rightarrow0$ limit. First I'll argue that decoherence provides the unique and interpretation-neutral way to understand the emergence of classical from quantum physics, I'll do this by reference to an account of emergence, and I'll argue that this account leaves interpretations with the more metaphysical task of articulating how ontology supervenes upon the decohered quantum description. Decoherence entails a form of dynamical independence between the branches of a superposition in the relevant basis to a very good degree of approximation. This means that the evolution in any such branch is screened off from the other branches and from the peculiarly quantum effects. The upshot of this is that in specific contexts classical dynamics are instantiated in fundamentally quantum systems. I'll claim that decoherence is thus responsible for the emergence of classical behaviour. Second I'll consider the role of the $\hbar$ limit. I’ll mention the history going back to Bohr and Dirac (see Bokulich (2008)) but focus on more recent analyses, especially Feintzeig’s work (though see also Landsman (2017)). Feintzeig’s rigorous treatise establishes a relation between the algebras of classical and quantum mechanics: he demonstrates that as $\hbar$ goes to zero, non-commuting observables commute. He argues that this is a reduction of classical by quantum mechanics, in the sense that it is “an explanation of the success of classical mechanics on the basis of quantum mechanics”. However, I’ll demonstrate that this limit is neither necessary nor sufficient for such explanations of classicality. First there are circumstances where classicality is instantiated but $\hbar$ is not small relative to the action; second there are circumstances where $\hbar$ is small relative to the action but classical behaviour is not instantiated. I discuss alternative accounts of what the $\hbar$ limit achieves and suggest that it may provide an empirical grounding for quantum mechanics in the sense that it tells us how to understand the theory, rather than informing us of the relation between the theory and its predecessors. I’ll conclude with some more general reasons why the distinction between the roles of decoherence and the $\hbar$ limit should be expected: limits are rather blunt tools in that they don’t provide the context-specificity that one finds in decoherence theory. Given that classical mechanics is only instantiated in certain contexts rather than brutely at large energy or timescales, the explanation of its success ought to depend on something more subtle than a limiting procedure. Bokulich, Alisa (2008). Reexamining the Quantum-Classical Relation: Beyond Reductionism and Pluralism. Cambridge University Press. Feintzeig, Benjamin H (2022). The classical–quantum correspondence. Cambridge University Press. Landsman, Klaas (2017). Foundations of Quantum Theory: From Classical Concepts to Operator Algebras. Springer Open.