Abstract
I distinguish two types of reduction within the context of quantum-classical relations, which I designate “formal” and “empirical”. Formal reduction holds or fails to hold solely by virtue of the mathematical relationship between two theories; it is therefore a two-place, a priori relation between theories. Empirical reduction requires one theory to encompass the range of physical behaviors that are well-modeled in another theory; in a certain sense, it is a three-place, a posteriori relation connecting the theories and the domain of physical reality that both serve to describe. Focusing on the relationship between classical and quantum mechanics, I argue that while certain formal results concerning singular \(\hbar \rightarrow 0\) limits have been taken to preclude the possibility of reduction between these theories, such results at most provide support for the claim that singular limits block reduction in the formal sense; little if any reason has been given for thinking that they block reduction in the empirical sense. I then briefly outline a strategy for empirical reduction that is suggested by work on decoherence theory, arguing that this sort of account remains a fully viable route to the empirical reduction of classical to quantum mechanics and is unaffected by such singular limits.
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Notes
For further discussion of the Nagel/Schaffner approach to reduction, see, for example Dizadji-Bahmani et al. (2010).
In (2007), Landsman offers a number of suggestions as to the particular measure of \(S_{cl}\) that might be adopted in various cases.
Another reason often cited for the inability of quantum theory to recover classical chaos is that classical chaos entails exponential divergence of closely spaced initial conditions in phase space, while on the usual association of classical phase space points with narrow wave packet states in quantum theory, the unitary nature of the Schrodinger evolution—which preserves the inner product between any two initial states throughout their evolution—precludes such a divergence between the corresponding wave packets. However, Zurek has argued that when we incorporate the effects of environmental decoherence, the effective quantum dynamics of the system in question is no longer unitary (it is only the total closed system consisting of the system in question and its environment that is assumed to evolve unitarily) and so this objection no longer applies.
For further discussion of the vagueness of existing formulations of the limit-based approach, see Rosaler (2015c).
See Landsman’s (2007) for examples of non-singular \(\hbar \rightarrow 0\) limiting relations.
Here, I take classical behavior to designate behavior that is accurately represented by some purely classical model. That is, I do not take it to include those systems whose behavior is well-described by semiclassical models that employ hybrids of quantum and classical concepts.
A projection-valued measure (PVM) on a Hilbert space \(\mathcal {H}\) is a set of self-adjoint operators \(\{ {\hat{P}}_{\alpha } \}\) on \(\mathcal {H}\) such that
$$\begin{aligned} \sum _{\alpha } {\hat{P}}_{\alpha } = {\hat{I}}, \end{aligned}$$(7)$$\begin{aligned} {\hat{P}}_{\alpha } {\hat{P}}_{\beta } = \delta _{\alpha \beta } {\hat{P}}_{\alpha }, \ \end{aligned}$$(8)where there is no summation over repeated indices in (8). The concept of a positive operator-valued measure (POVM) on \(\mathcal {H}\) generalizes the notion of a PVM by relaxing the requirement of orthogonality in (8). Thus, a positive-operator-valued measure (POVM) on a Hilbert space \(\mathcal {H}\) is a set \(\{ {\hat{\Pi }}_{\alpha } \}\) of positive operators such that
$$\begin{aligned}&\sum _{\alpha } {\hat{\Pi }}_{\alpha } = {\hat{I}}. \end{aligned}$$(9)Recall that an operator \({\hat{O}}\) is positive if it is self-adjoint and \(\langle \Psi | {\hat{O}}|\Psi \rangle \ge 0 \) for every \(|\Psi \rangle \in \mathcal {H}\). Note that every PVM is also a POVM.
It is important to note that the ensemble distribution \( \langle X | {\hat{\rho }}_{S}(t) | X \rangle \) reflects a distribution across branches, which originate in some initial narrow state \({\hat{\rho }}^{0}_{S}\).
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Acknowledgments
Thanks to David Wallace, Simon Saunders, Christopher Timpson, Jeremy Butterfield and Robert Batterman for many helpful discussions on the classical domain of quantum theory.
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Rosaler, J. “Formal” Versus “Empirical” Approaches to Quantum–Classical Reduction. Topoi 34, 325–338 (2015). https://doi.org/10.1007/s11245-015-9328-1
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DOI: https://doi.org/10.1007/s11245-015-9328-1