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One world is (probably) just as good as many


One of our most sophisticated accounts of objective chance in quantum mechanics involves the Deutsch–Wallace theorem, which uses state-space symmetries to justify agents’ use of the Born rule when the quantum state is known. But Wallace argues that this theorem requires an Everettian approach to measurement. I find that this argument is unsound. I demonstrate a counter-example by applying the Deutsch–Wallace theorem to the de Broglie–Bohm pilot-wave theory.

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  1. I borrow the latter analogy from Wallace (2012, pp. 427–428).

  2. I am not making any claims about how well these theories deal with locality. In fact, I hope this paper emphasizes locality as a deciding factor between interpretations by demonstrating the relative flexibility of the probability problem.

  3. Something like the symmetry theorem lies at the heart of several other recent derivations of the Born rule, including the self-locating uncertainty approach of Sebens and Carroll (2018) and the “envarience” approach of Zurek (2005, 2009). Both seem to require an Everettian approach. I focus exclusively on Wallace’s derivation to provide one clear account of how to disentangle many-worlds assumptions from the symmetry theorem.

  4. I use van Fraassen’s (1980, pp. 134–157) pragmatic model as a framework, but I do not require this model to give the final word on explanation.

  5. Recall that a projection-valued measure, for some measurable space \((X,\Sigma )\) and some subset \({\mathcal {E}}\) of \({\mathcal {P}}({\mathcal {H}}_S)\), is a map \(E: \Sigma \rightarrow {\mathcal {E}}\) that is non-negative, normalized, and countably additive. For ease of exposition, I will not be treating positive-operator-valued measurements (POVMs) in this paper. But note that we can recover all POVMs as PVMs on closed systems via Neumark’s theorem. See Busch et al. (1995) for a comprehensive discussion of these concepts.

  6. Nothing hinges on the choice of the Schrödinger picture, here; the aim is merely to get one well-defined notion of kinematics and dynamics on the table.

  7. Recall that a partial \(\sigma \)-algebra is a partial complemented lattice, i.e., a lattice with partial operations \(\bigvee \) and \(\bigwedge \) which denote the least upper bound and greatest lower bound of a set of elements, respectively, and the operation \(\lnot \), which denotes the complement of an element.

  8. I borrow this statement of Lewis’s PP from Wallace (2012, p. 141). PP is a specific formalization of the intuitions described by the credential and inferential links. The links themselves are ambiguous between Lewis’s formalization and, e.g., those of Hall (2004) and Ismael (2008). The differences among various formalizations are relevant for chance functions that are self-undermining, i.e., that do not assign \( ch (t)=1\) (Pettigrew , 2012). But on the operational approach, we assume that chances are not self-undermining.

  9. Note that while \({\mathcal {P}}({\mathcal {H}}_S)\) is a partial complemented lattice, it is not a Hilbert lattice; these latter lattices are complete, but non-intuitive from the standpoint of probability. For more technical details on Hilbert lattices, see Rédei (1998); for a critical perspective, see Kochen (2015).

  10. Note that by taking the domain of the chance function to be the partial \(\sigma \)-algebra \({\mathcal {P}}({\mathcal {H}}_S)\), Gleason’s theorem assigns probabilities to all projections on the Hilbert space—and so to all possible measurements, not just to those that can be done simultaneously. In particular, this choice of domain assumes that the probability values are measurement-neutral, or independent of a choice of additional, compatible measurements. For more on measurement-neutrality (which is also sometimes called “noncontextuality”), see Sect. 3.5 (and especially footnote 19).

  11. As Busch (2003) has shown, we can remove the parenthetical by attending to POVMs, which generalize PVMs. I will be sticking with PVMs for ease of exposition, but the proceeding arguments all generalize naturally to allow for POVMs.

  12. Wallace (2003) also emphasizes this point, and he notes that the symmetry theorem offers an explanation of measurement neutrality. I recover this explanation for the Bohmian in Sect. 3.5.

  13. In (2003, pp. 433–434), Wallace points to this underlying problem when he asserts that were we to try to use Gleason’s theorem to derive the Born rule, we would still need to appeal to the “games” developed in Deutsch’s original proof of the symmetry theorem. As Gill (2005) notes, it is possible to add one of Deutsch’s assumptions to Gleason’s theorem in order to derive the right bijection (viz., what appears below as the “normalization link”). But it is less clear that a similar strategy can offer an explanation of measurement neutrality.

  14. Wallace (2019) provides a rigorous, formal theory of subsystem-recursivity that informs my brief and qualitative discussion here.

  15. I borrow this terminology from Wallace (2012, p. 281). These views have natural analogs that cut branches into Siderian stages rather than Lewisian worms—Wallace refers to these as the Disconnected view and the Stage view, respectively (2012, p. 282). Tappenden (2011) has extensively developed the Stage view, and he saliently notes that the view yields post-measurement, pre-observation uncertainty that is quite similar to single-world uncertainty about chancy events. But since the Hydra and Lewisian views suffice to make my point about the symmetry theorem, I will not cover the Disconnected and Stage views here.

  16. Note well that I am not making any specific ontological claims here about \(\Psi \)! The dotted lines in Fig. 3c are meant to represent possible particle paths in (four-dimensional) spacetime, much like the solid lines in Fig. 3b are meant to represent worms in (four-dimensional) spacetime. This view of Bohmian particles is compatible with taking \(\Psi \) to be an object that lives in an ontic 3N-dimensional configuration space. It is compatible with taking \(\Psi \) to specify a multi-valued field in spacetime in the style of Romano (2021). It is also compatible with taking \(\Psi \) to be purely nomic, nothing more than a law governing the motions of particles in spacetime. The reader should feel free to choose whichever view of \(\Psi \) seems most natural! Nothing in the subsequent argument will hinge on this choice, so long as \(\Psi \)’s main role remains the specification of point-particle dynamics.

  17. See Sect. 5 and fn. 20 of DGZ (1992) for precise definitions and a brief discussion of how decoherence yields effective wave functions. For a longer discussion, see Romano (2016).

  18. In an early footnote, Wallace might concede that this argumentative strategy is open to the Bohmian. He notes that the relevant symmetries may only appear at the level of distributions on phase space (2003, p. 435, fn. 8). But using the symmetries in distributions raises the specter of circularity. My approach assigns only dynamical significance to the pilot wave \(\Psi \) (in the manner discussed in Sect. 3.2).

  19. Wallace (2012) refers to measurement neutrality as “noncontextuality.” But since this usage is at odds with the more prevalent notion of Kochen-Specker contextuality (1975), I will stick with the former term. The pilot-wave theory is both measurement-neutral and (Kochen-Specker) contextual (because the values of self-adjoint observables typically depend on the context of measurement).


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The author would like to thank Harvey Brown, Benjamin H. Feintzeig, James Read, and David Wallace for their invaluable discussions and suggestions. Many thanks as well to the thorough and rigorous criticisms of several anonymous referees, without which this paper would be in a much sorrier state (though any remaining errors are, of course, the author’s sole responsibility!). Finally, much thanks to the participants of the 2020 Michaelmas Oxford Philosophy of Physics Seminar for a fantastic discussion of this work. The author was supported during the completion of this work by the National Science Foundation under grants #1846560 and #2043089.

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Correspondence to Jer Steeger.

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Proof of the symmetry theorem

Proof of the symmetry theorem

Before proving the symmetry theorem, we prove as a lemma an important condition that follows from the structural links assumption.

Branching link. When \(\{{\mathcal {S}}^i\}\) is strictly \(\Psi \)-branching and \({\mathcal {T}}(\alpha _i, \beta _j) \ne 0\) for \(t_i < t_j\),

$$\begin{aligned} ch _{\left( \Psi ,\alpha _i\right) } \left( \beta _j \right) = \frac{ ch _\Psi \left( \beta _j\right) }{ ch _\Psi \left( \alpha _i \right) }. \end{aligned}$$

Proof of the branching link

First, we show that chances respect branching, i.e., \(\mathbf {ch}_\Psi (\gamma ) = 0\) whenever \(\gamma \) contains \(\alpha _i, \beta _j\) such that \({\mathcal {T}}(\alpha _i,\beta _j) = 0\). Explicitly,

$$\begin{aligned} \mathbf {ch}_\Psi ( \gamma ) \le \mathbf {ch}_{\Psi }\! \left( \bigvee \{ \gamma \,|\, \gamma _j = \beta _j \wedge \gamma _i = \alpha _i \} \right)&= \mathbf {ch}_{\Psi } \!\left( \bigvee \{ \gamma \,|\, \gamma _{j} = \beta _j \} \,\bigg | \, \bigvee \{ \gamma \,|\, \gamma _i = \alpha _i \} \right) \end{aligned}$$
$$\begin{aligned}&= \mathbf {ch}_{(\Psi ,\alpha _i)} \!\left( \bigvee \{ \gamma \,|\, \gamma _{j} = \beta _j \} \right) \end{aligned}$$
$$\begin{aligned}&= ch _{(\Psi ,\alpha _i)} \left( \beta _j \right) \end{aligned}$$
$$\begin{aligned}&= 0, \end{aligned}$$

where (32) follows from additivity and the definition of conditional probability, (33) follows from temporal link, i.e., (29), (34) follows from the definition (28) in the probability assumption, and (35) follows from normalization link, the rules of probability, and the fact that \({\mathcal {T}}(\alpha _i,\beta _j) = 0\). The rules of probability then imply that \(\mathbf {ch}_\Psi ( \gamma ) =0\).

Now suppose that \(\{{\mathcal {S}}^i\}\) is strictly \(\Psi \)-branching and \({\mathcal {T}}(\alpha _i, \beta _j) \ne 0\) for \(t_i < t_j\). Note that

$$\begin{aligned} ch _{(\Psi ,\alpha _i)} \left( \beta _j \right)&= \mathbf {ch}_{(\Psi ,\alpha _i)} \!\left( \bigvee \{ \gamma \,|\, \gamma _{j} = \beta _j \} \right) \end{aligned}$$
$$\begin{aligned}&= \mathbf {ch}_{\Psi } \!\left( \bigvee \{ \gamma \,|\, \gamma _{j} = \beta _j \} \,\bigg | \, \bigvee \{ \gamma \,|\, \gamma _i = \alpha _i \} \right) \end{aligned}$$
$$\begin{aligned}&= \frac{\mathbf {ch}_{\Psi }\! \left( \bigvee \{ \gamma \,|\, \gamma _j = \beta _j \wedge \gamma _i = \alpha _i \} \right) }{\mathbf {ch}_\Psi \! \left( \bigvee \{ \gamma \,|\, \gamma _i = \alpha _i \} \right) } \end{aligned}$$
$$\begin{aligned}&= \frac{\mathbf {ch}_\Psi \left( \bigvee \{ \gamma \,|\, \gamma _{j} = \beta _j \} \right) }{\mathbf {ch}_\Psi \! \left( \bigvee \{ \gamma \,|\, \gamma _i = \alpha _i \} \right) } \end{aligned}$$
$$\begin{aligned}&= \frac{ ch _\Psi \left( \beta _j \right) }{ ch _\Psi (\alpha _i )}, \end{aligned}$$

where (36) follows from probability, (37) follows from temporal link, (38) follows from the definition of conditional probability (and a bit of algebra), (39) follows from \(\mathbf {ch}_\Psi \) respecting branching and additivity (and our supposition), and (40) follows from (28). \(\square \)

With branching link in hand, we can quickly prove the symmetry theorem.

Proof of the symmetry theorem

Following (Wallace 2012, Ch. 4), we aim to prove that \( ch _\Psi (\alpha _i) = \langle \Psi (t_i), \alpha _i \Psi (t_i) \rangle \) in four steps. First (i), we generalize the intuitive re-labeling argument to show that projections with equal Born weights must have equal chances—i.e., we show that \( ch _\Psi \) must be a function of Born weights. Second (ii), we invoke some of our environmental degrees of freedom to show that this function is increasing. Third (iii), we invoke N environmental degrees of freedom to show that function must equal the Born weight when it is rational. Fourth (iv), we use a simple limiting argument (and the second and third steps) to obtain agreement for arbitrary Born weights.

  1. (i)

    Suppose \(\langle \Psi (t_i) , \alpha _i \Psi (t_i) \rangle = \langle \Psi (t_i), \beta _i \Psi (t_i) \rangle \).

    First, suppose that both sides equal zero. Then \(\Psi (t_i)\) lies in the range of both \(\lnot \alpha _i\) and \(\lnot \beta _i \), and so by normalization link, \( ch _\Psi (\lnot \alpha _i ) = ch _\Psi (\lnot \beta _i ) = 1\). Thus, by probability, \( ch _\Psi ( \alpha _i) = ch _\Psi ( \beta _i ) = 0\).

    Now suppose otherwise. By the above reasoning, we get that \( ch _\Psi ( \alpha _i ) \ne 0\) and \( ch _\Psi ( \beta _i ) \ne 0\). Next, we run the analog of the intuitive re-labelling argument. By decoherence availability, we can consider two different strictly branching history algebras for the next step of decoherence: one that gives a projection \(\alpha _{i+1}\) weight from \(\alpha _i\) and one that gives it weight from \(\beta _i\). To distinguish these, let \(\Psi \) evolve with the first dynamics, and let \(\Phi \) evolve with the second dynamics. Now define \(\Phi (t_j) =\Psi (t_j)\) for \(j \le i\) and let \(\Psi (t_{i+1}) = X \Psi (t_i)\) and \(\Phi (t_{i+1}) = Y \Psi (t_i)\) for the unitary operators

    $$\begin{aligned} X := V^\alpha \alpha _i + W^\alpha (1 - \alpha _i) \qquad Y := V^\beta \beta _i + W^\beta (1 - \beta _i) \end{aligned}$$

    where \(V^\alpha \) and \(V^\beta \) share the range of \(\alpha _{i+1}\) and \(W^\alpha \) and \(W^\beta \) share the range of some mutually orthogonal projection \(\beta _{i+1}\). By construction, \(\Psi (t_{i+1}) = \Phi (t_{i+1})\).

    Note, too, that \(X \alpha _i \Psi (t_i)\) and \(Y \beta _i \Psi (t_i)\) both lie in the range of \(\alpha _{i+1}\). So by branching link, we have

    $$\begin{aligned} ch _{(\Psi ,\alpha _i)} \left( \alpha _{i+1} \right) = \frac{ ch _\Psi (\alpha _{i+1} )}{ ch _\Psi (\alpha _i )} = 1, \end{aligned}$$

    and similarly

    $$\begin{aligned} ch _{(\Phi ,\beta _i)} \left( \alpha _{i+1} \right) = \frac{ ch _\Phi (\alpha _{i+1} )}{ ch _{\Phi } (\beta _i )} = 1. \end{aligned}$$

    By state supervenience, \( ch _\Phi (\alpha _{i+1} ) = ch _\Psi (\alpha _{i+1} ) \), and so \( ch _\Psi (\alpha _i \ ) = ch _\Phi (\beta _i )\). Applying state supervenience once more, we get

    $$\begin{aligned} ch _\Psi (\alpha _i ) = ch _\Psi (\beta _i ). \end{aligned}$$

    (Note that, again by state supervenience, this last equality holds even if our original history algebra was not strictly branching.)

  2. (ii)

    Suppose \(\langle \Psi (t_i) , \alpha _i \Psi (t_i) \rangle > \langle \Psi (t_i), \beta _i \Psi (t_i) \rangle \).

    By decoherence availability, we may assume that the \(i+1\) step of decoherence is given by a unitary Y such that, for \(\gamma _{i+1}\) and \(\omega _{i+1}\) two mutually orthogonal projections, we have

    $$\begin{aligned} \begin{aligned} \langle Y^\dagger \Psi (t_i) , (\gamma _{i+1} \vee \omega _{i+1}) Y\Psi (t_i) \rangle&= \langle \Psi ( t_i), \alpha _i \Psi (t_i)\rangle \\ \langle Y^\dagger \Psi (t_i), \gamma _{i+1} Y \Psi (t_i) \rangle&= \langle \Psi (t_i), \beta _i\Psi (t_i) \rangle \end{aligned} \end{aligned}$$

    where Y maps vectors in the range of \(\alpha _i\) to the range of \(\gamma _{i+1} \vee \omega _{i+1} \). Then, by step (i),

    $$\begin{aligned} ch _\Psi (\gamma _{i+1} \vee \omega _{i+1} ) = ch _\Psi ( \alpha _i ),\qquad ch _\Psi ( \gamma _{i+1}) = ch _\Psi ( \beta _{i} ). \end{aligned}$$

    By probability,

    $$\begin{aligned} ch _\Psi (\gamma _{i+1} \vee \omega _{i+1} ) \ge ch _\Psi (\gamma _{i+1} ) \end{aligned}$$

    and so

    $$\begin{aligned} ch _\Psi (\alpha _i ) \ge ch _\Psi (\beta _i ). \end{aligned}$$
  3. (iii)

    Suppose \(\langle \Psi (t_i), \alpha _i \Psi (t_i) \rangle \) is rational, i.e. equal to \(\frac{M}{N}\) for some positive integers MN.

    By decoherence availability, we may pick some \({\mathcal {H}}_{SE}\)-spanning sequence of orthogonal projections \(\gamma ^1, \ldots , \gamma ^m, \ldots , \gamma ^N\) that generate a \(\sigma \)-algebra containing \(\alpha _i\) such that, for some \(\Phi \) such that \(\Phi (t_i) = \Psi (t_i)\),

    $$\begin{aligned} \langle \Phi (t_i), \gamma ^m \Phi (t_i) \rangle = \frac{1}{N} \end{aligned}$$

    for all m. By (i), \( ch _\Phi (\gamma ^m )\) must be independent of m. Thus, by probability, \( ch _\Phi (\gamma ^m ) = \frac{1}{N}\).

    Now let \(\omega := \bigvee _{i\le M} \gamma ^i\). We have that \(\langle \Psi (t_i), \alpha _i \Psi (t_i) \rangle = \langle \Phi (t_i), \omega \Phi (t_i) \rangle = \langle \Psi (t_i), \omega \Psi (t_i) \rangle \), so by step (i), \( ch _\Psi (\alpha _i) = ch _\Psi (\omega )\). By probability, \( ch _\Psi (\omega ) = \sum _{m=1}^M ch _\Psi (\gamma ^m) = \frac{M}{N}\), and so we get that

    $$\begin{aligned} ch _\Psi ( \alpha _i) = \frac{M}{N}. \end{aligned}$$
  4. (iv)

    Suppose \(\langle \Psi (t_i), \alpha _i \Psi (t_i) \rangle = r \in [0,1]\), where r may not be rational.

    By (i), \( ch \) is a function of Born-rule weights, i.e.

    $$\begin{aligned} ch _\Psi (\alpha _i ) = f(\langle \Psi (t_i), \alpha _i \Psi (t_i) \rangle ) \end{aligned}$$

    for some \(f:[0,1] \rightarrow [0,1]\). By step (ii), f is increasing; by step (iii), \(f(M/N)=M/N\).

    So let \(\{a_i\}\) and \(\{b_i\}\) be, respectively, increasing and decreasing sequences of rational numbers in [0, 1] converging to r. Since \(f(b_i) = b_i \) for all i, \(f(r)\le r\). And since \(f(a_i) = a_i \) for all i, \(f(r)\ge r\)—thus, \(f(r) = r\), and so

    $$\begin{aligned} ch _\Psi ( \alpha _i ) = \langle \Psi (t_i), \alpha _i \Psi (t_i) \rangle . \end{aligned}$$

\(\square \)

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Steeger, J. One world is (probably) just as good as many. Synthese 200, 97 (2022).

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  • Deutsch-Wallace theorem
  • Born rule
  • Everettian quantum mechanics
  • pilot-wave theory
  • chance
  • principal principle