Abstract
I present what might seem to be a local, deterministic model of the EPR-Bohm experiment, inspired by recent work by Joy Christian, that appears at first blush to be in tension with Bell-type theorems. I argue that the model ultimately fails to do what a hidden variable theory needs to do, but that it is interesting nonetheless because the way it fails helps clarify the scope and generality of Bell-type theorems. I formulate and prove a minor proposition that makes explicit how Bell-type theorems rule out models of the sort I describe here.
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Notes
Bell himself took the upshot to be that no local model—deterministic or otherwise—could reproduce the measurement outcomes of such experiments (see [1, 2]). For an excellent overview of the state of the art on Bell-type theorems, see [25]. See also [18, 19] and [20] for particularly clear expositions of how Bell-type theorems work, with the relevant assumptions stated as precisely as possible. An insightful alternative approach to thinking about theorems of this sort can be found in [24]. I should note, though, that the version of Bell’s theorem I will present in Sect. 2 of the present paper follows [16] and is somewhat different from the versions discussed by Jarrett [18, 19] and Malament [20] (which are in the tradition of [15]) or Pitowsky [24]. I will make some contact between the present discussion and the Jarrett-Malament tradition of presenting Bell-type theorems in the final section of the paper.
The model I discuss here is inspired by, but not the same as, models discussed by Joy Christian. (See [5–12] and other papers available online; these are edited and collected with some new material in [13].) To my mind, at least, the present model is considerably simpler than Christian’s, and it avoids certain technical complications that his models encounter. (For more on these technical issues, see in particular [22] and references therein.) That said, although my engagement with Christian’s work will be from an oblique angle, what I say applies equally well to the models he discusses. I should also note that since the present paper first appeared online, Christian has posted a reply [14]. However, his reply is not responsive to the issues raised here and for this reason I do not engage with it.
In personal correspondence and elsewhere, Christian has criticized the model presented here as ad hoc and unrealistic. This charge is entirely just. But I should be clear that the model is not meant to be a realistic contender to explain Bell-violating correlations in quantum mechanics; rather, it is a simple toy model with certain suggestive formal properties that helps clarify the explanatory demands that a local, deterministic model of the EPR-Bohm experiment would have to meet. The argument here is not that the present model fails and thus Christian’s (unquestionably different) model also fails; it’s that a careful analysis of the present simple model reveals something important about what any successful model must do. I then show that no model of a certain quite general form can do this.
The character of the response offered here is related to a brief criticism of Christian’s proposal by Grangier [17], although the details are quite different.
Does this point need to be made in print? I am sympathetic with readers who might think that it does not. But the work by Christian alluded to above, and the significant controversy it has spawned in some circles (see for instance the discussion at Scott Aaronson’s blog, http://www.scottaaronson.com/blog/?p=1028), suggests that there remains considerable confusion regarding just what Bell-type theorems tell us, even among serious physicists.
The Pauli spin vector is actually a map from unit vectors in three dimensional Euclidean space to the space of operators on the two-dimensional Hilbert space representing spin states. But the abuse of notation should be harmless, since the inner product is shorthand for the obvious thing: σ⋅n=σ x n x +σ y n y +σ z n z . Note that we are working in units where ħ=2.
The set \(\mathbb{S}^{2}\subseteq\mathbb{R}^{3}\) is to be understood as the 2-sphere, i.e., the set of unit vectors in \(\mathbb{R}^{3}\).
In the following expressions, I am writing integrals over Λ without being fully specific about what Λ looks like, with the possible consequence that the interpretation of the expressions is ambiguous. It turns out, though, that Bell-type results are based on features of integrals that are so basic that it does not matter what kind of integral is being written.
At times I will continue to refer to a “local, deterministic hidden variable model,” but the “local” and “deterministic” modifiers will be simply for emphasis. As I will understand it in the rest of this paper, a hidden variable model is automatically local and deterministic.
There are several reasons why a reader might balk at this point. I admit that the argument given here is problematic; indeed, I will attempt to say, as precisely as possible, what goes wrong with this argument in the next section. In the meantime, I encourage a reader who anticipates the problems even at this stage to hold back her objections to see where the discussion goes. This is a productive garden path to wander down, even if one can already see that it ends in a coal pit.
At this point, I am changing notation, since the model I will presently describe is most naturally expressed using tensor fields defined on a manifold. Here and throughout I will use the abstract index notation explained in [23] and [21]. For present purposes, one can safely think of these as the counting indices of coordinate notation, with an Einstein summation convention assumed. I will use the following translation manual for vectors previously discussed: I will now use α a instead of a to represent the vector about which Alice chooses to make her measurement, and β a instead of b to represent the vector about which Bob chooses to make his measurement. These vectors still live in \(\mathbb{S}^{2}\).
What does it mean to say that representing states of rotation as antisymmetric rank 2 tensors respects the “algebraic and topological properties” of rotations in three space? The space of constant antisymmetric rank 2 tensors on three dimensional Euclidean space (understood as a vector space) is itself a three dimensional vector space over \(\mathbb{R}\) (i.e., it is closed under addition and scalar multiplication) and moreover forms a Lie algebra with the Lie bracket defined by, for any two antisymmetric rank 2 tensors \(\overset{1}{\omega}_{ab}\) and \(\overset{2}{\omega}_{ab}\), \([\overset{1}{\omega},\overset{2}{\omega}] _{ab}=\overset{1}{\omega}_{an}\overset{2}{\omega}{}^{n}{}_{b}-\overset{2}{\omega}_{an}\overset{1}{\omega}{}^{n}{}_{b}\). A short calculation shows that this Lie algebra is none other than \(\mathfrak{so}_{3}\), the Lie algebra associated with the Lie group of rotations of three dimensional vectors, SO(3). The elements of SO(3), meanwhile, can be represented by the length preserving maps between vectors in \(\mathbb{R}^{3}\) that are standardly identified with rotations. Note that since the Lie algebra \(\mathfrak{so}_{3}\) is isomorphic to the Lie algebra \(\mathfrak{su}_{2}\), the vector space generated by the Pauli spin matrices, which are the operators associated with spin for a spin 1/2 system, is also a representation of \(\mathfrak{so}_{3}\).
These are not “observables” in anything like the standard sense. They might be better characterized as complete representations of the system, and perhaps the apparatus, after measurement. I will at times use the term “observable” in this non-standard sense since these “observables” are intended to play a similar formal role to the observables in a Bell-type hidden variable model.
There are other ways one might do this, too. Christian, for instance, proposes that one represent rotations using bivectors in a Clifford algebra. There are various reasons to think that proposal is equally well-motivated as, or even better motivated than, the one I make here. But ultimately such distinctions will not matter. See the next section for more on this point.
A volume element can be understood to give an orientation to the entire three dimensional manifold in much the same way that antisymmetric rank 2 tensors give an orientation to a plane, by specifying the relative order of any three linearly independent vectors. For more on volume elements, see [21].
Actually, there is a slight abuse here: expectation values are usually defined to be real numbers; the “expectation values” I define presently are actually rank 2 tensors (specifically in this case the zero tensor). However, the interpretation of these generalized expectation values is unambiguous, and they clearly have the desired physical significance.
It is at this stage that the model I present diverges most substantially from Christian’s. In his proposal, the product expectation value is calculated using the geometric product of two Clifford algebra-valued observables. In mine, it is the inner product of two antisymmetric rank 2 tensor fields. Antisymmetric rank 2 tensor fields can generally be identified with bivectors in a Clifford algebra, but the products in the two cases are entirely different. It turns out, though, that this distinction just does not matter. Neither product is tracking the right experimental information, and indeed, as I will presently show, no such product could track the right experimental information and still yield a result that conflicts with Bell-type theorems.
This expectation value is a real expectation value, in the sense that it is real-valued, unlike the two single sided expectation values defined above.
The present point will become clear in what follows, but it is perhaps worth emphasizing here as well: the conclusion stated in the text above is false. That is, the model I have described in this section, and particularly the expressions for the product of the measurement results and its expectation value, do not reproduce the predictions of quantum mechanics. The goal of the remainder of the paper will be to articulate as clearly as I can where the problems lie.
To be clear, the Greek index on P is not a tensor index—rather, it indicates which measurement vector one is projecting relative to. In what follows, I will at times suppress this index to talk generally about a projection relative to some unspecified measurement vector.
It is worth observing that the observables defined above, evaluated for a given measurement vector, are in this set, relative to that choice of measurement vector.
We can now see that the argument in the previous section was a straightforward paralogism. It goes, (1) quantum mechanics predicts that the average value of the products of two sequences of integers will be the inner product of two vectors; (2) this alternative model predicts that the average value of the inner products of two sequences of tensors will be the inner product of two vectors; therefore (3) the alternative model reproduces the predictions of quantum mechanics. Since (1) and (2) make reference to two different sequences, (3) does not follow.
Calling a model of the sort I define above “generalized” is admittedly tendentious. There is a strong sense in which this sort of model is a specialization rather than a generalization of Bell’s notion of a hidden variable model, since (as will be clear in the proof of Corollary 4.2) it implicitly includes all of the pieces of a Bell-type hidden variable model, plus some additional (unnecessary) machinery. Nonetheless I am using the expression, since it captures the intended spirit of the (failed) model described in the body of the paper.
In principle, the family of maps could also be parameterized by values of Λ and everything below would go through with only minor modifications. I do not explicitly treat this possible generalization, however, because its significance is unclear: if one thinks of the projection as an operation that an experimenter performs, then it is hard to see how that operation could depend on a variable whose value the experimenter does not know.
I am now switching back to the notation used in Sect. 2, to emphasize the generality of these expressions. I am also explicitly indicating that the projection maps may depend on the measurement vectors.
I should emphasize that this result applies irrespective of X and of \(\mathcal{P}\), which means that it precludes models like the one Christian proposes in addition to the model I have discussed. It also means that one cannot get around the central claim with a clever choice of \(\mathcal{P}\).
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Acknowledgements
I am grateful to Shelly Goldstein, David Malament, John Manchak, Florin Moldoveanu, Abner Shimony, Howard Stein, and an anonymous referee for helpful comments on a previous draft of this paper. Thank you, too, to Jeff Barrett, Harvey Brown, David Malament, and Tim Maudlin for discussions on these and related topics and to David Hestenes for a helpful email exchange on Clifford algebras. I am especially grateful to Joy Christian for an informative correspondence and for his kind help on this project despite our continued disagreement. Finally, this paper has benefited from audiences at the Southern California Philosophy of Physics Group and the Maryland Foundations of Physics Group, and I am grateful to Christian Wüthrich and Jeff Bub for inviting me to make these presentation.
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Weatherall, J.O. The Scope and Generality of Bell’s Theorem. Found Phys 43, 1153–1169 (2013). https://doi.org/10.1007/s10701-013-9737-1
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DOI: https://doi.org/10.1007/s10701-013-9737-1