*Most interpretations of quantum mechanics have taken non-locality – “spooky action at a distance” – as a brute fact about the way the world is. But there is another way. Take seriously quantum theory’s higher-dimensional models, and we could make sense of the strange, "spooky" phenomenon of entanglement and restore some order to cause and effect. Perhaps the wave function is all there is, writes Alyssa Ney.*

Since the first Bell tests in the 1980s by Alain Aspect and his collaborators, experiments have confirmed again and again what quantum mechanics predicts: our three-dimensional reality is nonlocal. Particles or atoms - quantum systems - created in entangled states can continue to influence each other instantaneously, even when separated over great distances. For those wanting to understand what physics tells us about the nature of our world, these experiments cry out for explanation. How could the world fundamentally be so as to allow instantaneous effects over great spatial distances?

Philosophers of physics seek metaphysical models that are compatible with the reality of quantum entanglement, possible realities that can underlie these experimental appearances. Most models taken seriously today simply accept that there are brute correlations that can persist between objects in entangled states, even those that are separated by great distances. But another idea worth exploring is that the results of these Bell tests are trying to tell us that the three dimensions in which we experience the world are not the deepest reality. Rather, the nonlocality we observe over distances in *three dimensions *appears as a byproduct of a more fundamental spatial framework in higher dimensions. And in this higher-dimensional framework, the correlated subsystems of entangled particles are not separated at all.

The higher-dimensional approach along these lines that is most commonly discussed today is called *wave function realism*. This is the view that reality isn’t fundamentally a collection of objects – particles, atoms – spread out in three-dimensional space or even four-dimensional spacetime, but instead, reality is fundamentally a wave function, a field-like object that exists in some higher-dimensional quantum reality.

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Physicists have mostly tossed aside the idea that these higher-dimensional representations are tracking what is real.

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**Higher dimensions**

The idea of higher dimensions is familiar from string theory. String theorists speculate that the world contains seven extra dimensions in addition to the three with which we are already familiar. String theorists provide an explanation for why we don’t notice these extra dimensions, arguing they are curled up in a way that makes them too small for us to see. In quantum mechanics, the higher dimensions aren’t there as extra dimensions in addition to the usual three. Instead, in order to represent systems as entering into entangled states, quantum mechanics presents us with entirely distinct spatial frameworks.

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Configuration space**

One such framework is configuration space. Configuration space representations were introduced in the nineteenth century to provide more rigorous and elegant formulations of classical mechanics. However, they become even more indispensable in quantum mechanics. In configuration space, as the name suggests, each point corresponds to a total configuration: a complete specification of determinate locations for each particle in a given system. This means configuration spaces used to represent a system of particles with apparent locations in three-dimensional space will be 3N-dimensional, where N is the number of particles. That is to say that each point in space will be labelled by 3N numbers. For a system with two particles, the configuration space is six-dimensional and a point in space (a configuration) can be provided by six numbers where the first three correspond to the x, y, and z coordinates for the first particle and the second correspond to the x, y, and z coordinates for the second. For a system of three particles, we need a nine-dimensional configuration space; for four particles, a twelve-dimensional configuration space. And for our universe, which is thought to have about 10^{80} particles, configuration space is 3x10^{80}-dimensional.

In a classical situation, one in which the locations of all particles is determinate, a system can be represented by one simple point in its configuration space. In quantum mechanics, however, particles have indeterminate positions, and so a quantum system must be represented as a field smeared out over this configuration space. This is the quantum wave function. The wave function field will have amplitudes at points in configuration space that correspond to locations in three-dimensional space where these particles may be found.

**Hilbert space**

Another higher-dimensional framework for representing systems in quantum mechanics is Hilbert space. Each dimension in a Hilbert space corresponds to a determinate state of some observable (a position coordinate or spin along a given axis, for example). Here, total systems are represented as vectors or rays in Hilbert space. For example, to represent a system of two spin-1/2 particles, physicists will use a ray in a four-dimensional Hilbert space, with two dimensions corresponding to the spin of the first particle being up or down along some dimension, and two corresponding to the spin of the second particle being up or down along that or some other dimension. When we consider observables like position coordinates that can take an infinite number of possible values, the Hilbert spaces become infinite-dimensional.

Though higher-dimensional representations like these are ubiquitous in quantum mechanics, usually physicists do not take them seriously as representing what our world is like. In quantum mechanics, physicists have mostly tossed aside the idea that these higher-dimensional representations are tracking what is real.

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I myself am more sympathetic to wave function realism, a higher-dimensional interpretation of quantum mechanics that picks up where Schrödinger left off

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**Interpreting Schrödinger’s wave equation**

This distaste for higher dimensions arguably partly explains the rise of the Copenhagen interpretation of quantum mechanics. In 1926, as is well known, Schrödinger reformulated quantum mechanics using his famous wave equation. This formulation lent the promise of allowing not only a simpler and more familiar *mathematical *statement of the theory, but also a formulation that would be more capable of providing a clear account of the nature of the world according to quantum mechanics, at least more capable than Heisenberg’s matrix formulation. Schrödinger’s formulation allowed one to see quantum systems as waves or fields evolving smoothly and continuously over time in accordance with his wave equation. However, Schrödinger quickly discarded this hope, as it was repeatedly pointed out that the representation of these waves’ evolution (for N particle systems) required configuration spaces of 3N dimensions. It seemed obvious that these waves in higher dimensions couldn’t be interpreted realistically. And so, Schrödinger joined his colleagues in setting aside the hope for a realistic picture of the world according to quantum mechanics, and instead adopted the Copenhagen interpretation.

Not all physicists today are so resistant to taking higher-dimensional interpretations of quantum mechanics seriously. For example, the physicist Sean Carroll, in his recent book on quantum mechanics, *Something Deeply Hidden*, argues that the best interpretation of quantum mechanics takes it to represent the world fundamentally as a ray in Hilbert space. There are interesting arguments that lead to a view like that, however, I myself am more sympathetic to wave function realism, a higher-dimensional interpretation of quantum mechanics that picks up where Schrödinger left off and interprets the world fundamentally as a wave or field. In this case, the world is not a simple ray in Hilbert space, but instead a field, the wave function, spread out over configuration space.

The idea that we take configuration space representations in quantum mechanics seriously as tracking physical or real, fundamental fields in 3N dimensions traces at least to John Bell in the 1970s. Bell was a key figure trying to pull the physics community back to the practice of treating its theories as realistic depictions of the world. As the philosopher of physics David Albert has been pressing since the 1990s, given the indispensability of higher-dimensional representations in quantum mechanics, a realistic attitude toward these theories seems to require we take wave functions as real, even if this means they inhabit not the three-dimensional world of our appearances, but a more fundamental realm of higher dimensions.

As I argue in my recent book, *The World in the Wave Function*, however, even if higher- dimensional frameworks like configuration space or Hilbert space might be indispensable to the formulation of quantum mechanics, that does not mean that those wanting to interpret these theories realistically must take them as describing a world with hidden dimensions. Since Bell, many different interpretational frameworks have appeared which provide alternative ways of making sense of the quantum formalism.

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This leaves us with a way of resurrecting Schrödinger’s hope of providing a smooth and continuous realist interpretation of quantum mechanics

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**Explaining spooky action at a distance**

However, this does not mean there is not a good argument for preferring higher-dimensional, wave function realist interpretations over these others. It is here that we can return to quantum nonlocality and the surprising results of the many Bell experiments. Recall what these experiments show is that when we consider quantum systems spread out in three (or four) dimensions, there are persistent correlations that suggest immediate interactions over long distances (“spooky action at a distance”). The wave function realist argues that there is a deeper explanation for why we see these correlations. They are a manifestation of a field that is evolving in a higher-dimensional space in which there is no instantaneous action at a distance.

What is interesting is that the way alternative low-dimensional metaphysical frameworks have proven adequate to providing an interpretation of quantum mechanics is by giving up on fundamental locality. But in doing so, they have given up on trying to explain nonlocality. They just take it as brute, as given. In a sense, this is to give up on a very basic principle of science: that we explain correlations in terms of a common cause.

If one follows the wave function realist in trying to explain nonlocality in terms of a common, higher-dimensional cause – the wave function – this leaves us with a way of resurrecting Schrödinger’s hope of providing a smooth and continuous realist interpretation of quantum mechanics. The higher dimensions are surprising. But they are not superfluous. They play an explanatory role in making sense of the persistent and puzzling results of Bell tests. This does not mean we can know with certainty these higher dimensions are real. But at this early stage of grappling with these experimental results, it is *an *explanation, and this makes it worth pursuing.

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