### Intro

There has recently been a lot of talk about spacetime being emergent rather than fundemental. While physicists and philosophers have given many interesting strategies for how this could work, there is one in particular I'd like to discuss in this essay. As part of a more general strategy for solving the measurement problem in quantum mechancis, David Albert and Barry Loewer have suggested that the configuration space in Bohmian mechancins is physically real, rather than just a mathematical representation, where the true dimensionality of space is several times the number of elementary degrees of freedom. On this view, the manifest world should be understood as something like a projection or shadow of the world particle in configuration space, structured in such a way that it appears three dimensional to observers.The tenability of this view depends on several factors, the most important of which is whether it’s possible for a world like ours to emerge from this higher dimensional space. There are clear instances where configuration space should be understood as a mathematical tool rather than as a description of physical space. To take an example from classical mechanics, consider a rigid body with its location defined relative to an origin and it’s orientation defined relative to some frame. The configuration for this rigid body is six dimensional because it has a 6 degree of freedom, but no one would talk this to imply that this physical phenomenon represented by this configuration space must necessarily be an actual point particle in higher dimensional space.

There is some modivation for thinking that the configuration space in Bohmian mechanics should be taken more seriously as a representation of the world than in classical mechanics. One difference is that the higher dimensional space seems indispensable to the quantum mechanical picture in a way it isn’t in classical mechanics. In the rigid body example above, the six dimensions can easily be separated into two three dimensional spaces, one of which describes the location of the object and the second of which describes the orientation of the object in motion. If we generalize this strategy, we might claim that a dimensional configuration space should be taken as fundamental when it is no longer separable into multiple lower dimensional spaces. This example, if correct, shows why this higher-dimensional space is indispensable in quantum mechanics. If we consider again a singlet state and the configuration space associated with it, attempting to separate this space into individual subspaces describing each particle cannot capture information about the correlations between the particles. Another important point is that in the classical mechanical case, because of the way velocity will determine position, the total possible states of the system given some initial state is really only a subspace of the total configuration space, while in the quantum mechanical case, the whole space physically possible. This is further reason that the configuration space is indispensable in the quantum mechanical case but not the classical mechanical one.

How might it be possible for the manifestly three dimensional world to emerge from this higher dimensional space? There have been several attempts to explain this, one of which is due to Bell. Because in Bohmian mechanics there is, in addition to the wave function, a hidden variable that determines the results of our measurements, as long as the state of any Bohmian particle can be interpreted in a three-dimensional space, we can expect that the entirety of our experience can be interpreted three-dimensionally. Bell points out that it is the “hidden variables” we experience, while the wave function remains hidden. There is, however, a problem with this view. For any given coordinate in this space, there is nothing in the coordinate itself that specifies how the parameters should be mapped onto the three-dimensional space. This could pose a problem for interpreting configuration space as something real, as I will attempt to show below.

Consider again, a rigid body in four dimensional space. Clearly the orientation of the axis relative to which the coordinates specifying the state of the rigid body are a matter of definition. So if the position of the rigid body is represented by the coordinate (1, 1, 1, 1) relative to some axis, there may be some other axis relative to which the coordinate is (2, 2, 2, 2). And the same way that the axis position can be arbitrarily defined, the rotation of these axes can be arbitrarily defined, meaning that there’s some axis relative to which the coordinate is (2, 1, 2, 1). Clearly in all these situations the actual state of the object is the same, which means that the state of the object is invariant under transformations of the coordinate system. This rigid body is analogous to the world particle in configuration space.

But if I now say this 4 dimensional coordinate actually “projects” down to two particles in two different locations, much the same way that the world particle in configuration space is supposed to “project” to our three dimensional world, we find that the coordinates are no longer invariant over transformations of the axis. Combined with the fact that there is no preferred way to orient the axis, it would follow that the single point in higher dimensional space fails to fully determine the locations of the two particles. If this analogy also holds in the Bohmian case, it would follow that the world particle in configuration space doesn’t fully specify the state of the manifest three dimensional world, which provides evidence that the configuration space should be understood as a mathematical construct rather than something physically fundamental.

There is, however, another suggestion due to David Albert (1996) that may resist this objection. One could claim that the wavefunction state does not need to determine a three dimensional state at each moment of time, but rather that it is the effect of the evolution over time that creates our experience of a three dimensional world. Albert’s claim is that the laws that govern the evolution of the system are what creates the impression that the world is three dimensional. This means that because the dynamics of the world particle are restricted, a specific grouping of the coordinates, while artificial, will seem more compatible with the physical facts than others. Peter Lewis (2004), claims that for any two particles approaching each other, there is a grouping of coordinates into N groups of 3 so that when these groups approach each other, there are higher velocities of approach relative to the other groupings. This allows us to only consider a subset of the total coordinates, because these subspaces are more approximate representations of the total space.

Because these 3 groupings are more natural, one might suppose that evolution has equipped us psychologically such that the world appears three dimensional, even though the wave function exists fundamentally and in higher dimensional space. Here I think Albert has made a serious proposal for how we could come to interpret a higher dimensional space in lower dimensions. While I think this proposal could explain how we could come to interpret a higher dimensional space in lower dimensions, I don’t think it solves the problem that a particle in higher dimensional configuration space underdetermines the state of the 3 dimensional world that should emerge from it. As with the coordinate transformation example I gave previously, since how the 3d emergent world emerges from the world particle isn’t invariant under coordinate transformations, additional information must be given for how to map the dimensions of the configuration space onto features of the actual world. But it isn’t clear from the world particle alone how to do this, so the world particle as a physical phenomenon would underdetermine the state of the actual world.

While, if it could avoid the objection I have above, this view is an effective way of reconciling entanglement and separability (not the separability of space-time points, but fundamental state separability), reducing the physical world to a particle in high dimensional configuration space is such a radical shift in ontology that one could reasonably question what would remain of Humean Supervenience in this picture. In the 4D view, Lewis takes causes to supervene on the counterfactual relations between points in spacetime. So even though causation isn’t fundamental, it supervenes on the relations between the fundamental entities. If we wanted our notion of cause to supervene on the fundamental entities of the higher dimensional view, it would supervene on the relations between the single world particle that exists at each moment of time. A full discussion of how the higher dimensional picture could change our views on causation would require another paper, but I’ll make a few observations here. First off, it seems that overdetermination wouldn’t pose a problem for counterfactual theories of causation amongst others in fundamental physics (though one still might want to deal with overdetermination to make sense of ordinary language talk about causation), because only a single world particle exists at each moment in time. For the same reason it doesn’t seem that preemption cases apply to fundamental relations. On the other hand, there seems to be some intuitive appeal for taking a causal process approach.

From these arguments, I conclude that the manifest world would fail to emerge from a physically real configuration space.

### Citations

Albert, D. Z. (1996). Elementary Quantum Metaphysics. Bohmian Mechanics and Quantum Theory: An Appraisal Boston Studies in the Philosophy of Science, 277–284. doi: 10.1007/978-94-015-8715-0_19Dewar, Neil. (2016). What the Humean cannot say about entanglement. Retrieved from http://philsci-archive.pitt.edu/12046/.

Dewar, Neil (2018). La Bohume. Synthese. doi: 10.1007/s11229-018-1800-1

Esfeld, M. (2014). Quantum Humeanism, or: Physicalism without Properties. The Philosophical Quarterly, 64(256), 453–470. doi: 10.1093/pq/pqu030

Lewis, P. J. (2004). Life in Configuration Space. The British Journal for the Philosophy of Science, 55(4), 713–729. doi: 10.1093/bjps/55.4.713

Loewer, B. (1996). Humean Supervenience. Philosophical Topics, 24(1), 101–127. doi: 10.5840/philtopics199624112