How does pilot wave theory explain "identical particle" interference?

Question

Pilot wave theory says that there exist waves in 3D space which carry particles. This explains, say, the double slit experiment.

But this does not explain the behavior of identical particles. According to standard QM, a system of two identical particles will have quantum interference. But this interference does not take place in the real world 3D space, but rather in an abstract space. The wavefunction of two identical particles looks like $\psi (x_1,x_2)$. But the points $(x_1,x_2)$ live in an abstract space, as in, $(x_1,x_2)$ is not to be identified with a location in the real world 3D space. Rather, it is to be identified with a configuration of the system.

So, since the waves in pilot wave theory live in the real world space, how can it explain the wave-like interference in abstract spaces?

Answer 1

To quote Bohm's 1952 paper:

In the two-body problem, the system is described therefore by a six-dimensional Schroedinger wave and by a six-dimensional trajectory, specifying the actual location of each of the two particles. The velocity of this trajectory has components $\nabla_1 S/m$ and $\nabla_2 S/m$, respectively, in each of the three-dimensional surfaces associated with a given particle.

The "Schrodinger wave" is what we now call the pilot wave, which is written $\psi= e^{iS} R$ for $\mathbb R$-valued functions $S$ and $R$.

So the straightforward answer to your question is that it's founded on an incorrect premise - the pilot wave is defined on the configuration space of the system (in this case, $\mathbb R^3\times \mathbb R^3$) not simply on $\mathbb R^3$. If you read the paper, it is made quite plain that Bohm is proposing an alternative interpretation of the mathematics which underlies the standard formulation of quantum mechanics, not a new theory. The equations you need to solve in de Broglie-Bohm picture are exactly equivalent to the equations you need to solve in ordinary QM, you're just wrapping different words around them.

Answer 2

The pilot wave theory is mostly a strategy for denying the implications of quantum theory. The pilot wave theory takes the wavefunction and adds particles on top of it. Pilot wave theorists then have two options. (1) They can deny that the wavefunction is real, in which case it makes no sense that the particles are influenced by it and the theory can't explain single particle interference let alone more complicated issues like EPR correlations. (2) They say the wavefunction is real in which case they still have to deal with the issue of how to understand the fact that it looks approximately like a collection of parallel universes under conditions of decoherence. So pilot wave theory doesn't solve the measurement problem and also adds structure to quantum theory so it has parallel universes and other stuff on top.

Also, contrary to one of the answers given above, pilot wave theory does make predictions different from those of quantum theory for the added particles.