Monday, August 22, 2016

Partial location, quantum mechanics and Bohm

The following seems to be intuitively plausible:

  1. If an object is wholly located in a region R but is not wholly located in a subregion S, then it is partially located in R−S.
  2. If an object is partially located in a region R, then it has a part that is wholly located there.
The following also seems very plausible:
  1. If the integral of the modulus squared of the normalized wavefunction for a particle over a region R is 1, then the particle is wholly located in the closure of R.
  2. If the integral of the modulus squared of the normalized wavefunction for a particle over a region R is strictly less than one, then the particle is not wholly located in the interior of R.
But now we have a problem. Consider a fundamental point particle, Patty, and suppose that Patty's wavefunction is continuous and the integral of the modulus squared of the wavefunction over the closed unit cube is 1 while over the bottom half of the cube it is 1/2. Then by (3), Patty is wholly contained in the cube, and by (4), Patty is not wholly contained in the interior bottom half of the cube. By (1), Patty is partially located in the closed upper half cube. By (2), Patty has a part wholly located there. But Patty, being a fundamental particle, has only one part: Patty itself. So, Patty is wholly located in the closed upper half cube. But the integral of the modulus squared of the wavefunction over the closed upper half cube is 1−1/2=1/2, and so (4) is violated.

Given that scenarios like the Patty one are physically possible, we need to reject one of (1)-(4). I think (3) is integral to quantum mechanics, and (1) seems central to the concept of partial location. That leaves a choice between (2) and (4).

If we insist on (2) but drop (4), then we can actually generalize the argument to conclude that there is a point at which Patty is wholly located. Either there is exactly one such point--and that's the Bohmian interpretation--or else Patty is wholly multilocated, and probably the best reading of that scenario is that Patty is wholly multilocated at least throughout the interior of any region where the modulus squared of the normalized wavefunction has integral one.

So, all in all, we have three options:

  • Bohm
  • massive multilocation
  • partial location without whole location of parts (denial of (2)).
This means that either we can argue from the denial of Bohm to a controversial metaphysical thesis: massive multilocation or partial location without whole location of parts, or we can argue from fairly plausible metaphysical theses, namely the denial of massive multilocation and the insistence that partial location is whole location of parts, to Bohm. It's interesting that this argument for Bohmian mechanics has nothing to do with the issues about determinism that have dominated the discussion of Bohm. (Indeed, this argument for Bohmian mechanics is compatible with deviant Bohmian accounts on which the dynamics is indeterministic. I am fond of those.)

I myself have independent motivations for embracing the denial of (2): I believe in extended simples.

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2 comments:

  1. Michael GonzalezAugust 22, 2016 at 12:03 PM

    Is it right to talk about locations and parts derived from the wave-function (which is in configuration space) as though they were a location/part in actual space-time? The particle is obviously not located in some part of configuration space. It's located somewhere in actual space-time. Maybe my question is ill-thought-out; I just feel like something is incorrect about moving from discussing parts and locations in physical space to discussing locations in a configuration space that deals with probability distributions.

    In any case, I favor a Bohmian account anyway, since I think it's the only one that preserves the intuitions which allow us to say things like "I (a physical thing composed of particles) got this particular readout (ditto) in this laboratory (ditto)...". Especially when you look at the hydrodynamic experiments where phenomena like the two-slit experiment were replicated at the macro level with droplets on the surface of a fluid. Nothing is multi-located or "wave/particle dualistic" about it; there is a straightforward, intuitive explanation for what's happening.

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  2. Alexander R PrussAugust 23, 2016 at 3:43 PM

    In talking of integrals, I was thinking of the special case of a single particle, in which case we have the same region U of space and in configuration space. In the case of n particles, we can integrate over the region of configuration space that *corresponds* to U (a product of U with 3(n-1) copies of R).

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