## Thursday, August 18, 2016

### Degrees of location?

If collapse versions of quantum mechanics are right, then objects typically don't have location simpliciter. Instead, they have a wavefunction the square of whose modulus describes the probability that the object will collapse to a given location. Perhaps right after a collapse, the objects have a single definite location, but the single definite location doesn't last beyond that moment.

Suppose we take all this seriously as metaphysics. I think there are several options. The first is that we should take the wavefunction, or the square of its modulus, as providing the whole story about an object's location. In that case, it is rarely if ever correct to say that an object has a particular location. Instead, one should say that objects have their locations to various degrees. (This degreedness of location is different from the way in which we can say that a person who has one leg in a room is to a lesser degree in the room than someone who has an arm and a leg in it.) Location, at least as exhibited in the actual world, is a degreed property.

The second option is that an object is wholly in a location when and only when the wavefunction assigns unit probability to its being there. This has the odd consequence that if a point particle is wholly in a region R, then it is also in every punctured region of the form R−{x}, since the integral of the modulus squared of the wavefunction over R and over R−{x} will be the same. But this means that the particle is wholly absent from every point of the region R, even while wholly present in the region R. That seems problematic.

A third option is that being wholly located is metaphysically primitive, and there is a law of nature that makes it be the case that when the integral of the modulus squared of the (normalized) wavefunction of a particle over a region R is 1, and the region R is "nice" (e.g., equals the interior of its closure), the particle is wholly in R.

I like the first option most...

Unknown said...

Surely paragraph three is even worse, I can choose a countably dense subset of measure zero in R and find that the particle is not in it. That seems truly perverse to me.

Alexander R Pruss said...

In option 3, I assumed that the law of nature was restricted to the case where R was "nice", and I offered the suggestion that "nice" might mean equal to the interior of the closure (= "regular open"). The only dense subset of R that is equal to the interior of its closure is R itself.

Unknown said...

Oh yes, very reasonable. I was just adding the (not very interesting extension) that a thing is at no particular place in the sense that if it is in R it is in R-{x} i.e. the probability for any particular outcome for a continuous random variable is zero, with the extension from {x} to a set of measure zero. If we are talking about 'regions' in any meaningful sense it probably makes good sense to exclude such weird stuff.

Alexander R Pruss said...

On the other hand, it seems to be possible for a particle to have a precise single-point location (the wavefunction would be a delta function; I don't think any realistic measurement could produce this outcome, though, so maybe we could say it's physically impossible??). If so, the restriction to regular open sets doesn't work. Maybe better: If the integral of the square of the modulus of the normalized wavefunction over a region R is 1, then by law the particle is wholly present within the closure of R. And when the integral of the square of the modulus of the normalized wavefunction over a region R is 0, then by the law the particle is wholly absent from the interior of R.

Michael Gonzalez said...

Are there not metaphysical problems to any of these options if we add that larger objects have locations which are not "degreed", even though they are constructed of all these "location-degreed" particles?

Alexander R Pruss said...

Why think larger objects have locations that are not degreed?

Michael Gonzalez said...

Aside from our intuitions, there's the matter of exactly what experimental data we think we have, which leads us to postulate all these different physical theories of quantum mechanics in the first place. If the point or ink dot or computer pixel isn't actually in the specific location we perceive it to be in, then we don't actually have the data we think we have, and thus anything we think we've confirmed experimentally about quantum mechanics is (at least) suspect. John Bell often talked about this: How can you embrace a metaphysics in which the fundamental pieces have no location, when you are counting on pointers and printouts and laboratories and researchers (all of which are built from these fundamental pieces) to have particular locations?

Alexander R Pruss said...

All you need is that the pointers have their particular locations *to a very high degree*, I think.

Michael Gonzalez said...

Maybe. I don't know. It seems like the whole theory is predicated on particular experimental results (viz, positions on pointers), so the only reason we're even talking about the possible interpretations (including degreed locations) is because we want to explain these results. If we don't actually get those results (because every piece of every point is actually only sort-of located where it is), doesn't that saw off the branch that the theory was resting on?

It's like with "Flashy" GRW or Albert's "Many Minds"... If the world we believe in doesn't exist, then why are we trying to explain the constitution of that world... especially in terms of parts that don't actually align they need to in order to construct the world being explained?