Let us continue with our exercise of trying to infer natural philosophy conclusions from general features of the laws of physics. Again, our method will be to assume that a symmetry in physical laws indicates that some of the states we conceptually distinguish don’t represent real differences, and that this is telling us something about the underlying objects.
The statistics of indistinguishable particles in quantum mechanics is an especially fertile field for contemplation. As Stephen Barr once put it in an argument over the resurrection of the dead at First Things, elementary particles don’t have haecceity. You can’t trace out the history of “this particular electron”. How can this be?
Consider a helium atom with one electron in state |A> and one electron in state |B>. How do we construct the state vector for the 2-particle system? Well, why not just say “the first electron is in state A, and the second electron is in state B”, where the labeling is arbitrary, but once we’ve made it, we can stick to it, calling the state of the system |AB>? Those of you who took undergraduate quantum mechanics know that this is wrong. Electrons are indistinguishable, so the state vector must be symmetric or antisymmetric, and since electrons are fermions, it must be antisymmetric. The correct answer is (|AB>-|BA>)/sqrt(2). Getting the right answer has important consequences, such as the Pauli exclusion principle: if A=B, you get zero, so two identical fermions in the same state never happens. Now, this is not just a bookkeeping point. In classical mechanics, one could still freely choose which electron gets to be called “the first electron” and which gets to be called the second, but once you tag them at a single time, you can keep the same labels on them as they evolve. You can say, for instance, that the electron that started out at A is now at C, while the electron that started at B is now at D. In quantum mechanics, you can’t do this, even in principle. The state vector has all the information in the system, and if it changes from |AB>-|BA> to |CD>-|DC>, all you can say is that there were electrons in the A and B states, and now there are electrons in the C and D states. You can’t say which went to which. Swapping the two electrons at any time doesn’t change the state; it doesn’t really mean doing anything.
Now, it could be that such information exists outside of the state vector, i.e. as a “hidden variable”, but in this game I’m playing I always assume the absence of unneeded structure. In this case, there really is a problem with saying “this particular electron” (or photon, or up quark, or Z boson, etc). Just as in my last post, the system as a whole–in this case, the system of every instance of a particular particle type (since we can still think of the state vector as having a separate factor for each distinct particle type, assuming there is ultimately more than one)–seems to be logically co-primordial with the particles themselves.
This certainly relates to my physics vs. reductionism musings, but I’d like to talk about something else instead. An Aristotelian confronted with the properties of fermions and bosons is more likely to feel perplexed than affirmed. The Aristotelian sees material objects as composed of form and matter. Multiple material objects of the same type can exist because they are composed of different matter. Now, there’s a common perception that, of the two principles, matter is rather straightforward–it’s the “stuff” we encounter all the time–whereas form is a mysterious thing, something that natural scientists are accused of ignoring, and something that the scholastic philosopher must concentrate most of his energy on defending. In fact, I think the situation is the reverse. Form–natures of things, functions in organisms, distinctive levels of intelligibility in composite bodies–comes naturally to our minds. It’s what our intellect is for, after all, and it’s all over the sciences. Matter is the mysterious thing. It’s very hard to describe what it actually contributes. All the properties of a thing are formal properties. Matter, on the other hand, is supposed to be a kind of potency. If form is intelligibility, potency is lack of intelligibility. It is the property of a thing that it is partially undetermined by its own nature. For example, being a human being means I must have certain properties, but not that I should be a certain age or have a certain profession. Those properties are not themselves material–they are accidental forms–but they are only possible because I am not identical with my nature (humanity), and matter (potency) is the principle of this separation. In fact, rather than saying that forms organize matter, Thomists are more likely to talk about matter limiting form, restricting it to a particular instance, a particular subject. You see what I mean when I said that matter, not form, is where the obscurity is? I don’t think this is a flaw of the Aristotelian/scholastic system. I think it’s just that the problem of individuation is really subtle.
Okay, what about the fermions and bosons? If each electron were a composition of the form of electronness plus its own parcel of primary matter to distinguish it from its nature, then electrons would in principle be distinguishable. It would make sense to say “this electron here, not that one there”. Let’s follow our no-hidden-variables rule, and say that we can’t do this. Are electrons immaterial objects? That’s absurd. First, they must have potency, since the state of the electrons (the whole aggregate, let’s say) doesn’t flow from any sort of necessity from their nature. And this potency must be material as the scholastics understand that term, in that the state of the electrons does have spatial information. So, instead we should say that the whole particle species is individuated by primary matter, not each one by its own parcel. The scholastics usually liked to think of macroscopic substances (e.g. living organisms) as being the recipients of primary matter, with the formed subparts of the substance having a sort of derived existence. We are actually no closer to this view. Rather than each individual particle in my body being its own separately existing thing, we are now in danger of giving primacy to each particle species as a whole. If there is ultimately only one elementary species, e.g. strings with different excitation states, then it will be easiest to think of the whole system of the universe as being informed by primary matter, with substances in the universe having a derived existence from the total state. Monism is a bigger danger than reductionism.
Strange as this sounds, the Aristotelian will be bothered by one point in particular. Matter is supposed to allow any object to have multiple instances, but in this case, it’s the whole species of particle that is individuated. Can we individuate another entirety of the species by informing some distinct primary matter? That’s a weird claim, since “the entirety of the species X”, by definition, doesn’t sound like a thing that can be multiply instantiated. In that sense, it’s like Saint Thomas’ speculations on the angels, who are compositions of act and potency, but there can only be one of them per species. However, maybe in principle why there could be two instantiated “whole species” of a particle with certain properties. Just imagine two distinguishable species of particles with slightly different properties, and take the limit as the differences in properties go to zero. A mix of, say, electrons of one species instantiation and electrons of the other species instantiation would then not obey Fermi-Dirac statistics. This sort of thing certainly doesn’t seem to have happened in the real world, but if it is even imaginable, then “the species of photons” and “the species of strange quarks” would be material in the usual way. On the other hand, Thomists will probably not like this argument, because it might also work with angels, I would think. Why couldn’t there be two instantiations of a single angelic species, informing distinct subjects?