A couple of weeks ago I talked about the thermodynamics of indistinguishable particles, which are different from distinguishable particles, and this allows us to know that helium atoms, for example, are truly indistinguishable, not merely very similar. There are a wealth of other phenomena that demonstrate that particles of the same kind are truly indistinguishable, in violation of the “identity of indescernibles” princple that lies at the foundation of a great deal of metaphysics in modern philosophy. At its deepest level, the indistinguishability of particles restricts the most fundamental forms of our quantum-mechanical description of nature.
This is going to get a bit technical, but it’s important to understand how deep and powerful the notion of indistinguishability is. People often react to it with a kind of “Oh there must be something wrong with that you just haven’t figured it out yet” kind of attitude. And sure, maybe: we’ve been wrong before. Doubt is the essence of knowlege. But given that literally all of quantum mechanics would have to be wrong if identical particles were distinguishable, it seems fairly unlikely in this case, doesn’t it?
One way of looking at quantum mechanics is that it is the physics of things we cannot see.
Einstein famously asked his biographer the physicist Abraham Pais, "Do you really believe the moon isn't there when no one is looking at it?"
This is the wrong question, completely unrelated to what matters, which is: What is the ontological status of the moon when no one can look at it?
We know it isn't the same as the ontological status of the moon when someone, somewhere can look at it, because we have a completely different dynamical theory that describes the motion of objects we cannot look at: quantum mechanics. Objects we can look at--whether or not we do--are described by classical mechanics. They do not exhibit interference phenomena, multiple classical pathways, or the counting statistics of indistinguishable particles. Objects we cannot look at do.
Now as it happens, for a large object like the moon we can always look at it. The quantum veil can only hide very weakly interacting things or things with very simple structures. Precisely why this is so is a hard question that I'll come back to in time.
But for sufficiently simple objects--which are generally small and/or cold--it is possible to put them into states where we cannot see them, and the act of "seeing" or interacting with them forces them into a different state, which is one way of looking at the Uncertainty Principle: the act of observation disturbs the system in such a way as to change it. I don't think this is a very good way of looking at it, mind, because what we know now is that the system was not "some particular way" to be disturbed out of before observation. It could be many ways, at least as our classical minds are constrained to understand it by the laws of non-contradiction and causality.
Objects we cannot see are described by wavefunctions, and the probability they are one way or another when we can see them again is given by the square of the wavefunction.
When we have a single particle in one dimension we have a wavefunction of one variable, the particle position: f(x). Another particle might have a different wavefunction, g(x). If these are indistinguishable particles there are two of them, but they don't necessarily have the same wavefunction: f(x) and g(x) can be different, which reflects the possibility that the particles might have different, but unknowable, positions.
That is, we can have one indistinguishable particle over here at x1 and the other over there at x2. We can swap them and nothing will happen to the wavefunction... but if f(x) and g(x) are different functions this means that their simple product f(x)*g(x) can't describe our pair of particles, because then f(x1)*g(x2) won't be equal to f(x2)*g(x1). We would be able to tell if the particles changed places, which means we’d be able to tell one from the other in the same way if we lay two coins side-by-side we can tell the difference between the left on being heads and the right one being tails, and the left one being tails and the right one being heads. That is precisely the situation we have to avoid in any description of indistinguishable particles, becuase it makes them… distinguishable.
One way to create a two-particle state that correctly describes indistinguishable particles is to take {f(x1)*g(x2)+f(x2)*g(x1)}/sqrt(2) where the factor of sqrt(2) is to ensure the probability never goes over 1: it is just a normalization factor.
This is called a symmetric state, because we can swap particles without it changing, and if we have more than two particles we can build up more complicated multi-particle symmetric states using sums of products of single-particle states.
So far, so good.
This isn't the only way of doing it, though: the only things we can observe are the probabilities of various outcomes, and those depend on the square of the wavefunction, which means there is no way to tell the difference between a system described by one particular wavefunction and one that's described by the negative of that wavefunction. The square is the same in both cases.
This means we can also describe indistinguishable particles by the anti-symmetric combination of their product, which changes sign when they are swapped around: {f(x1)*g(x2)-f(x2)*g(x1)}/sqrt(2) For more complicated states we can again construct a more complicated sum that changes sign but otherwise is identical when any who particles are swapped.
Wavefunctions with these specific mathematical forms are the only ones that can describe indistinguishable particles, and the fact that they are the only wavefunctions that actually describe the world we live in is very strong evidence that particles of the same kind really are indistinguishable.
The foundations of this idea for symmetric states were worked out by the Indian physicist Satyendra Bose with later contributions by Einstein. Italian physicist Enrico Fermi, along with English physicist Paul Dirac, figured out anti-symmetric states. In both cases, what they were concerned with was how symmetry or anti-symmetry affected the statistics of occupied states, and found that for collections of particles described by a symmetric wavefunction there was a tendency for them to all be in the same state, and for collections of particles described by an anti-symmetric wavefunction no two of them could ever be in the same state.
For reasons not as fully understood as we would like there is a relationship between which particles are described by symmetric or anti-symmetric wavefunctions, and their intrinsic angular momentum. Elementary particles behave like little spinning tops, and there is a "natural" unit of angular momentum at the sub-atomic level that we just call "spin 1". Some particles, like electrons, have precisely half this natural unit. Others have integer multiples of it, including zero.
Particles with half-integer spin like electrons have anti-symmetric wavefunctions and obey Fermi-Dirac statistics (no two particles allowed in the same state), and particles with integer spin like helium atoms and photons have symmetric wavefunctions and obey Bose-Einstein statistics (all particles "want" to be in the same state).
As is often the case in physics, there is a simple argument for one part of this situation, and no simple argument for the other: in the anti-symmetric state, if the individual particle wavefunctions f(x) and g(x) are identical, which would be the case if both particles were in the same state, then the anti-symmetric wavefunction would be {f(x1)*f(x2)-f(x2)*f(x1)}/sqrt(2), which is obviously zero. That means particles described by this kind of wavefunction have zero probability of being in the same state.
There is no comparably simple argument for why integer-spin particles all want to be in the same state, but the phenomenon is enormously important. Photons, for example, which are light particles, have integer spin, and the tendency this gives them to all want to be in the same state is what is responsible for lasers. A laser-beam consists of a very large number of photons that are all in identical states of motion, which is why the beam is so narrow and straight. In an ordinary light source the photons move outward in a cone, but in a laser the quantum cascade that produces them ensures they are all moving in precisely the same direction, up the limits of the Uncertainty Principle.
To take another example, in metals there are many electrons that are free to move within the material. These "conduction electrons" should contribute to the heat capacity of the metal, which is a measure of how many ways any thermal energy added to a block of material can be divided up across the available internal states. Naively this contribution should be quite large, but the experimentally observed value is less than one percent of the naive value. This is not a small effect, and like lasing it happens at room temperature.
These effects are strictly due to the indistiguishability of the particles.
If there were any way whatsoever for us to know which particle was which, they would not happen. They do happen, so we know that identical particles aren't just "very similar" but genuinely the same.
Except there are two of them: something unknowable on the far side of the quantum veil is distinguishing them.
We can just never under any circumstances know what it is.