# Is the “particle exchange” argument for fermions vs. bosons in introductory quantum textbooks legitimate?

In most of my introductory quantum mechanics textbooks the "particle exchange" operator is used to establish the distinction between fermions and bosons. The argument is made that if identical particles change places once, then the system is physically the same so the wavefunction may only change by a phase. If they change places again then the system is back exactly where it was and the wavefunction must be exactly the same. Thus, particle exchange changes the wavefunction by either +1 (we call these bosons) or -1 (and we call these fermions). (Because (+1)^2 and (-1)^2 both equal 1.)

I have trouble with this argument because it seems to me that $$identical$$ particles changing places once is no different than changing places $$n$$ times.

Then I found this paper discussing the possibility of $$anyons$$ in 2D.
https://www.ifi.unicamp.br/~cabrera/teaching/referencia.pdf In this paper the fermion/boson argument is made by using "continuity of the wavefunction" arguments to find that two exchanges must return the wavefunction to its original state. I find this argument convincing - but it only applies in 3D. The paper goes on to say that any phase change is possible after two exchanges in 2D.

However, there is nothing in the typical "intro to QM" argument that shouldn't apply to 2D just as well as 3D. Thus, I am worried that the "two exchanges brings us back to the original wavefunction" argument is, in fact, invalid. Am I right?

• Indeed, that textbook argument is really misleading and leads to a lot of confusion. I said a bit about it in an extended rant here. – knzhou Mar 5 at 16:53
• thanks @knzhou - that rant also leads to many other relevant discussions, it is nice to know that I am not the only one confused by this - and it is frustratingly difficult to use keywords to search for this topic ... – Paul Young Mar 5 at 17:04
• The argument is incorrect because it is already after one exchange that the states are supposed to be the same. The sign is a projective, as they always pop up for fermions. This is also what i understand of the paper... are you familiar with the subtleties of "states are rays" instead of "states are vectors"? – Lorenz Mayer Mar 5 at 17:34
• no, I am unfamiliar with "states are rays" – Paul Young Mar 5 at 17:45