Heraclitis said that you can not step in the same river twice.
Wolfgang Pauli said that you can not be an identical fermion twice
that is that there can’t be two identical fermions.
Some Heraclitian said that you can’t step in the same river once.
This is clearly silly.
Similarly, I think that an honest application of the basic assumptions of quantum mechanics (as listed by Von Neuman) would imply that there can’t be one fermion.
A Fermion is a particle with spin + or minus hbar/2. For example, electrons are fermions. The word spin suggests that it refers to well spin as in the Earth spins around its axis and elementary particles spin around. Spin has the logical implication that charged particles with spin have magnetic fields. The magnetic properties of atoms are logically implied by the spin of their electrons, the orbital angular momentum of electrons and confusing stuff going on in the nucleus making extremely weak magnetic fields.
The spin of photons makes perfect sense if it is interpreted as an intrinsic angular momentum. I mean you know spinning. The strongest reason to interpret spin as angular momentum is that, if it is so interpreted, total momentum is conserved as particles are created and destroyed.
The Pauli exclusion principal (no two identical fermions) could have been accepted as a mere fact. However, physicists aim to explain results in terms of more fundamental principles. In this case, they note that if there are two identical fermions a and b and one switches a and b, then the wave form describing the system of particles is multiplied by -1 (I think the proof is hard in any case I never made head nor tail of it). Now clearly switching two identical particles changes nothing. That means the wave is equal to minus itself, which means it is zero. That means the probability of their being two identical fermions is zero.
OK back to spin and angular momentum. I argued at length that spin is, as the name suggests, a kind of angular momentum, because quantum mechanics has a clear interpretation of momentum and therefore angular momentum. The momentum operator is the gradient with respect to space of the wave form. Thus angular momentum is the derivative with respect to the angle.
Now what sort of wave form does a particle with angular momentum hbar/2 have ? well the derivative of the wave form with respect to the angle measured from a point is angular momentum around that point divided by h. That means that if one rotates the waveform of a fermion 360 degrees it is mulitplied by e to the power i times Pi which is equal to -1.
As far as I can tell, the only natural quantum mechanical interpretation of spin implies that the wave form of a fermion is multiplied by -1 if it is rotated 360 degrees. Clearly nothing changes if it is rotated 360 degrees. That seems to me to mean that the wave form of a fermion must be equal to zero, that is, the probability that there is a fermion is zero.
So why is the same argument a proof of the true Pauli exclusion principle and not a proof of the false claim that electrons don’t exist ?
One way to avoid the conclusion that quantum mechanics can prove that electons don’t exist is to say that the spin operator can not be interpreted as any known operator operating on a wave form. That is one can conclude that, if the core hypotheses of quantum mechanics as listed by Von Neuman are indeed the core hypotheses of quantum mechanics, then quantum mechanics is false. This is the approach of David Saxon in “Elementary Quantum Mechanics” except he doesn’t note that he is concluding that Von Neuman or quantum mechanics is wrong.
The other way is to just ignore it. The same reasoning leads to a result which is interpreted as proof that something is impossible in one case and which is interpreted as proof of nothing in the other.
I think that physicists pretend that they have reduced their theories to a few core principles when they haven’t. The reasoning from the core principles to correct predictions does not seem to me to follow fixed rules (that would be among the core principles). Instead arguments are accepted because their conclusions are empirically correct.
If I am right, this is nothing horrible. Even if the Pauli exclusion principle is an empirical fact not explained by deaper theories it is very deep, that is a very simple rule which explains a huge amount of data. Still I think one ought to admit which things one has figured out from other discoveries and which one has learned empirically but not explained with deaper theories.
Of course I never got past elementary quantum mechanics, so I’m probably just confused.