Electron spin that takes two rotations to complete one cycle

Some people mistakenly say "I've changed 360 degrees" when they mean "I've changed 180 degrees," and get a laugh when someone points out, "That means you're back to where you started!" However, in physics, there are things that do not return to their original state even after a 360-degree rotation.

It is known that for elementary particles called fermions, such as electrons, protons, and neutrons, rotating them 360 degrees around any spatial axis results in a negative sign being applied to their wave function or field quantity. This property is described by the mathematical concept of a spinor. This property of electrons is linked to Fermi statistics, where the wave function becomes completely antisymmetric when many electrons are gathered, and it is one of the sources that give rise to various physical properties.

However, even among those who know physics, it is sometimes misunderstood that the negative sign that appears after a 360-degree rotation can never be observed. This is because they mistakenly believe it is merely a phase factor applied to the entire wave function. Yet, when using neutrons, which are fermions like electrons, that negative sign can actually be observed physically.

By applying a magnetic field to neutrons, one can experimentally rotate them a full 360 degrees. The wave function and field quantity of the neutron, being a fermion, do not return to their original state, but instead acquire a negative sign.

Therefore, we can conduct the following experiment. We apply a magnetic field only to the path of one of the neutrons split in a double-slit experiment, rotating the neutron passing through it by 360 degrees. The interference fringes observed by detectors A and B below will then shift from the interference fringes observed without the magnetic field. In the world of quantum mechanics, even if you rotate something 360 degrees, it does not return to its original state.

However, if you rotate the neutron one more time for a total of 720 degrees, the interference fringes return to their original state. It is a case of minus 1 times minus 1 equals plus 1. This property was also touched upon in Section 12.2.1 of Chapter 12 in the textbook below.

Furthermore, I believe it should be widely known that in hidden variable theories such as Bohmian mechanics or Nelsonian mechanics, the spinor nature of this spin rotation cannot be reconciled with reality. Seeking a classical realistic picture for spin is inherently a lost cause.