**Fermions** like **electrons**, **protons**, and **neutrons** inhabit a 720° world: 360° rotations negate their quantum states, but 720° rotations restore them.

A simple macroscopic model of such **spinors** is an arrow translating on a **Möbius strip**: as the center circle rotates, the attached arrow flips after 360° but flips back after 720°.

In **Dirac notation**

but

R_{4\pi}|\psi \rangle = +|\psi \rangle= e^{i 2\pi}|\psi \rangle,where the **ket** |\psi\rangle is the state, the exponentials are **phase factors, **and their arguments are **phase shifts**.

To detect a relative phase shift, send a neutron via two paths, rotate it along one path with a magnetic field (coupled to its magnetic dipole moment), and observe **destructive** **interference** for 360° rotations and **constructive** **interference** for 720° rotations. (The experiment is harder with charged electrons and protons, whose translation is deflected by the magnetic field.)

Thanks, Mark! I enjoy reading your posts as well.