Spinors

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°.

As the circle rotates, the attached arrow flips after 360° and flips back after 720°

In Dirac notation

$$R_{2\pi}|\psi \rangle = -|\psi \rangle = e^{i \pi}|\psi \rangle,$$

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.)

About John F. Lindner

John F. Lindner was born in Sleepy Hollow, New York, and educated at the University of Vermont and Caltech. He is an emeritus professor of physics and astronomy at The College of Wooster and a visiting professor at North Carolina State University. He has enjoyed multiple yearlong sabbaticals at Georgia Tech, University of Portland, University of Hawai'i, and NCSU. His research interests include nonlinear dynamics, celestial mechanics, and neural networks.
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