## Mobius strip representation of spin 1/2

Physics

The elementary particles known as fermions (electron, muon, tau and u, d, c, s, b, t quarks ) and their antiparticles are the constituents of matter. A fermion will impart $\pm \hbar /2$ units of spin angular momentum when it interacts with other particles. It does not matter what the energy of the fermion is; a fermion always imparts the same amount of spin angular momentum when it interacts. The absolute value of this constant spin angular momentum in units of  $\hbar$ is used as a short hand notation to indicate the fermion nature. We say that fermions are spin=1/2 particles.

The elementary particles known as gauge bosons (photons, gluons and W, Z) are the field quanta  that facilitate interaction among fermions.  A gauge boson also imparts a constant spin angular momentum independent of its energy. A gauge boson always imparts $\pm \hbar$ units of spin angular momentum. We say that gauge bosons are spin=1 particles.

The $\hbar$ is not a variable. The $\hbar$ is a constant of nature. It is a constant number.

The spin=1/2 property of fermions cannot be explained by classical physics. In Quantum Mechanics there is no explanation of the spin=1/2 behavior either but Quantum Mechanics provides a mathematical treatment of the spin=1/2 particles and makes probabilistic predictions based on this treatment.

The quantum mechanical wavefunction of a spin=1/2 particle is represented by a mathematical object known as the spinor which requires 2 full rotations to come back to its original state.

Mathematics of spinors

My favorite spinor tutorial is Introduction to Spinors by Andrew Steane of Oxford University.

You can also take a look at the Wikipedia article titled Spinor.

Geometrical representation of spinor

In my opinion, the most interesting and also the most relevant geometrical representation of spinor is the Mobius strip. Imagine yourself traveling on the surface of a Mobius strip. After one full rotation – around the imaginary axis that passes through the center of the Mobius strip – your orientation is reversed. You have to go around one more time to come back to your original orientation. Spinors have the same property. That’s why a spinor can be represented by a Mobius strip.

There is an interesting web page titled Dirac’s Belt at MathPages which talks about the Mobius strip representation of spinors.

If you search the arXiv you will find papers examining various aspects of the correspondence between the spin 1/2 and the Mobius strip. Obviously, this has been on physicists’ mind for a while now.