Topology of the Quantum Dimension – Twists

In order to better understand the difference between bosons and fermions, we will now take a look at the complex phase. We will begin with the simplest state, namely j=0. We associate a rotating wheel with each angle, with constant radius – yet, what about the phase?

The phase can take any value, without the probability being changed. We are modelling these different phases for each solid angle using a paper strip. The direction of the arrows on the paper strip indicates the phase at the corresponding angle.

Cut open, the constant phase in the paper strip model simply looks like this. The simplest change of the topology of the phase is one twist. We can twist either clockwise or anti-clockwise. The superposition of the phases of the clockwise and anti-clockwise twist correspond to the sum of e^(+/- i ϕ/2), that is, cos(ϕ/2).

What happens if we first cut open the paper strip of the state j=0, then twist it, then glue it together again? Topologically speaking, a Möbius strip thus arises. The Möbius strip has only one surface. Following a rotation through 360°, we arrive at the reverse side, and only after 720°, do you get back to where you started!

The superposition of the clockwise and anti-clockwise Möbius strip again leads to cos(ϕ/2), where the angle phi runs from 0° to 720°. In such a way, the double-valuedness of the spin state with a single nodal point emerges.

If we repeat the “cutting open”, “twisting” and “gluing together” operations, all other spin states can be constructed starting with the state j=0 with trivial topology. The fermions with an odd number of twists are all Möbius strips with only one surface – in contrast to the bosons, which have an even number of twists. This is the main difference of the phase of the quantum states from bosons and fermions: only the fermions completely fill up the 720° space; the bosons are already satisfied with 360°.