Topology of the Quantum Dimension – Spin and Polarisation

Viewed from a topological perspective, there is an interesting connection between the spin of electrons and the polarisation of photons. Let us first of all compare the experiments: the Stern-Gerlach experiment measuring the spin of electrons as shown in U2-7, and the experiment measuring the polarisation of photons as shown in U1-8.

Let’s start with the spin: if the spin is aligned parallel to the magnetic field, the probability for “spin up” is 100%. The probability for “spin up” decreases if the angle between the initial spin direction and the magnetic field increases. After a rotation of 180°, the initial spin is aligned anti-parallel to the magnetic field. In this case, the probability for “spin down” is 100%.

Compared to the case of polarisation, there is a small, but important difference: the “horizontal” polarisation only describes an axis, and, unlike the spin, does not have any direction. The horizontal and vertical polarisation axes are thus at an angle of 90° to one another. The probabilities for spin “up” and “down” differ compared to the probabilities for reflection and transmission in the angle by a factor of two.

In U2-9 we have already discussed the corresponding amplitudes in the quantum dimension: in case of the spin, the standing wave is wound up on the circle twice, leading to one node in 360°. In the case of the polarisation, the amplitude is described by a single-valued standing wave with two nodes in 360°. Recall that all amplitudes originate from the 720°-quantum dimension, being unobservable for us until a projection has been made into the 360°-world.

This doubling of the angle is the most important track in our further search for a topological interpretation of particles and interactions, as we will elaborate further in the next few slides.