Topology of the Quantum Dimension – Complex Quantum Dimension

What is the relation between amplitudes in the quantum dimension and observable probabilities? The square of the absolute value of the rotating wheel defines the probability – this is a reduction of the wheel to its radius. The angle – in other words, the phase information – is lost in the process.

An infinite number of different possibilities lead to the same probability – and even this is in fact only a probability, not a certain outcome! What is the conclusion? Digital data can, for instance, simply be copied; but the states in the quantum dimension cannot be copied!

For we do not know which of the infinitely many possibilities we are supposed to copy, and, if we want to know, we once again arrive at the right-hand side, the digital data without any phase information.

If we arrange the amplitudes like a string of pearls, and relate them to solid angles, we obtain a representation for Omega. Omega has many applications – for example, as part of the s-wave of a bound state of an electron in the atom.

If we run once in a circle at a fixed radius – for example, at the Bohr radius – the probability density of the electron is constant. However, the rotating wheels do not need to turn synchronously, as long as the radii remain the same – for each of these representations leads to the same probabilites.

This is a simple example of the so-called gauge principle: the phase of the wave function can be rotated locally to any value, and each of these infinitely many representations lead to the same observable probabilities.