Topology of the Quantum Dimension – Complexified Function

What does the complex extension of a real function f(x) look like, for example for the function f(x) defined as 1/(1+x^2)? First, we extend the real number x by the imaginary component i y.

Each real function f(x) can be extended to the complex plane, with x being replaced by x+ i y, as can be seen in this example. From f(x+i y), firstly, the real part emerges on the complex plane, which we are visualising here as a mountain landscape coloured in blue. An astonishingly rich mathematical structure arises. Two singularities can clearly be identified, at plus and minus i.

Secondly, we display the imaginary part of f(x+i y) on the complex plane. Next, we are showing the real and imaginary parts together. There is an interesting connection between the real part and the imaginary part of f: consider the altitude lines of the mountain landscape here, to start with, for the real part. Near the singularities at plus/minus i the altitude lines become increasingly dense.

… and this take shows the altitude lines of the mountain landscape for the imaginary part. It can be seen that the altitude lines of the real and imaginary parts always intersect at right angles.

This also means, however, that the information of the real part is contained in the imaginary part – and vice versa! And the entire complex structure is already determined by the function f(x) on the real axis – even if the deeper structure is somewhat concealed here, in particular that of the singularities.

This complex extension of the function f(x) helps us to understand this deeper structure much better. What about the quantum dimension? What is the situation there? Is everything specified by the real probabilities here as well?

No, not at all! Here the route is reversed, from the complex dimension into the real world. The invisible phase shows that there is no one-to-one connection between the complex wave function and real probabilities. We will be showing you the dramatic consequences of this fact in the next few slides.