# Can you get Born probabilities that are irrational from an experiment?

Is it possible to design an experiment getting irrational number predictions (in practice, you can't actually measure infinite numbers of decimal points, obviously) for Born probabilities?

If you could, it would mean that any simple theory of observers in a finite number of multiworlds couldn't account for them (except perhaps in a limit).

A simple search didn't show up any webpages directly answering the question.

It is well-known (but surprisingly hard to find a good reference for) that the probability of a photon being transmitted through some device is the fraction of the incident power of the classically transmitted through it. (See, for example, here).

If you first create a linearly polarized wave/photon beam by sending it through a linear polarizer, and then position a second linear polarizer at angle $\theta$ behind it, the fraction of power transmitted will be $\cos^2(\theta)$.

Now, for rational $\theta$, the sine and cosine are necessarily irrational. But this is in radians, we usually adjust rational degress, i.e. fractions of $\pi$. Shockingly, it can be shown that rational degress always produce algebraic values of the cosine, but fortunately not all of these are of degree two or simple, so we can (see linked paper), for example choose $\theta = 36°$ and get

$$\cos^2(36°) = \frac{1}{16}(1 + \sqrt{5})^2 = \frac{1}{16}(1 + 2\sqrt{5} + 5) = \frac{3 + \sqrt{5}}{8}$$

which is certainly no rational prediction.

Note that we produce irrational predictions for the probability for every rational angle that is not $\theta \in \{45°,30°,135°,150°\}$.

Probably the same can be done for rational values for the angle in radians, but then we would have to adjust the angle not by the normal degree scale, but e.g. with a strip of known rational length laid around the circumference of the polarizer or somesuch.