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.

• in theory, you can't actually measure infinite numbers of decimal points neither in practice. – jinawee Aug 10 '14 at 15:51
• "irrational ratio"...um, what? – ACuriousMind Aug 10 '14 at 15:52
• @jinawee But could you get an experiment that in theory produced, say sqrt(1/2), and if the experiment confirmed that to ten decimal points or so, it would be evidence. – ike Aug 10 '14 at 15:53
• @ACuriousMind Some experiment that could go two or more ways on each run, and is expected under QM to produce a probability of at least one outcome that is irrational. And the experiment can be set up only using existing equipment, that is you can't use something that's turned at an irrational angle, because we couldn't do that precisely without starting with irrationals in our equation. Is that clearer? – ike Aug 10 '14 at 15:56
• Related: physics.stackexchange.com/q/64101/2451 and links therein. – Qmechanic Aug 10 '14 at 16:04

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.

• Why is it shocking, by the way? And degress should be degrees. – ike Aug 10 '14 at 17:12
• @ike: I'm always amazed at such algebraic results. It need not be shocking for anyone else ;) – ACuriousMind Aug 10 '14 at 17:16
• There's an algebraic formula that you can use to calculate any rational sine, so it seems obvious to me. Say you have x/y, with x,y integer. You can make a formula for sin(π(x/y+x/y+...) y times) which will result in sin πx and have an algebraic relation to sin πx/y. If sin(π) is algebraic, then any rational degree is, also. I think this can be formulized rather shortly. – ike Aug 10 '14 at 17:20
• Can you give experiments where irrational results were measured to several decimal points? – ike Aug 10 '14 at 17:52
• @ike: I'm afraid I know no concrete experimental results - not because they don't exist (they certainly do), but because I am not particularly interested in them. (I'm also at a loss how to reliably search for that) – ACuriousMind Aug 10 '14 at 18:31