rational or real

Physics as we know it is based on real (or complex) numbers, but it is interesting to ask what (if anything) would change if we would consequently replace real valued variables with rational numbers. After all one could make the case that any measurement can only result in rational numbers and probabilities derived from counting outcomes are rational numbers as well.

Obviously, it would not be an easy change, e.g. one would have to replace differential equations with difference equations (with arbitrary base). One would have to consider if and how it affects e.g. the Lorentz group and one would have to deal with the fact that the rational numbers do not constitute a Hilbert space. In other words, it is not immediately clear that one gains anything using Q instead of its natural completion R.

However, if one thinks that it is obvious that using Q instead of R would only complicate things but not make any real difference, I suggest to read this paper: "We explicitly evaluate the free energy of the random cluster model at its critical point for 0 < q < 4 using an exact result due to Baxter, Temperley and Ashley." The authors find that the free energy of the system depends on whether a certain function of the continuous parameter q is "a rational number, and if it is a rational number whether the denominator is an odd integer".

This is one of the weirdest things I have ever seen in statistical mechanics.

update: RZ (see comments) points out that it is not immediately clear from the paper that the numerical value of the free energy is indeed different for rational numbers, just because the form of the free energy function is not the same.
However the sentence "This implies that the free energy
of the random cluster model, if solved, would also share this property" on p.3 would then be highly misleading.

In any case the quantum kicked rotator, also mentioned by RZ, might be a better example of what I had in mind.


RZ said...

I haven't read the paper but resonances that occur when parameters are commensurate are not such rare things.

wolfgang said...

I guess 'resonance' is not a bad way to visualize what is happening here.
But to get such a behavior from a continuous (coupling) parameter is still weird, at least for me, and it is not such an exotic model either...

RZ said...

I don't know. The quantum kicked rotator and KAM theory both have drastic behaviour changes when parameters change from rational to irrational.

RZ said...

I browsed through the paper. It is not clear to me that the *numerical* values of the free energy would jump when approaching a rational point, only the *form* of the expression changes. Consider the difference between n!=1X2X3..n and
gamma(x+1) for x non integer.

wolfgang said...

The reason I find it weird is that q determines basically the weight of the clustering and my intuition is that the effect should be continuous with continuous changes of q.

>> only the *form* of the expression changes

This is a good point, it seemed to me that the two expressions were quite different but it would make sense to check the numerical values.

wolfgang said...

They write "This implies that the free energy of the random cluster model, if solved, would also share this property" on p.3
which implies a difference also for the numerical value of the free energy as I understood it.

But at the same time it makes clear that they did not check it!

I updated the blog post to reflect this ambiguity.

PS: Thanks for the comments!