Musings over algebra and topology

One open problem in quantum mechanics reconstruction is the complete classifications of the realization of the algebraic properties. We know quantum mechanics can be formulated over the reals, complex, or quaternionic numbers, but is this all there can be? The problem is solved in the finite dimensional case, but is open in the infinite case. But why is this a hard problem? I think a recent unrelated cute recreational math video shows the heart of the mater. The video attempts to prove that:

1+2+4+8+16+... = -1

This is not the same as 1+2+3+...= -1/12 The rigorous treatment of the Riemann zeta function cannot be covered in only one post (you have to go past the usual Ramanujan tricks), but the current case is much simpler.

Formally, the algebraic manipulations are trivial:

if \(S = 1+q+q^2 + \cdots\) then \(Sq = q+q^2 + q^3 + \cdots = S-1\) and so \(S = 1/(1-q)\) which for q=2 results in -1.

But does it make sense to have this kind of algebraic manipulation? We learn in school that this is allowed only for convergent series which means that \(q\in (-1, 1)\) and 2 is outside the radius of convergence. Case closed, right? Wrong!

We have do dig deeper into what it means that an algebraic manipulation makes sense. The easiest thing to do is to consider the existence of a metric, which is a positive function which assign a number between any two points subject to the usual properties. The most important property of a metric is the triangle inequality and once we have it we can have the usual epsilon-delta arguments.

If you study functional analysis most of the concepts come from considering a metric. Take for example the notions of continuity or that of compactness. A function is continuous if when we approach a point from both ends the value of the function converges. Also a set is compact if it is bounded and closed. One difficulty in learning topology is in generalizing those common sense ideas and using only open sets. For example a function is continuous if and only if it returns open sets into open sets, and a set is compact if from any covering with open sets we can extract a finite covering.

But once we freed ourselves from the usual metric intuition we can see and appreciate things in a different light. For the problem above the key idea is to reorder the numbers to create a different topology and a different metric where the sum does converge. The trivial algebraic manipulation suggests how to do it and one arrives at the p-adic numbers.

So it looks that there is flexibility in messing with ordering and neighborhoods to satisfy algebraic identities. But how much flexibility is allowed by nature in the case of quantum mechanics? 

First, is there a p-adic quantum mechanics? Some publications claim there is, but they are all nonsense. p-adic numbers violate the so-called Archimedean property. While it is conceivable to mathematically imagine universes where probability predictions violate the Archimedean property, a non-Archimedean quantum mechanics must violate the Archimedean property for the Jordan algebra as well and this is where you get in trouble from the physical point of view.

p-adic quantum mechanics is not physical, but can we twist the order of the real numbers in a different way which respects the Archimedean property and yet we get a district topology? To me this looks highly unlikely but I do not have a proof for this impossibility. I did not even began to scratch the surface of topology and algebra in this post, but I hope I succeeded in highlighting the main issue: topology is not as rigid as naive metric epsilon-delta functional analysis proofs from college would made us believe. Categorical arguments nail the algebraic structure of quantum mechanics, but they have nothing to offer on the topological side.