Can quanta of unlimited genus exist in theory?
What a fascinating question. I've only seen finite topologies with finite genus mentioned.
This brings to mind the tendency of physicists to extrapolate from finite mathematics to infinite.
Can one answer:
1. What genus does a free electron exhibit?
2. What genus does an atomically bound electron exhibit?
3. What genus does an electron sea exhibit?
4. What genus does an electron in an atomic corral exhibit?
Otherwise, does the concept of differing topologies not necessarily apply to quanta with their probabilistic manifestation?
Hopefully someone who has studied this more will comment. I believe that it is possible to model free electrons, atomic electrons and electron seas without resorting to any topology other than the usual Minkowski, which has genus zero.
The topology arguments have to do with GR, which can be applied to spaces more complicated than Minkowski. What you're asking, I believe, is how does one combine quantum mechanics with general relativity. Of course no one knows how to do that yet.
There is an interesting attempt at applying nontrivial topologies to the problem of representing elementary particles. Mark J. Hadley has written a series of articles on the subject, but you should start with his dissertation:
Wouldn't an inverted infinite quantum well display spacetime properties of topological genus one?
I wouldn't be surprised if Hadley knows Jeffrey Bub, under whom I took a course on quantum interpretation at UMD, himself a student of Bohm.
Quantum theory itself takes place on a fixed topological spatial background. What you seem to be asking is wether topology of space can change dynamically in a particle creation/annihilation processes or when particles form a bound state. As just said, this requires that you go beyond conventional quantum theory and actually engage yourself in a specific approach to quantum gravity which allows for topology change. Now, allowing for dynamical topology change in a non perturbative framework is extremely difficult and is not accomplished yet in the form you might imagine yourself (the only paper where one could treat topology change rigorously was in 2-D dynamical triangulations quantum gravity, but there the ``holes´´ are infinitesimally small and extension to higher dimensions and/or finite holes seems unlikely). However, it is possible to unreavel the statistics such particles should satisfy, this is done in 3-D for topological geons by Sorkin, Dowker, Surya et al.
My opinion about these things is you should first limit yourself to understanding classical/quantum dynamics on the simplest topological background (ie R^4) before you go to such exotic routes (and you will discover that the latter might take you a lifetime!)
And if you or anyone else is interested in discussing any of Hadley's work, you are welcome to do so at the yahoo! group QM_from_GR -- Hadley is one of the active members there ;).
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