Quantum physics and the Gromov-Hausdorff distance: more than discrete or continuous

In face of the question “is reality digital or analog?”, my first reaction was “how can one be so unimaginative?”. Actually, I know that there is at least another possibility, thanks to some mathematical results, like Pansu’ Rademacher theorem for Carnot groups and to the Gromov-Hausdorff distance.
I saw this question announced as the subject of a FQXI essay contest and I decided to give a try to explain a bit what I have in mind. It was interesting to do this. I think it was also a bit disconcerting, because it feels a lot like social interacting during the participation in a MMORPG.

Anyway, my contribution was this (see arXiv link also)

More than discrete or continous: a bird’s view

The idea is that any experiment is like making a map of a part of the reality (say at the quantum scale, for example) into another (the results, as we read them in the lab). Estimating distances are proposed as a good analogy of a measurement. Or, it may be possible that there are a priori limits of the precision (accuracy, etc) of the map (any map obtained by a exploiting a physical process) due to the fact that at different scales the space is different, like metric spaces are.

Up to some normalization, the Planck length may be just (proportional to) the Gromov-Hausdorff distance between the (unit ball in ) the Heisenberg group (as a Carnot group) and the euclidean phase space of the same system.

Reality (whatever that means, say the “real” phase space of a system) may be discrete, continuous, both or none, but at different scales, up to a (small) error, the Heisenberg group or the euclidean space, respectively, may be close to it (to the small part of reality explored). Then, as a consequence of Pansu’ Rademacher theorem, it follows that one cannot make a perfect (read “bi-lipschitz”) map of a part of reality into the other, but instead, there is a lower limit of the accuracy of any such map, quantitatively proportional with a Gromov-Hausdorff distance.

In order to explain this in more detail, I took the view that anyway the differential calculus (which is the mathematical base of physics) is an abstraction of the activity of map-making, or chorography.

The “bird’s view” is a play with a quote from Plato, I let you discover it, at the end of the paper.


5 thoughts on “Quantum physics and the Gromov-Hausdorff distance: more than discrete or continuous”

    1. Thank you for the link! But where is the resemblance? In the Finkelstein paper there are still used Hilbert spaces and riemannian metrics. On one side one may always replace (complex) Hilbert spaces by Heisenberg groups (by doing a semidirect product with a circle) and then unitaries lift to linear (in Carnot groups sense) and volume preserving maps, but what about riemannian metrics? On the other side, the main idea is that of experiments as map-making activities…

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