Konwersatorium im. Leopolda Infelda

Sea stars orient themselves in the plane with five arms. The latter, when suitably weighted, form an overcomplete frame which solves the identity, and so allow the echinoderm to have a non-commutative point of view (with 2_2 matrices) of any function on a set of 5 elements. Hence, sea stars arms proceed with a frame quantization of C5.

Frame quantization or Coherent state(s) quantization are generic phrases for naming a certain point of view in analyzing functions on a set X, e.g. the fivefold orientations for sea stars, equipped with a measure. The approach matches what physicists understand by quantization when the observed measure space X is the phase space of a mechanical system. It matches also well established approaches by signal analysts, like wavelet analysis. The set X can be finite, countably infinite, or uncountably infinite. The approach is generically simple, of Hilbertian essence, and always the same: one builds a family C of vectors jxi (the frame vectors or "coherent states) in a Hilbert space H , which are labelled by elements of X and which resolve the unity operator inH . This is the departure point for analyzing the original set and functions living on it from the point of view of the frame (in its true sense) C.

We end in general with a non-commutative algebra of operators inH. Changing the frame family C produces another quantization, another point of view, possibly equivalent to the previous one, possibly not. Starting from the sea star orientations, various examples of such explorations will be presented.

References
[1] J.P. Gazeau 2009 Coherent States in Quantum Physics, Wiley-VCH