Micheal Cowling - University of New South Wales di Sydney (Cattedra Fubini) - CERTAINTY ABOUT UNCERTAINTY

Abstract: Heisenberg's Uncertainty Principle is the affirmation that one cannot know exactly the position and the momentum of a quantum particle. This physical principle may be modelled mathematically by saying that a function and its Fourier transform cannot both be localised. We examine two ways in which this mathematical statement may be made precise, one due to Weyl, in which uncertainty is measured as a variance of a probability distribution, and another due to Hardy, in which it is shown that Gaussians have important extremal properties. We then proceed to show that this principle is manifested in other areas of analysis. For example, eigenfunctions of Laplacians on manifolds have very precisely specified spectral properties, and hence cannot be localised on the manifold, and kernel operators cannot be localised close to the diagonal if their spectra are also localised.

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