Given a quantum particle on a line, its momentum and position are described by a pair of Hermitean operators (p, q) which satisfy the canonical commuta-tion relation. There is a third observable r, say, contained in the Heisenberg algebra generated by p and q, which simultaneously satisfies canonical com-mutation relations with both position and momentum. The Heisenberg triple of the observables (p, q, r) is not only unique (up to unitary equivalences) but also maximal (no four equi-commutant observables exist). Being invariant under a cyclic permutation, the triple (p, q, r) endows the Heisenberg algebra with an interesting threefold, largely unexplored symmetry. I will briefly sketch why these considerations are important in the context of so-called mutually unbiased bases, and that they suggest to rethink Heisenberg's uncertainty relation by first generalizing it to an expression involving the pro-duct of three variances, and then to even more general functions thereof.
Triples of Pairwise Canonical Observables and generalized uncertainty relations
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Nom de l'orateur
Stefan Weigert
Etablissement de l'orateur
Department of Mathematics, University of York/UK
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