• Paul R. Halmos
Part of the Graduate Texts in Mathematics book series (GTM, volume 19)


A mathematical formulation of the famous Heisenberg uncertainty principle is that a certain pair of linear transformations P and Q satisfies, after suitable normalizations, the equation PQ - QP = 1. It is easy enough to produce a concrete example of this behavior; consider L2(-∞, +∞) and let P and Q be the differentiation transformation and the position transformation, respectively (that is, (Pf)(x) = f′(x) and (Qf)(x) = xf(x)). These are not bounded linear transformations, of course, their domains are far from being the whole space, and they misbehave in many other ways. Can this misbehavior be avoided?


Hilbert Space Compact Operator Commutator Subgroup Abnormal Operator Normed Algebra 
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© Springer-Verlag New York Inc. 1982

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  • Paul R. Halmos

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