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arXiv · 2208.07388

Bounded perturbations of the Heisenberg commutation relation via dilation theory

Malte Gerhold·Orr Shalit
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Abstract

We extend the notion of dilation distance to strongly continuous one-parameter unitary groups. If the dilation distance between two such groups is finite, then these groups can be represented on the same space in such a way that their generators have the same domain and are in fact a bounded perturbation of one another. This result extends to d-tuples of one-parameter unitary groups. We apply our results to the Weyl canonical commutation relations, and as a special case we recover the result of Haagerup and Rordam that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation are a bounded perturbation of a pair of strongly commuting selfadjoint operators. We also recover Gao's higher-dimensional generalization of Haagerup and Rordam's result, and in typical cases we significantly improve control of the bound when the dimension grows.

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Malte Gerhold, Orr Shalit. 2022-08-15. Bounded perturbations of the Heisenberg commutation relation via dilation theory. https://doi.org/10.1090/proc%2F16456

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