Bounded perturbations of the Heisenberg commutation relation via dilation theory
Malte Gerhold, Orr Shalit
TL;DR
The paper extends dilation distance to $d$-tuples of strongly continuous one-parameter unitary groups and proves that finite distance $d_{ m D}(u,v)$ yields equivalent representations on a common space with generators sharing domain and differing by a bounded perturbation. It establishes explicit norm bounds for the proximity of unitary-group realizations and their generators, linking dilation proximity to bounded perturbations of the associated self-adjoint generators. Applying this framework to Weyl canonical commutation relations, the authors recover Haagerup–Rordam's bounded-perturbation result and Gao's higher-dimensional generalization, with dimension-free constants. Using Weyl operators on symmetric Fock space, they derive a bound $ orm{P_k^ fty-Q_k} le frac{5}{ sqrt{2}} orm{ heta- heta'}^{1/2}$ for $ heta$-commuting unitary groups, highlighting dilation theory as a robust route to CCR perturbations and connecting to Stone–von Neumann phenomena.
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.