  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.
