  It often goes unnoticed that, even for a finite number of degrees of freedom,
the canonical commutation relations have many inequivalent irreducible unitary
representations; the free particle and a particle in a box provide examples
that are both simple and well-known. The representations are unitarily
inequivalent because the spectra of the position and momentum operators are
different, and spectra are invariant under unitary transformations. The
existence of these representations can have consequences that run from the
merely unexpected to the barely conceivable. To start with, states of a single
particle that belong to inequivalent representations will always be mutually
orthogonal; they will never interfere with each other. This property, called
superseparability elsewhere, is well-defined mathematically, but has not yet
been observed. This article suggests two single-particle interference
experiments that may reveal its existence. The existence of inequivalent
irreducibile representations may be traced to the existence of different
self-adjoint extensions of symmetric operators on infinite-dimensional Hilbert
spaces. Analysis of the underlying mathematics reveals that some of these
extensions can be interpreted in terms of topological, geometrical and physical
quantities that can be controlled in the laboratory. The tests suggested are
based on these interpretations. In conclusion, it is pointed out that
mathematically rigorous many-worlds interpretations of quantum mechanics may be
possible in a framework that admits superseparability.
