  An n-particle 3-dimensional Wigner quantum oscillator model is constructed
explicitly. It is non-canonical in that the usual coordinate and linear
momentum commutation relations are abandoned in favour of Wigner's suggestion
that Hamilton's equations and the Heisenberg equations are identical as
operator equations. The construction is based on the use of Fock states
corresponding to a family of irreducible representations of the Lie
superalgebra sl(1|3n) indexed by an A-superstatistics parameter p. These
representations are typical for p\geq 3n but atypical for p<3n. The branching
rules for the restriction from sl(1|3n) to gl(1) \oplus so(3) \oplus sl(n) are
used to enumerate energy and angular momentum eigenstates. These are
constructed explicitly and tabulated for n\leq 2. It is shown that measurements
of the coordinates of the individual particles gives rise to a set of discrete
values defining nests in the 3-dimensional configuration space. The fact that
the underlying geometry is non-commutative is shown to have a significant
impact on measurements of particle separation. In the atypical case, exclusion
phenomena are identified that are entirely due to the effect of
A-superstatistics. The energy spectrum and associated degeneracies are
calculated for an infinite-dimensional realisation of the Wigner quantum
oscillator model obtained by summing over all p. The results are compared with
those applying to the analogous canonical quantum oscillator.
