  We argue that the notion of identical particles is no longer well defined in
quantum systems governed by non-commutative deformations of space-time
symmetries. Such models are characterized by four-momentum space given by a
non-abelian Lie group. Our analysis is based on the observation that, for
states containing more than one particle, only the total momentum of the system
is a well defined quantum number. Such total momentum is obtained from the
non-abelian composition of the particles individual momenta which are no longer
uniquely defined. The main upshot of our analysis is that all previous attempts
to construct Fock spaces for these models rested on wrong assumptions and
indeed have been unsuccessful. We also show how the natural braiding of
momentum quantum numbers which characterizes the exchange of factors in the
tensor product of states is covariant under relativistic transformations thus
solving a long standing problem in the field.
