Abstract: Elementary particles are identified as unitary irreducible
representations of the Poincare group, the symmetry group of Minkowski space(time).
This talk is concerned with the availability of viable position operators for
such representations. It turns out that an argument of Wightman together with a theorem
of Mackey yields a simple criterion for the existence of physically viable position operators. Explaining
this beautiful, but little known, story from the very beginnings of infinite-dimensional representation
theory is the main point of the talk. At the end, however, I'll pose some related questions for present day