On the integrabillty of certain symmetric representations of the Lie algebra of SO₀(4,1)

A proof of the existence of an essentially self-adjoint extension of a symmetric SO₀(4,1) Nelson operator, which is constructed out of the generators of a positive mass, arbitrary spin unitary irreducible representation of the Poincaré group, is presented. Our analysis of SO₀(4,1) and its Lie algebra provides us with an example of an observation of Harish-Chandra: There exist subspaces of the space of differentiable vectors of a representation of a noncompact group which are invariant under the Lie algebra, but the closures of the subspaces are not invariant under the group. The chief results of this paper should hold true for SO₀(n,1). In particular, we should have a realization of an arbitrary principal series irreducible unitary representation of SO₀(n,1) on the direct sum of two identical unitary irreducible representation spaces of the motion group in an n-dimensional Minkowski space, which has one timelike dimension.