THE INDECOMPOSABLE REPRESENTATION OF SO0(2,2) ON THE ONE-PARTICLE SPACE OF THE MASSLESS FIELD IN 1+1-DIMENSION

The group SO0(2,2) is the finite-dimensional conformal group of the 1 + 1-dimensional Minkowski spacetime M. We identify the indecomposable representation of SO0(2,2) approximate to SO0(2, 1) x SO0(2, 1) that acts on the one-particle physical space of the massless scalar field on M. We accomplish this by realizing this space as a space K of positive energy distributional solutions to the massless Klein-Gordon equation, on which the Klein-Gordon inner product is well defined and positive semi-definite. We then use the analyticity properties of these solutions in the forward tube to show that SO0(2, 2) acts naturally on K, preserving the inner product. On fight-moving solutions, one copy of SO0(2, 1) acts trivially, whereas the restriction of the representation to the other copy is the unique one-dimensional extension of the first term of the discrete series of representations of SO0(2, 1). Similar results hold for left-moving solutions.