Abstract
The group SO0(2, 2) is the finite-dimensional conformal group of the 1 + 1-dimensional Minkowski spacetimeM. We identify the indecomposable representation of SO0(2, 2) ≅ SO0(2, 1) × SO0(2, 1) that acts on the one-particle physical space of the massless scalar field onM. We accomplish this by realizing this space as a space ℋ 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 ℋ, preserving the inner product. On right-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.
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De Bievre, S., Renaud, J. The indecomposable representation of SO0(2, 2) on the one-particle space of the massless field in 1 + 1 dimension. Lett Math Phys 34, 385–393 (1995). https://doi.org/10.1007/BF00750070
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DOI: https://doi.org/10.1007/BF00750070