An orthogonality theorem is obtained for square integrable representations on homogeneous spaces of a locally compact group. From this, a series of lemmas is derived showing the informational completeness of natural covariant localization operators, as well as of the generalized Wigner distributions (matrix elements of the group). Some of these results give explicit reconstruction formulas for the quantum state from its expectation values against these families of operators. The results are applied to special (phase space) representations of the Heisenberg, affine, and Galilei groups.
We obtain phase space representations of the Poincaré group for zero mass particles of all helicities, including photons. A natural quantization scheme for massless particles arises, and a covariant phase space localization operator is found.
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