The Massive Coherent States of the Poincaré Group and Fuzzy Quantization of Space-time

In obtaining a fuzzy quantization of space-time, one works with the full Poincaré group P(1, 3) = P, obtains a phase space Γ for a massive particle, and then takes L2(Γ). In order to obtain a purely quantum mechanical approach to measurement theory, one picks a wave function η for a particle in an irreducible representation space H of P, and then intertwines H with L2(Γ). Every state has averages of momentum, position, and spin, which in turn produces a point in phase space. So, take the coherent state {U(g)η | g ∈ P} against which one measures every state ψ by computing the transition probabilities against the coherent state. That will be the quantity that one considers here. These coherent states result in a fuzzy quantization of space-time; i.e., this give the probability of measuring "distance" in the sense of the Poincaré group.