Quantization of Time of Arrival Functions via Harmonic Analysis on the Euclidean Motion Group on the Plane

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Conference Proceeding

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This work presents a generalized Weyl quantization for the time of arrival function for a system consisting of a free particle confined to move on a line segment. The generalized Weyl quantization relies crucially on the phase space of the classical system being quantized and on the choice of the group of symmetries of this space, which is non-unique. The Hilbert space of the corresponding quantum system is the representation Hilbert space of the unitary irreducible representations of the symmetry group. For the system considered here, the phase space of our time of arrival function is the cylinder S1×R and the symmetry group E(2) is the Euclidean motion group of the plane. We consider two coordinate systems leading to two distinct quantization.