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Fri, Apr 09, 2021
Graduate Student Seminar Covariant Integral Quantization with the Poincaré group in (1+1)-Space-time Dimensions Haridas K. Das Contact mishty.ray@okstate.edu for the zoom link. Abstract: This talk will illustrate a general formalism, named covariant integral quantization for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The example that we examine here concerns the Poincaré group in (1+1)-space-time dimensions, denoted $Ƥ_+^1 (1,1)$. The cotangent bundle of the quotient of $Ƥ_+^1 (1,1)$ by the affine group has the natural structure of a physical phase space. We do an integral quantization of functions on this phase space, using coherent states coming from a certain representation of $Ƥ_+^1 (1,1)$. The representation in question corresponds to the “zero-mass'' or “light-cone'' situation, which when restricted to the affine subgroup gives the unique unitary irreducible representation of that group. This representation is also the one naturally associated to the above mentioned coadjoint orbit. The coherent states are labelled by points of the affine group and are obtained using the action of that group on a specially chosen vector in the Hilbert space of the representation. They satisfy a resolution of the identity, which can be computed using either the left or the right Haar measure of the affine group. The covariant integral quantization is implemented using both choices and we obtain a relationship between the two quantized operators corresponding to the same phase space function
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