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De Sitter to Poincaré contraction and relativistic coherent states. (English) Zbl 0706.22018
The paper under review continues the analysis of the relativistic phase space [begun in ibid. 51, No.1, 23-44 (1989; Zbl 0702.22024)] by identifying it with the quotient of the Poincaré group, \(P_+^{\uparrow}(1,1)\), in one space and one time dimensions, by the time translation group, and further analyzing the sections \(\beta\) : \({\mathbb{R}}^ 2\to P_+^{\uparrow}(1,1)\). It is well-known that for two- dimensional space-times, one of the possible relativities is the (anti) de Sitter relativity with the kinematical group \({\mathcal S}{\mathcal O}_ 0(2,1)_{\pm}.\)
In quantum theory, elementary systems are associated to projective unitary, irreducible representations of the kinematical group. In the case of \(P_+^{\uparrow}(1,1)\), they are denoted by P(m), for P(m) giving the description of a system with mass m. The representation P(m) arises from the contraction of a representation \({\mathcal D}_+({\mathcal E}_ 0)\) belonging to the discrete series of the anti de Sitter group \({\mathcal S}{\mathcal O}_ 0(2,1)_+.\)
In Section 2 the authors give a review of properties of \({\mathcal D}_+({\mathcal E}_ 0)\). Further, they demonstrate that the contraction singles out the aforementioned section in the Poincaré group, with a unique left and right invariant measure, and the representation P(m) becomes square integrable over the coset space. Finally, the authors make a construction of a new set of Poincaré coherent states, or more generally, weighted coherent states. The latter have all common properties, resolution of the identity, overcompleteness, reproducing kernels, and the orthogonality relations.
Reviewer: E.Kryachko

MSC:
22E70 Applications of Lie groups to the sciences; explicit representations
81R30 Coherent states
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
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