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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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