Discrete frames of Poincaré coherent states in 1+3 dimensions.
Let \(G\) be a locally compact group and \(H\) a closed subgroup such that the quotient manifold \(X=G/H\) admits a \(G\)-invariant measure \(\nu\). Let \(U\) be a continuous unitary irreducible representation of \(G\) in a separable Hilbert space \(\mathfrak{H}\) and \(\sigma: X\rightarrow G\) be a Borel section. The representation \(U\) is called square integrable mod \((H,\sigma )\), if there exists a nonzero vector \(\eta\in \mathfrak{H}\) such that the operator \(\int_X| \eta_{\sigma (x)}><\eta_{\sigma (x)}| d\nu (x)\) converges (weakly) to a bounded, positive, invertible operator \(A_{\sigma}\), where \(\eta_g:=U(g)\eta\), \(g\in G\) [cf. \textit{S.T. Ali; J-P Antoine; J-P Gazeau}, Coherent states, wavelets and their generalizations, Springer, New York (2001; Zbl 1064.81069)]. If \(A^{-1}_{\sigma}\) is bounded, then the family of coherent states \(\mathcal{F}_{\sigma}= \{ \eta_{\sigma (x)}, x\in X\}\) is called a frame of rank one. Similarly, one introduces the rank-\(n\) frames if one considers instead of one vector \(\eta\in\mathfrak{H}\), \(n\)-vectors \(\eta_j\), \(j=1,\dots ,n\), and one considers \(A_{\sigma}:=\Sigma_{j=1}^n\int_X| \eta_{\sigma (x)}^j> <\eta_{\sigma (x)}^j| d\nu (x)\). Starting with the rank-\(n\) frames one can construct coherent states on homogeneous spaces more general than the standard constructions [\textit{R. Ghilmore}, Geometry of symmetrized states, Ann. Phys. 74, 391--463 (1972); \textit{A. M. Perelomov}, Comm. Math. Phys. 272--236 (1972; Zbl 0243.22016)]. Next step is to consider instead of the continuous rank-\(n\) frame the discrete subset \(\Lambda\) such that the vectors \(\{\eta^j_{\sigma (x_m)}\}, j=1, \dots ,n, x_m\in\Lambda\) constitute a discrete frame. In the paper under review the authors give a positive answer to the discretization problem in the case of the Poincaré group in \(1+3\) dimensions [cf. \textit{S. T. Ali, J.-P. Antoine; J.-P. Gazeau}, Ann. Inst. Henri. Poincaré, Théor. 55, 829--855 (1991; Zbl 0752.22002); \textit{S.T. Ali, J-P. Antoine; J-P. Gazeau}, Ann. Inst. Henri Poincaré, Théor. 66, 857--890 (1991; Zbl 0752.22003)], while the tightness tends to be lost. The construction is done for two types of admissible vectors with compact support in a parallelipipedic box or a spherical shell. The continuous formalism is recalled in Section 2, Section 3 deals with the case of the \(1+1\)-dimensions, while Section 4 is quite technical. In [\textit{A. L. Hohouéto; K. Thirulogasanthar; S. T. Ali; J.-P. Antoine}, Coherent state lattices and square integrability of representations, J. Phys. A, Math. Gen. 36, No. 47, 11817--11835 (2003)] it was proved the general result that when the group \(G\) is a semidirect product of a vector space and a semisimple connected group the coherent state transform is discretizable and one can obtain discrete frames.
scientific article; zbMATH DE number 1921127