We show two independent theoretical methods which lead to generate eigen values of a composite two-particles system associated with the non-locality of EPR effect which was experimentally proved by Aspect and others.One method is to use the quantum groups (Woronowicz and Podles)and the other is to use the two-dimensional kappa-Poincare groups.The representation of Hilbert spaces of these groups has eigen functions and eigen values with real numbers on one dimensional spaces.The range of eigen values is real number,so that the composite two particles system in quantum groups can macroscopicly exists and it has a nonlocal effect.
Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order
Nκ−1 N κ - 1 , for some
κ>0 κ > 0 . For the range
κ∈[0,431) κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for
κ∈[0,201) κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).
Complete determination of the polarization state of light requires at least four distinct projective measurements of the associated Stokes vector. Stability of state reconstruction, however, hinges on the condition number κ of the corresponding instrument matrix. Optimization of redundant measurement frames with an arbitrary number of analysis states, m, is considered in this Letter in the sense of minimization of κ. The minimum achievable κ is analytically found and shown to be independent of m, except for m=5 where this minimum is unachievable. Distribution of the optimal analysis states over the Poincaré sphere is found to be described by spherical 2 designs, including the Platonic solids as special cases. Higher order polarization properties also play a key role in nonlinear, stochastic, and quantum processes. Optimal measurement schemes for nonlinear measurands of degree t are hence also considered and found to correspond to spherical 2t designs, thereby constituting a generalization of the concept of mutually unbiased bases.