From the Rotation Group to the Poincaré Group

S. Twareque Ali, Eduard Prugovečki

Abel Camacho, A. Camacho-Galván

Ademir Eugenio de Santana, A. Matos Neto, J. D. M. Vianna +1 more

Shinichi Saito

Polarisation is described by an SU(2) state due to the coherence of photons emitted from a ubiquitous laser source. We propose a polarisation modulator, called as a Poincaré rotator, which enables dynamic control over a polarisation state by executing an arbitrary rotation on the Poincaré sphere. The Poincaré rotator works as an arbitrary operator in the SU(2) Lie group, combining two U(1) operations to change the phase and the amplitude of the state, respectively. By modulating both the phase and amplitude, we have achieved distinguishable 4×4=16, 8×8=64, and 10×10=100 states on the Poincaré sphere. As a locus of the polarisation states on the sphere, we have successfully drawn the molecular structure of Buckminsterfullerene (C60) and the coastline of the Earth.

Shinzo Nakai, Muneo Sasaki

Amétépé Lionel Hohouéto

We show that there exists a V -admissible-type subspace in the carrier space of the Wigner representation of the Poincaré group in 1+3 dimensions. This subspace is built up from an orthonormal set of Sturmian-type functions which verify naturally the assumption of rotational invariance under which relativistic coherent states frames were previously obtained. Then, we propose an extension of the concept of V -admissibility for the class of semi-direct product groups, which takes into account the relativity groups which were not covered by the existing approaches. In the light of this extension, the square-integrability modulo an appropriate subgroup of the Wigner representation of the group of Poincaré is revisited and some physical implications are discussed.