Decoherence and the Poincaré Sphere

The two-by-two density matrix constitutes a representation of the Lorentz group. However, the Lorentz group preserves the determinant of the density matrix and therefore it cannot describe the evolution of the decoherence process. Stokes vectors obtained from the density matrix are known to be a four-vector in the Lorentzian regime. Consequently, within this regime the radius of the associated Poincaré sphere has a constant value. It is noted that the O(3, 2) group contains two Lorentz subgroups. The change in the determinant in one Lorentz subgroup can be compensated by the other. It is thus possible to describe the decoherence process as well as the radius variation of the Poincaré sphere, as a symmetry transformation in the O(3, 2) space. It is shown also that these two coupled Lorentz groups provide a concrete example of Feynman’s rest of the universe. The two-by-two matrix representations of the energy-momentum four-vectors in the two Lorentz subgroups of the O(3, 2) are also related through the sum of their determinants. These matrix representations are mathematically analogous to those of the density matrices. While density matrices provide a suitable description of decoherence, the energy-momentum matrices deal with the change of mass. It is indicated that, the coherency parameter depends on the environment, thus is not a fundamental quantity, while mass is most fundamental.