de Sitter group as a symmetry for optical decoherence

Abstract

Stokes parameters form a Minkowskian 4-vector under various optical transformations. As a consequence, the resulting two-by-two density matrix constitutes a representation of the Lorentz group. The associated Poincaré sphere is a geometric representation of the Lorentz group. Since the Lorentz group preserves the determinant of the density matrix, it cannot accommodate the decoherence process through the decaying off-diagonal elements of the density matrix, which yields to an increase in the value of the determinant. It is noted that the O(3, 2) de Sitter group contains two Lorentz subgroups. The change in the determinant in one Lorentz group can be compensated by the other. It is thus possible to describe the decoherence process as a symmetry transformation in the O(3, 2) space. It is shown also that these two coupled Lorentz groups can serve as a concrete example of Feynman's rest of the universe.

Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere

“…Here we are interested in the transition from Equation (6) to Equation (8), via Equation (10). For convenience, we start from µ greater than ω with µ ′ given by Equation (9).…”

“…Its underlying language is the two-by-two coherency matrix. This coherency matrix contains the symmetry of SL (2, c) isomorphic to the the Lorentz group applicable to three space-like and one time-like dimensions [4,6,7].…”

“…The Poincaré sphere for polarization optics contains the SL(2, c) symmetry isomorphic to the four-dimensional Lorentz group applicable to the Minkowski space [3][4][5][6][7]. Thus, the Poincaré sphere extends Wigner's picture into the three space-like and one time-like coordinates.…”

Symmetries Shared by the Poincaré Group and the Poincaré Sphere

“…After making this transition, we can come back to the original frame to obtain the four momentum matrix of Equation (23).…”

“…It is of course possible to construct a larger group in which this variable plays a role in a group transformation [23], but in this paper, we are more interested in its role in a particle gaining a mass. With this point in mind, let us diagonalize the coherency matrix of Equation (69).…”