Poisson commutator-anticommutator brackets for ray tracing and longitudinal imaging via geometric algebra

We use the vector wedge product in geometric algebra to show that Poisson commutator brackets measure preservation of phase space areas. We also use the vector dot product to define the Poisson anticommutator bracket that measures the preservation of phase space angles. We apply these brackets to the paraxial meridional complex height-angle ray vectors that transform via a 2x2 matrix, and we show that this transformation preserves areas but not angles in phase space. The Poisson brackets here are expressed in terms of the coefficients of the ABCD matrix. We also apply these brackets to the distance-height ray vectors measured from the input and output sides of the optical system. We show that these vectors obey a partial Moebius transformation, and that this transformation preserves neither areas nor angles. The Poisson brackets here are expressed in terms of the transverse and longitudinal magnifications.