Symmetries of the pseudo-diffusion equation and related topics

We show in details how to determine and identify the algebra g = {A(i)} of the infinitesimal symmetry operators of the following pseudo-diffusion equation (PSDE) LQ ae<inverted exclamation> Q(x, p, t) = 0. This equation describes the behavior of the Q functions in the (x, p) phase space as a function of a squeeze parameter y, where t = e (2y). We illustrate how G (i)(lambda) ae<inverted exclamation> exp[lambda A (i)] can be used to obtain interesting solutions. We show that one of the symmetry generators, A (4), acts in the (x, p) plane like the Lorentz boost in (x, t) plane. We construct the Anti-de-Sitter algebra so(3, 2) from quadratic products of 4 of the A (i), which makes it the invariance algebra of the PSDE. We also discuss the unusual contraction of so(3, 1) to so(1, 1)aOE h(2). We show that the spherical Bessel functions I (0)(z) and K (0)(z) yield solutions of the PSDE, where z is scaling and "twist" invariant.