Symmetries of the pseudo-diffusion equation and related topics

Abstract

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 ≡ \(\left[ {\frac{\partial }{{\partial t}} - \frac{1}{4}\left( {\frac{{{\partial ^2}}}{{\partial {x^2}}} - \frac{1}{{{t^2}}}\frac{{{\partial ^2}}}{{\partial {p^2}}}} \right)} \right]\)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(λ) ≡ exp[λA_i] can be used to obtain interesting solutions. We show that one of the symmetry generators, A₄, 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)∌ h². We show that the spherical Bessel functions I₀(z) and K₀(z) yield solutions of the PSDE, where z is scaling and “twist” invariant.