Symmetries of the pseudo-diffusion equation, and its unconventional 2-sided kernel

TL;DR

This paper analyzes the symmetry algebra of the pseudo-diffusion equation, constructs transformations to simplify it, and develops a two-sided kernel, revealing its Lie algebraic structure and physical implications.

Contribution

It identifies the symmetry algebra of the pseudo-diffusion equation, constructs a transformation to normalize variable coefficients, and develops a two-sided kernel based on its factorization property.

Findings

The symmetry algebra is isomorphic to that of the constant coefficient version.

A local point transformation maps the variable coefficient to a constant coefficient form.

A two-sided kernel for the PSDE is constructed using its factorization property.

Abstract

We determine by two related methods the invariance algebra $\g$ of the \emph{`pseudo-diffusion equation'} (PSDE) $$ L~Q \equiv \left[\frac {\partial}{\partial t} -\frac 1 4 \left(\frac {\partial^2}{\partial x^2} -\frac 1 {t^2} \frac {\partial^2}{\partial p^2}\right)\right]~Q(x,p,t)=0, $$ which describes the behavior of the $Q$ functions in the $(x,p)$-phase space as a function of a squeeze parameter $y$, where $t=e^{2y}$. The algebra turns out to be isomorphic to that of its constant coefficient version. Relying on this isomorphism we construct a local point transformation which maps the factor $t^{-2}$ to 1. We show that any generalized version $u_t-u_{xx}+ b(t) u_{yy}=0$ of PSDE has a smaller symmetry algebra than $\g$, except for $b(t)$ equals to a constant or it is proportional to $t^{-2}$. We apply the group elements $G_i(\ga) := \exp[\ga A_i]$ and obtain new solutions of the PSDE from simple ones, and interpret the physics of some of the results. We make use of the `factorization property' of the PSDE to construct its \textit{`2-sided kernel'}, because it has to depend on two times, $t_0 < t < t_1$. We include a detailed discussion of the identification of the Lie algebraic structure of the symmetry algebra $\g$, and its contraction from $\su(1,1)\oplus\so(3,1)$.