Gaussian creation and annihilation operators
= Gaussian creation and annihilation operators
{c}
{title2=$a^-=D,\quad a^+=x-D$}
On polynomials in $L^2$ of standard <Gaussian measure>, <Gaussian integration by parts> makes $D$ and $x-D$ adjoints, with commutator $[a^-,a^+]=I$. The <Probabilists' Hermite polynomials> are obtained by applying $a^+$ repeatedly to $1$, and $a^-\operatorname{He}_n=n\operatorname{He}_{n-1}$. These give a Gaussian-space realization of <creation and annihilation operators>.