Quantum canonical transformations as integral transformations

Abstract

We discuss how the Hamiltonian changes in quantum canonical transformations. To the operator $\hat{\mathcal{H}}(\hat{p},\hat{q})$ one can associate (in a given ordering rule) a $c$-number function $\mathcal{H}\mathrm{}(p,q)\mathrm{}$. It is this function that appears in the action of the phase-space path integral. A quantum canonical transformation $\hat{\mathcal{H}}\to {\hat{\mathcal{H}}}^{\prime }$ can now be expressed as an integral transformation ${\mathcal{H}}^{\prime }(\overline{p},\overline{q})=\int \mathrm{dp}\mathrm{dq}\mathcal{T}(\overline{p},\overline{q};p,q)\mathcal{H}\mathrm{}(p,q)\mathrm{}$. The kernel $\mathcal{T}$ is constructed explicitly for point transformations and for the $p=-\overline{q}$, $q=\overline{p}$ reflection by studying changes of variables in the path integral. The ordering dependence of $\mathcal{T}$ is displayed. The invariance of commutation rules is also discussed.