Non-hermitian radial momentum operator and path integrals in polar coordinates

A salient feature of the Schr{o}dinger equation is that the classical radial momentum term $p_{r}^{2}$ in polar coordinates is replaced by the operator $hat{P}^{dagger}_{r} hat{P}_{r}$, where the operator $hat{P}_{r}$ is not hermitian in general. This fact has important implications for the path integral and semi-classical approximations. When one defines a formal hermitian radial momentum operator $hat{p}_{r}=(1/2)((frac{hat{vec{x}}}{r}) hat{vec{p}}+hat{vec{p}}(frac{hat{vec{x}}}{r}))$, the relation $hat{P}^{dagger}_{r} hat{P}_{r}=hat{p}_{r}^{2}+hbar^{2}(d-1)(d-3)/(4r^{2})$ holds in $d$-dimensional space and this extra potential appears in the path integral formulated in polar coordinates. The extra potential, which influences the classical solutions in the semi-classical treatment such as in the analysis of solitons and collective modes, vanishes for $d=3$ and attractive for $d=2$ and repulsive for all other cases $dgeq 4$. This extra term induced by the non-hermitian operator is a purely quantum effect, and it is somewhat analogous to the quantum anomaly in chiral gauge theory.