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 $d\geq 4$. 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.