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 ${\widehat{P}}_r^{†}{\widehat{P}}_r$, where the operator ${\widehat{P}}_r$ 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 $\hat{p}{r}=(1/2)((\frac{\hat{\vec{x}}}{r}) \hat{\vec{p}}+\hat{\vec{p}}(\frac{\hat{\vec{x}}}{r}))$,therelation$\hat{P}^{\dagger}{r} \hat{P}{r}=\hat{p}{r}^{2}+\hbar^{2}(d-1)(d-3)/(4r^{2})$holdsin$d$-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 $d\ge 4$. 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.