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

A salient feature of the Schrödinger equation is that the classical radial momentum term pr in polar coordinates is replaced by the operator P̂ † r P̂r, where the operator P̂r 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 p̂r = (1/2) ( ( ~̂xr )~̂ p + ~̂p( ~̂x r ) ) , the relation P̂ † r P̂r = p̂ 2 r + h̄ (d− 1)(d− 3)/(4r2) 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 ≥ 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.