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

A salient feature of the Schrodinger equation is that the classical radial momentum term P(r)(2) in polar coordinates is replaced by the operator (P) over cap (+)(r)(P) over cap (r), where the operator (P) over cap (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 (p) over cap (r) = (1/2) ((x/r)p + p(x/r)), the relation (P) over cap (+)(r)(P) over cap (r) = (p) over cap (2)(r) + h(2) (d-1)(d-3)/(4r(2)) holds in d-dimensional space and 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.