Journal of Mathematical Physics
Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central‐force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse‐square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.
D. Peak and A. Inomata, "Summation Over Feynman Histories in Polar Coordinates," Journal of Mathematical Physics 10(8), 1422-1428 (1969). doi:10.1063/1.1664984