On the Best Quadratic Approximation in Feynman's Path Integral Treatment of the Polaron

The best quadratic approximation to the retarded polaron action due to Adamowski {\it et al.} and Saitoh is investigated numerically for a wide range of coupling constants. The non-linear variational equations are solved iteratively with an efficient method in order to obtain the ground state energy and the effective mass of the polaron. The virial theorem and expansions for small and large couplings are used to check the high accuracy of the numerical results. Only small improvements over Feynman's (non-optimal) results are observed. For a moving polaron it is shown that the most general quadratic trial action may contain anisotropic terms which, however, do not lead to improvements for the ground state energy and effective mass.