Direct path-integral treatment of the polaron problem

Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that $n\mathrm{th}$-order terms in the polaron coupling constant are obtained from $n\mathrm{th}$-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a $2n\mathrm{th}$-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Padé approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\alpha =10\mathrm{}$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.