Ground-State Theorem for Free Polarons

A ground-state theorem for the Fröhlich polaron is derived. We show that the zeroth moment of the current-current correlation function is proportional to the kinetic energy of the polaron. While the electron-phonon interaction does not depend on the electron mass, the coupling constant $\alpha$ does; this allows us to relate the kinetic energy with the ground-state energy using the Feynman-Helmann theorem. As an example we compare the polaron ground-state energy on variational grounds with the ground-state energy obtained from the function $\chi \left(\omega \right)$ calculated by Feynman, Hellwarth, Iddings, and Platzman. For $\alpha <5\mathrm{}$ both ground-state energies turn out to be identical, as can be concluded from numerical calculations.