  A theory of electronic friction is developed using the exact factorization of
the electron-nuclear wavefunction. No assumption is made regarding the
electronic bath, which can be made of independent or interacting electrons, and
the nuclei are treated quantally. The ensuing equation of motion for the
nuclear wavefunction is a non-linear Schr\"{o}dinger equation including a
friction term. The resulting friction kernel agrees with a previously derived
mixed quantum-classical result by Dou, Miao \& Subotnik (\emph{Phys. Rev.
Lett.} \textbf{119}, 046001 (2017)), except for a \emph{pseudo}-magnetic
contribution in the latter that is here removed. More specifically, it is shown
that the electron dynamics generally washes out the\emph{ gauge} fields
appearing in the adiabatic dynamics. However, at T=0 K, the
\emph{pseudo}-magnetic force is fully re-established in the typical situation
where the electrons respond rapidy on the slow time-scale of the nuclear
dynamics (Markov limit). Hence, we predict Berry's phase effects to be
observable also in the presence of electronic friction, and non-trivial
geometric phases should be attainable for molecules on metallic magnetic
surfaces.
