  A variety of compounds, for example doped paraelectrics and polar metals,
exhibit both ferroelectricity and correlated electronic phenomena such as
low-density superconductivity and anomalous transport. Characterizing such
properties is tied to understanding the quantum dynamics of inversion symmetry
breaking in the presence of itinerant electrons. Here, we present a
comprehensive analysis of the normal state properties of a metal near a quantum
critical transition to a ferroelectric state, in both two and three dimensions.
Starting from a minimal model of electrons coupled to a \emph{transverse} polar
phonon via a Rashba-type spin-orbit interaction, we compute the dynamical
response of both electrons and phonons. We find that the system can evince both
Fermi and non-Fermi liquid phases, as well as enhanced pairing in both singlet
and triplet channels. Furthermore, we systematically compute corrections to
one-loop theory and find a tendency to quantum order-by-disorder, leading to a
phase diagram that can include second order, first order, and finite-momentum
phase transitions. Finally, we show that the entire phase diagram can be
controlled via application of external strain, either compressive or
volume-preserving. Our results provide a map of the dynamical and
thermodynamical phase space of quantum ferroelectic metals, which can serve in
characterizing existing materials and in seeking applications for quantum
technologies.
