  We study interacting theories of $N$ left-moving and $\overline{N}$
right-moving Floreanini-Jackiw bosons in two dimensions. A parameterized family
of such theories is shown to enjoy (non-manifest) Lorentz invariance if and
only if its Lagrangian obeys a flow equation driven by a function of the
energy-momentum tensor. We discuss the canonical quantization of such theories
along classical stress tensor flows, focusing on the case of the root-$T
\overline{T}$ deformation, where we obtain perturbative results for the
deformed spectrum in a certain large-momentum limit. In the special case $N =
\overline{N}$, we consider the quantum effective action for the root-$T
\overline{T}$-deformed theory by expanding around a general classical
background, and we find that the one-loop contribution vanishes for backgrounds
with constant scalar gradients. Our analysis can also be interpreted via dual
$U(1)$ Chern-Simons theories in three dimensions, which might be used to
describe deformations of charged $\mathrm{AdS}_3$ black holes or quantum Hall
systems.
