  We study the evolution of correlation functions of local fields in a
two-dimensional quantum field theory under the $\lambda T\bar T$ deformation,
suitably regularized. We show that this may be viewed in terms of the evolution
of each field, with a Dirac-like string being attached at each infinitesimal
step. The deformation then acts as a derivation on the whole operator algebra,
satisfying the Leibniz rule. We derive an explicit equation which allows for
the analysis of UV divergences, which may be absorbed into a non-local field
renormalization to give correlation functions which are UV finite to all
orders, satisfying a (deformed) operator product expansion and a
Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing
that the Fourier-transformed renormalized two-point functions behave as
$k^{2\Delta+2\lambda k^2}$, where $\Delta$ is their IR conformal dimension. We
discuss in detail deformed Noether currents, including the energy-momentum
tensor, and show that, although they also become non-local, when suitably
improved they remain finite, conserved and satisfy the expected Ward
identities. Finally, we discuss how the equivalence of the $T\bar T$
deformation to a state-dependent coordinate transformation emerges in this
picture.
