  We study the $T\overline{T}$ deformation of two dimensional quantum field
theories from a Hamiltonian point of view, focusing on aspects of the theory in
Lorentzian signature. Our starting point is a simple rewriting of the spatial
integral of the $T\overline{T}$ operator, which directly implies the deformed
energy spectrum of the theory. Using this rewriting, we then derive flow
equations for various quantities in the deformed theory, such as energy
eigenstates, operators, and correlation functions. On the plane, we find that
the deformation merely has the effect of implementing successive
canonical/Bogoliubov transformations along the flow. This leads us to define a
class of non-local, 'dressed' operators (including a dressed stress tensor)
which satisfy the same commutation relations as in the undeformed theory. This
further implies that on the plane, the deformed theory retains its symmetry
algebra, including conformal symmetry, if the original theory is a CFT. On the
cylinder the $T\overline{T}$ deformation is much more non-trivial, but even so,
correlation functions of certain dressed operators are integral transforms of
the original ones. Finally, we propose a tensor network interpretation of our
results in the context of AdS/CFT.
