On the flow of states under $T\overline{T}$

TL;DR

This paper analyzes the $T\overline{T}$ deformation of 2D quantum field theories from a Hamiltonian perspective, revealing how it affects energy spectra, operators, and symmetries, with implications for AdS/CFT correspondence.

Contribution

It provides a Hamiltonian framework for understanding $T\overline{T}$ deformations, including flow equations and the preservation of symmetries on the plane, and introduces dressed operators with preserved algebraic structures.

Findings

Deformation implements successive canonical transformations.

Dressed operators satisfy original commutation relations.

On the plane, conformal symmetry is preserved under deformation.

Abstract

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.