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On the flow of states under $T\overline{T}$
by Jorrit Kruthoff, Onkar Parrikar
Submission summary
As Contributors:  Jorrit Kruthoff · Onkar Parrikar 
Preprint link:  scipost_202007_00053v2 
Date submitted:  20201013 18:12 
Submitted by:  Kruthoff, Jorrit 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
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 nonlocal, '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 nontrivial, 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.
Current status:
Author comments upon resubmission
Dear Editor,
We thank the referees for their careful reading of our manuscript and their valuable comments. The manuscript is modified quite a bit to address the concerns expressed in both the reports, although the main results have remained unaffected. See below for our reply to both reports.
Report 1
We thank Prof. Cardy for his valuable questions and comments.
1) The first clarification the referee suggested was about the analysis of the ultraviolet (UV) divergences. We agree that our original manuscript lacked a careful discussion on this issue and we deeply appreciate this comment made by the referee. In order to address it we have, at various places in the revised manuscript, discussed the UV divergences. First, when we introduce the Green function $G$ in equation 2.4 and below, we mention that the coincident point limit needs to be regulated in order for equation 2.7 to make sense. We mentioned this below 2.7. The next place where UV divergences play a role is around equation 3.8. Since our flow equation for the undeformed operators agrees with that of Prof. Cardy's from [17], we make use of his analysis of the UV divergences at this point to address the need for operator renormalization. Similarly, we also discuss the dressed operators and their divergencies below equation 3.17.
2) The second clarification sought by the referee is about the definition of dressed operators. Here we mention that we are not saying that the $T\overline{T}$ deformed theory can always be written as a dressed version of the undeformed theory. Let us first consider the classical case for simplicity. In the classical theory on the plane (and only on the plane) and within the finite energy sector, it so happens that the deformed Hamiltonian can be obtained from a canonical transformation of the undeformed Hamiltonian, see eq. 3.11. Classically, this means that the deformation is merely implementing a canonical transformation on the phase space coordinates. It is this fact, which allows us to define the dressed operators  they are the observables written in the canonically transformed coordinates. Already at this stage, we point out that this is \emph{not} the same as a statement along the lines suggested by the referee, i.e., 'interacting QFT is a dressed version of a free theory''. In that case, the interacting Hamiltonian is not a canonical transformation of the free Hamiltonian. Quantum mechanically, the deformation implements a Bogoliubov transformation, but we can nevertheless define dressed observables. These dressed operators do indeed have UV divergences in their flow equation  we expect these divergences to be related to operator renormalization. Importantly, the correlation functions of these operators are finite. We have tried to reiterate some of the points above in the draft, for instance below equation 3.17.
Let us emphasize that on the cylinder this statement about the canonical/Bogoliubov transformation is not true and we have no precise claim to make. We added a comment about this below 3.26.
Report 2
We thank the referee for her/his valuable questions and comments.
1) We have done so below 4.6 in the revised manuscript 2) Here, the referee sought a clarification for whether the dressed operators are nontrivial. The dressed operators are not the same as the original, undeformed CFT operators. Indeed, these latter ones are discussed in the subsection titled ``Undeformed operators'' on p 15. Consider first the classical theory. It makes sense to define dressed observables only because the deformed Hamiltonian is a canonical transformation of the original Hamiltonian; see equation 3.11. The canonical transformation (generated by $\mathcal{X}$) defines a new set of phase space coordinates at every point along the flow  these are the dressed observables. Note that this definition only makes sense within the finite energy sector on the plane . On the contrary, for the cylinder, or even for finite energy density states on the plane, the $T\overline{T}$ deformation is not a pure canonical transformation as we cannot drop the $\mathcal{Y}$ term in this case, and so it is not clear how to define dressed observables. 3) We have discussed the issue of UV divergences in the revised manuscipt, see our response to report number one (point number one thereof). 4) Thank you for spotting all the typos, we have corrected all of them in the revised manuscript.