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

Jorrit Kruthoff, Onkar Parrikar

TL;DR

This work develops a Hamiltonian-based understanding of the $T\overline{T}$ deformation in 2D QFTs, showing that the deformation induces a bi-local split of the Hamiltonian into a canonical transformation and an energy-shifting part. It derives explicit flow equations for energy levels and states, introduces dressed operators that remain causal and whose correlation functions are flow-invariant on the plane, and provides kernel and worldsheet interpretations via a Cauchy-string picture. On the cylinder, correlation functions become integral transforms of the seed theory, while the S-matrix acquires the characteristic CDD phase, and a 3D gravity path integral interpretation connects the TTbar flow to holographic ideas and tensor networks. The paper also discusses broader implications for entanglement, Virasoro symmetry, and generalizations to other deformations, highlighting links to AdS/CFT and the surface-state/tensor-network program. Overall, it offers a coherent framework to study nonlocal TTbar effects, their holographic meaning, and their observable consequences across spectra, operators, and correlators.

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.

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

TL;DR

This work develops a Hamiltonian-based understanding of the deformation in 2D QFTs, showing that the deformation induces a bi-local split of the Hamiltonian into a canonical transformation and an energy-shifting part. It derives explicit flow equations for energy levels and states, introduces dressed operators that remain causal and whose correlation functions are flow-invariant on the plane, and provides kernel and worldsheet interpretations via a Cauchy-string picture. On the cylinder, correlation functions become integral transforms of the seed theory, while the S-matrix acquires the characteristic CDD phase, and a 3D gravity path integral interpretation connects the TTbar flow to holographic ideas and tensor networks. The paper also discusses broader implications for entanglement, Virasoro symmetry, and generalizations to other deformations, highlighting links to AdS/CFT and the surface-state/tensor-network program. Overall, it offers a coherent framework to study nonlocal TTbar effects, their holographic meaning, and their observable consequences across spectra, operators, and correlators.

Abstract

We study the 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 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 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.

Paper Structure

This paper contains 17 sections, 136 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Energy eigenstates and their flow
  3. Rewriting the $T\overline{T}$ operator
  4. Energy eigenvalues and eigenstates
  5. A kernel for $U$
  6. Flow of operators and correlation functions
  7. On the plane
  8. On the cylinder
  9. Further developments
  10. $S$-matrix
  11. $3d$ gravity interpretation of $U$
  12. Surface-state correspondence and tensor networks
  13. Discussion
  14. Lagrangian vs. Hamiltonian definitions
  15. Pressure term in the Energy flow
  16. ...and 2 more sections

Figures (2)

  • Figure 1: We can interpret the unitary $U$ as making the Cauchy slice a dynamical surface parametrised by $X^a(\lambda,\sigma)$.
  • Figure 2: A cartoon of a tensor network. The black dots are initial Hilbert space degrees of freedom, say spins. The blue dots denote tensors which act on the spins as disentanglers while the red dots act as isometries. The emergent geometry of the network is reminiscent of the bulk geometry in AdS/CFT.