$T\overline{T}$ in AdS$_2$ and Quantum Mechanics

TL;DR

The paper develops a one-dimensional analogue of TT̄ deformations for quantum mechanics, tying the flow to coupling with worldline gravity and thereby isolating infrared AdS$_2$ interiors within holographic contexts.It derives the deforming operator from both dimensional reduction of TT̄ in AdS$_3$ and a dilaton-gravity Hartman-type analysis, applies it to JT gravity and the Schwarzian theory, and demonstrates that key features like maximal chaos persist under finite-cutoff deformations.A unified picture emerges in which the UV limit is a universal worldline action with a curved target-space, and a nonperturbative definition via coupling to 1D gravity reproduces the deformation as an integral transform of the seed theory's partition function.The framework connects bulk finite-cutoff physics to boundary quantum mechanics, offers exact results for spectra and thermodynamics, and opens avenues for holography in more general spacetimes and for related matrix-model/SYK contexts.

Abstract

Quantum field theory in zero spatial dimensions has a rich infrared landscape and a universal ultraviolet, inverting the usual Wilsonian paradigm. For holographic theories this implies a rich landscape of asymptotically AdS$_2$ geometries. Isolating the interiors of these spacetimes suggests a study of the analog of the $T\overline{T}$ deformation in quantum mechanics, which we pursue here. An equivalent description of this deformation in terms of coupling to worldline gravity is proposed, and applications to quantum mechanical systems - including the Schwarzian theory - are studied.

Figures (1)

• Figure 1: Shockwaves in JT gravity from finite cutoff. The aqua green and orange regions correspond to black hole exterior spacetimes of mass $M$ and slightly larger mass $\widetilde{M}$, respectively. The gray region is the extension of the Penrose diagram of the heavier black hole into the past. At point $A$ an ingoing massless particle is released. At point $B$ it passes through the finite cutoff boundary in the initial black hole geometry and at $\widetilde{B}$ in the new black hole geometry. The black dashed lines are lines of constant and equal values of the dilaton. Points $C$ and $\widetilde{C}$ are where the outgoing signal meets these lines.