$T\overline{T}$ in AdS$_2$ and Quantum Mechanics (1907.04873v1)

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.

TT in AdS$_2$ and Quantum Mechanics: An Overview

The paper under review addresses significant aspects of quantum field theory in zero spatial dimensions, particularly within the context of holography and deformations analogous to the $T\overline{T}$ deformation in one-dimensional quantum mechanics. The research explores the complex relationship between spacetime geometries and quantum mechanical systems, analyzing how a class of deformations can elucidate aspects of holography and potentially uncover new pathways for understanding quantum gravity.

Core Concepts and Methodology

The paper begins by discussing the unexpected richness of the infrared landscape in zero-dimensional quantum field theories and the universality in their ultraviolet behavior, reversing the typical Wilsonian paradigm. The authors focus on a specific class of integrable deformations, parameterized by transformations of the Hamiltonian, $H \rightarrow F(H)$, which maintain the integrability of quantum mechanical models by directly calculating deformed eigenvalues while keeping eigenfunctions unaltered. This approach affords a clear pathway to exploring quantum mechanics without the complications arising from spatial locality.

In exploring $T\overline{T}$ deformations, originally prominent in two-dimensional field theories, the research extends these ideas to one-dimensional quantum mechanics, proposing an equivalent description in terms of coupling to worldline gravity. The authors derive a deforming operator in the context of Jackiw-Teitelboim (JT) gravity in AdS$_2$, using a dimensional reduction approach to transition from $d = 2$ to $d = 1$. Furthermore, they provide solutions and flow equations applicable to general dilaton-gravity theories, including those with additional matter fields, highlighting their impact on quantum mechanical systems and holographic principles.

Key Numerical Results and Implications

The calculations reveal that the energies under the $T\overline{T}$ flow in one dimension display behaviors similar to the solutions of gravitational theories at finite cutoff. This alignment suggests that certain one-dimensional deformations might not only capture but isolate infrared geometrical features—a process pivotal for the paper of wider-reaching holographic models.

The paper also explores chaos in deformed quantum mechanics and its persistence even with the imposition of finite cutoffs, finding that the Lyapunov exponent remains maximal. This robustness highlights the fundamental nature of chaotic behavior in such quantum mechanical frameworks.

Theoretical and Practical Implications

The theoretical implications of this paper are twofold. Firstly, the exploration of $T\overline{T}$-like deformations in quantum mechanics contributes to understanding how spacetime geometries arise in quantum systems, providing a potential bridge to higher-dimensional frameworks. Secondly, these deformations may offer insights into the non-perturbative realms of quantum mechanics, given their ability to maintain integrability and provide worldline action interpretations in the ultraviolet regime.

Practically, the results could extend to computational models of quantum gravity and holography, offering methods to simulate and understand complex multi-dimensional systems via simple quantum mechanical analogs. The theoretical framework might also inspire physical realizations in controlled laboratory conditions, leveraging quantum simulators to explore the rich dynamical landscapes predicted by the models.

Future Developments

The findings in this paper pave the way for further research into deformations in broader settings, including extensions to systems with spatial dimensions and more intricate interactions. Investigating the connections between these quantum mechanical flows and string theory embeddings, along with potential applications to matrix models and supersymmetric systems, would significantly enhance the understanding of the interplay between quantum mechanics and gravitational theories. Furthermore, exploring these deformations in specific models like the Sachdev-Ye-Kitaev (SYK) model could illuminate new aspects of quantum chaos and its foundational role in holography.

In summary, this paper constitutes a substantial contribution to theoretical physics by expanding the scope and applicability of $T\overline{T}$ deformations to quantum mechanics, thereby enriching the dialogue between quantum fields, gravity, and holography.