Hamiltonian deformations in quantum mechanics, $T\bar T$, and SYK

TL;DR

This paper introduces a broad class of solvable Hamiltonian deformations inspired by $Tar T$, allowing exact computation of finite-temperature correlators and exploring implications for AdS/CFT, SYK, and the Schwarzian theory.

Contribution

It proposes a new class of solvable quantum mechanical deformations that preserve certain dynamical features and provides explicit formulas for their correlation functions.

Findings

Deformations map Hamiltonians to functions of themselves.

Finite-temperature correlators are exactly computed in deformed theories.

Maximal Lyapunov exponent remains unchanged under deformation.

Abstract

Motivated by $T\bar T$, we introduce and study a wide class of solvable deformations of quantum-mechanical theories. These deformations map the Hamiltonian to a function of itself. We solve these theories by computing all finite-temperature correlation functions of the deformed theory in terms of the correlators of the undeformed theory. Applications to AdS/CFT, SYK, and the Schwarzian theory are considered. We write down the deformed Schwarzian action for an arbitrary Hamiltonian deformation and find that the maximal Lyapunov exponent is unchanged.