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

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.

30 pages + appendices. 3 figures. v2: Typos corrected, updated and corrected the discussion in section 2.3 and references added