Temporal correlations and chaos from spacetime kernel

TL;DR

This paper introduces a generalized spacetime density kernel (GSDK) framework that connects higher-point correlation functions with spectral form factors, providing new insights into quantum chaos and dynamical properties.

Contribution

The paper develops a finite-dimensional formulation of timelike entanglement and introduces GSDK, linking higher-point correlators with spectral form factors and chaos diagnostics.

Findings

GSDK relates higher-point functions to spectral form factors.

Haar-averaged 2N-point functions yield moments of the spectral form factor.

Higher-point correlators reveal fine-grained dynamical information.

Abstract

We develop a finite-dimensional formulation of the recently introduced notion of ``timelike entanglement'', defined in terms of two-point functions between operators supported on different Cauchy slices. Using a local orthonormal operator basis, we recast this construction in terms of a generalized response tensor. Building on this, we introduce a generalized spacetime density kernel (GSDK) corresponding to higher-point correlation functions, including time-ordered as well as out-of-time-ordered correlators. We show that the Haar-averaged $(2N)$-point function yields the $(2N)$-th moment of the spectral form factor (SFF), evaluated at an $N$-enhanced effective temperature. The correlation functions of the GSDK operators also yield the SFF, with an effective $(1/N)$-reduction of the physical time-scales. The GSDK places both scrambling diagnostics and spectral statistics on a similar footing and clarifies how higher-point correlators and non-trivial time ordering capture fine-grained dynamical information of a quantum system.