Temporal correlations and chaos from spacetime kernels

A bstract 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. Motivated by the structure of Haar probe averaging, we contract the (2 N )-leg GSDK with cyclic permutation operators and thereby define a kernel form factor . This quantity is computable even when Haar averaging is not feasible, and it reduces to the (2 N )-th moment of the spectral form factor, 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 early time scrambling diagnostics and late time 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.