# Spacetime Density Matrix: Formalism and Properties

## TL;DR

This paper introduces the spacetime density matrix as a generalization of the standard density matrix, capturing correlations across different Cauchy surfaces, with formalism, properties, and a perturbative approach for weakly coupled subsystems.

## Contribution

It develops the formalism, properties, and equations of motion for the spacetime density matrix, including a super-operator framework and a perturbative method for moments.

## Key findings

- Universal short-time behavior of the second moment derived
- Coupling between subsystems influences nontrivial results
- Perturbative method for weak coupling systems developed

## Abstract

In this paper, we develop the general formalism and properties of the spacetime density matrix, which captures correlations among different Cauchy surfaces and can be regarded as a natural generalization of the standard density matrix defined on a single Cauchy surface. We present the construction of the spacetime density matrix in general quantum systems and its representation via the Schwinger Keldysh path integral. We further introduce a super-operator framework, within which the spacetime density matrix appears as a special case, and discuss possible generalizations from this perspective. We also show that the spacetime density matrix satisfies a Liouville von Neumann type equation of motion. When considering subsystems, a reduced spacetime density matrix can be defined by tracing over complementary degrees of freedom. We study the general properties of its moments and, in particular, derive universal short time behavior of the second moment. We find that coupling between subsystems plays a crucial role in obtaining nontrivial results. Assuming weak coupling, we develop a perturbative method to compute the moments systematically.