# Different instances of time as different quantum modes: quantum states across space-time for continuous variables

## TL;DR

This paper introduces a unified quantum framework for representing states across space and time, treating different time instances as separate quantum modes, and explores their properties and measurement methods.

## Contribution

It proposes six definitions of spacetime quantum states in continuous variables, unifying space and time in quantum theory and extending the pseudo-density matrix approach.

## Key findings

- Six possible definitions for spacetime states in continuous variables.
- Different measurement processes lead to desirable properties of spacetime states.
- Operational proposal for tomography of spacetime states.

## Abstract

Space-time is one of the most essential, yet most mysterious concepts in physics. In quantum mechanics it is common to understand time as a marker of instances of evolution and define states around all the space but at one time; while in general relativity space-time is taken as a combinator, curved around mass. Here we present a unified approach on both space and time in quantum theory, and build quantum states across spacetime instead of only on spatial slices. We no longer distinguish measurements on the same system at different times with measurements on different systems at one time and construct spacetime states upon these measurement statistics. As a first step towards non-relativistic quantum field theory, we consider how to approach this in the continuous-variable multi-mode regime. We propose six possible definitions for spacetime states in continuous variables, based on four different measurement processes: quadratures, displaced parity operators, position measurements and weak measurements. The basic idea is to treat different instances of time as different quantum modes. They are motivated by the pseudo-density matrix formulation among indefinite causal structures and the path integral formalism. We show that these definitions lead to desirable properties, and raise the differences and similarities between spatial and temporal correlations. An experimental proposal for tomography is presented, construing the operational meaning of the spacetime states.