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Space-time is one of the most essential, yet most mysterious concepts inphysics. In quantum mechanics it is common to understand time as a marker ofinstances 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 aroundmass. Here we present a unified approach on both space and time in quantumtheory, and build quantum states across spacetime instead of only on spatialslices. We no longer distinguish measurements on the same system at differenttimes with measurements on different systems at one time and constructspacetime states upon these measurement statistics. As a first step towardsnon-relativistic quantum field theory, we consider how to approach this in thecontinuous-variable multi-mode regime. We propose six possible definitions forspacetime states in continuous variables, based on four different measurementprocesses: quadratures, displaced parity operators, position measurements andweak measurements. The basic idea is to treat different instances of time asdifferent quantum modes. They are motivated by the pseudo-density matrixformulation among indefinite causal structures and the path integral formalism.We show that these definitions lead to desirable properties, and raise thedifferences and similarities between spatial and temporal correlations. Anexperimental proposal for tomography is presented, construing the operationalmeaning of the spacetime states.

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