  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.
