  In fundamental theories that accounts for quantum gravitational effects, the
spacetime causal structure is expected to be quantum uncertain. Previous
studies of quantum causal structure focused on finite-dimensional systems. Here
we present an algebraic framework that incorporates both finite- and
infinite-dimensional systems including quantum fields. Thanks to the absence of
a definite spacetime causal structure, Lagrangian quantum field theories can be
studied on a quantum superposition of spacetimes with a point identification
structure.
