  In all our well-established theories, it is assumed that events are embedded
in a global causal structure such that, for every pair of events, the causal
order between them is always fixed. However, the possible interplay between
quantum mechanics and general relativity may require a revision of this
paradigm. The process matrix framework keeps the validity of quantum physics
locally but does not assume the existence of a global causal order. It allows
to describe causal structures corresponding to a quantum superposition of 'A is
before B' and 'B is before A'. So far, the framework has been developed only
for finite-dimensional systems, and a straightforward generalization to
infinite dimensions leads to singularities. Such generalization is necessary
for continuous-variable systems and is a prerequisite for quantum fields on
indefinite causal structures. Here we develop the process-matrix framework for
continuous-variable systems. We encounter and solve the problems of
singularities. Moreover, we study an example of a process in infinite
dimensions, and we derive correlations exhibiting intereference due to
processes in which A is before B and B is before A.
