  One of the major concerns of Schr\"odinger, Lorentz, Einstein, and many
others about the wave function is that it is defined on the
$3\mathbf{N}$-dimensional configuration space, rather than on the
$3$-dimensional physical space. This gives the impression that quantum
mechanics cannot have a three-dimensional space or spacetime ontology, even in
the absence of quantum measurements. In particular, this seems to affect
interpretations which take the wave function as a physical entity, in
particular the many worlds and the spontaneous collapse interpretations, and
some versions of the pilot wave theory.
  Here, a representation of the many-particle states is given, as multi-layered
fields defined on the $3$-dimensional physical space. This representation is
equivalent to the usual representation on the configuration space, but it makes
it explicit that it is possible to interpret the wave functions as defined on
the physical space. As long as only unitary evolution is involved, the
interactions are local. I intended this representation to capture and formalize
the non-explicit and informal intuition of many working quantum physicists,
who, by considering the wave function sometimes to be defined on the
configuration space, and sometimes on the physical space, may seem to
researchers in the foundations of quantum theory as adopting an inconsistent
view about its ontology. This representation does not aim to solve the
measurement problem, and it allows for Schr\"odinger cats just like the usual
one. But it may help various interpretations to solve these problems, through
inclusion of the wave function as (part of) their primitive ontology.
  In an appendix, it is shown how the multi-layered field representation can be
extended to quantum field theory.
