  Field theories place one or more degrees of freedom at every point in space.
Hilbert spaces describing quantum field theories, or their finite-dimensional
discretizations on lattices, therefore have large amounts of structure: they
are isomorphic to the tensor product of a smaller Hilbert space for each
lattice site or point in space. Local field theories respecting this structure
have interactions which preferentially couple nearby points. The emergence of
classicality through decoherence relies on this framework of tensor-product
decomposition and local interactions. We explore the emergence of such lattice
structure from Hilbert-space considerations alone. We point out that the vast
majority of finite-dimensional Hilbert spaces cannot be isomorphic to the
tensor product of Hilbert-space subfactors that describes a lattice theory. A
generic Hilbert space can only be split into a direct sum corresponding to a
basis of state vectors spanning the Hilbert space; we consider setups in which
the direct sum is naturally decomposed into two pieces. We define a notion of
direct-sum locality which characterizes states and decompositions compatible
with Hamiltonian time evolution. We illustrate these notions for a toy model
that is the finite-dimensional discretization of the quantum-mechanical
double-well potential. We discuss their relevance in cosmology and field
theory, especially for theories which describe a landscape of vacua with
different spacetime geometries.
