  Quantum cellular automata (QCA) are discrete models of space and time
homogeneous quantum field theories (QFTs) and regarded as natural candidates
for quantum simulation. Description of a QCA over the separable Hilbert space
of finite, unbounded configurations (UFC Hilbert space) with unitary state
evolution is the {\it Schr\"{o}dinger template}, and over the incomplete
infinite tensor product algebra (ITPA) with evolution by algebra automorphism
is the {\it Heisenberg template}. Whether every Heisenberg template admits an
equivalent Schr\"{o}dinger template is a foundational question, and one that
has persisted as an open problem. In the present paper we prove that for every
Heisenberg template an equivalent Schr\"{o}dinger template exists. We frame the
question from a representation theory standpoint, using constructs and results
from the representation theory of finite and countably infinite dimensional
vector spaces and from category theory to answer it. With the previously known
existence of a Heisenberg template for every Schr\"{o}dinger template, our
result establishes the equivalence of the templates.
