  How would the world appear to us if its ontology was that of classical
mechanics but every agent faced a restriction on how much they could come to
know about the classical state? We show that in most respects, it would appear
to us as quantum. The statistical theory of classical mechanics, which
specifies how probability distributions over phase space evolve under
Hamiltonian evolution and under measurements, is typically called Liouville
mechanics, so the theory we explore here is Liouville mechanics with an
epistemic restriction. The particular epistemic restriction we posit as our
foundational postulate specifies two constraints. The first constraint is a
classical analogue of Heisenberg's uncertainty principle -- the second-order
moments of position and momentum defined by the phase-space distribution that
characterizes an agent's knowledge are required to satisfy the same constraints
as are satisfied by the moments of position and momentum observables for a
quantum state. The second constraint is that the distribution should have
maximal entropy for the given moments. Starting from this postulate, we derive
the allowed preparations, measurements and transformations and demonstrate that
they are isomorphic to those allowed in Gaussian quantum mechanics and generate
the same experimental statistics. We argue that this reconstruction of Gaussian
quantum mechanics constitutes additional evidence in favour of a research
program wherein quantum states are interpreted as states of incomplete
knowledge, and that the phenomena that do not arise in Gaussian quantum
mechanics provide the best clues for how one might reconstruct the full quantum
theory.
