  We describe a new exact lattice bosonization of matrix quantum mechanics
(equivalently of non-relativistic fermions) that is valid for arbitrary rank N
of the matrix, based on an exact operator bosonization introduced earlier in
[1]. The trace identities are automatically incorporated in this formalism. The
finite number N of fermions is reflected in the finite number N of bosonic
oscillators, or equivalently to the finite number N of lattice points. The
fermion Hamiltonian is exactly mappable to a bosonic Hamiltonian. At large N,
the latter becomes local and corresponds to the lattice version of a
relativistic boson Hamiltonian, with a lattice spacing of order 1/N. The finite
lattice spacing leads to a finite entanglement entropy (EE) of the bosonic
theory, which reproduces the finite EE of the fermionic theory. Such a
description is not available in the standard bosonization in terms of fermion
density fluctuations on the Fermi surface, which does not have a built-in short
distance cut-off (see, however, [2]). The bosonic lattice is equipped with a
geometry determined by the matrix potential or equivalently by the shape of the
Fermi surface. Our bosonization also works in the double scaled c=1 model,
where the bosonic EE again turns out to be finite, with the short distance
cut-off turning as g_s l_s, and reproduces the matrix result. Once again, such
a short distance cut-off cannot appear in the conventional dual of c=1 in terms
of the 2D string ``tachyon'', where the expected short distance scale is l_s.
This indicates our bosonization as a possibly different dual description to the
c=1 matrix model appropriate for ``local physics'' like quantum entanglement,
by contrast with the conventional duality to the eigenvalue density which works
well for asymptotic observables like S-matrices. We briefly discuss possible
relation of our bosonization to D0 branes.
