Exact lattice bosonization of finite N matrix quantum mechanics and c = 1

A bstract 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. It is based on the exact operator bosonization of non-relativistic fermions introduced earlier in [1]. The trace identities, which characterize finite rank matrices, are automatically incorporated in the bosonic theory. The finite number N of fermions is reflected in the finite number N of bosonic annihilation-creation operators, and 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, a recent description of the finiteness of the fermionic EE in a collective field theory formalism [2]). The bosonic lattice constructed in our work is equipped with a geometry that is determined by the matrix potential or equivalently by the shape of the Fermi surface. Our bosonization also works in the double scaled c = 1 matrix model; in particular the bosonic EE again turns out to be finite, with the short distance cut-off turning into g s l s , and reproduces the matrix result. Once again, this is to be contrasted with the conventional dual 2D string theory, where the bosonic EE is naturally identified with that of the “tachyon”, the massless string mode, where one may imagine the short-distance cut-off to be the string length l s . This appears to indicate our bosonization as a 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.