D=0 Matrix Model as Conjugate Field Theory

The D=0 matrix model is reformulated as a 2d nonlocal quantum field theory. The interactions occur on the one-dimensional line of hermitian matrix eigenvalues. The field is conjugate to the density of matrix eigenvalues which appears in the Jevicki-Sakita collective field theory. The classical solution of the field equation is either unique or labeled by a discrete index. Such a solution corresponds to the Dyson sea modified by an entropy term. The modification smoothes the sea edges, and interpolates between different eigenvalue bands for multiple-well potentials. Our classical eigenvalue density contains nonplanar effects, and satisfies a local nonlinear Schr\"odinger equation with similarities to the Marinari-Parisi $D=1$ reformulation. The quantum fluctuations about a classical solution are computable, and the IR and UV divergences are manifestly removed to all orders. The quantum corrections greatly simplify in the double scaling limit, and include both string-perturbative and nonperturbative effects.