A Proposal to Solve Finite N Matrix Theory: Reduced Model Related to Quantum Cosmology

Abstract

The SU(N) invariant model of matrix theory that emerges as the regularization of the 11-dimensional super membrane is studied. This matrix model is identified with M theory in the limit N→∞. It has been conjectured that matrix models are also relevant for finite N where several examples and arguments have been given in the literature. By the use of a Dirac-like formulation usually developed in finding solutions in Supersymmetric Quantum Cosmology, we exhibit a method that could solve, in principle, any finite N model. As an example of our procedure, we choose a reduced SU(2) model and also show that this matrix model contains relevant supersymmetric quantum cosmological models as solutions. By these means, our solutions constitute an example in order to consider why the finite N matrix models are also relevant. Since the degrees of freedom of matrix models are, in some limit, identified with those of Super Yang Mills Theory SYM with a finite number of supercharges, our methodology offers the possibility, through some but yet unspecified identification, to relate the quantization presented here with that of SYM theory for any finite N.