Computations in Large N Matrix Mechanics

TL;DR

This paper introduces a practical numerical method for large N matrix mechanics based on algebraic and variational techniques, enabling efficient computation of ground and excited states with rapid convergence.

Contribution

It develops a new algebraic/variational computational scheme for large N matrix models, including novel boundary conditions and unexpected theoretical insights.

Findings

Rapid convergence of numerical schemes in matrix models

Calculation of all moments of the ground state

Discovery of a large d expansion and a new selection rule

Abstract

The algebraic formulation of Large N matrix mechanics recently developed by Halpern and Schwartz leads to a practical method of numerical computation for both action and Hamiltonian problems. The new technique posits a boundary condition on the planar connected parts X_w, namely that they should decrease rapidly with increasing order. This leads to algebraic/variational schemes of computation which show remarkably rapid convergence in numerical tests on some many- matrix models. The method allows the calculation of all moments of the ground state, in a sequence of approximations, and excited states can be determined as well. There are two unexpected findings: a large d expansion and a new selection rule for certain types of interaction.