Large-N Limit as a Classical Limit: Baryon in Two-Dimensional QCD and Multi-Matrix Models

Abstract

This thesis concerns the large-N limit, a classical limit where fluctuations in gauge-invariant variables vanish. The large dimension limit for rotation-invariant variables in atoms is given as an example of a classical limit other than hbar vanishing. Part I concerns the baryon in Rajeev's reformulation of 2d QCD in the large-N limit: a non-linear classical theory of color-singlet quark bilinears. 't Hooft's meson equation is the linearization around the vacuum on the curved grassmannian phase space. The baryon is a topological soliton. Its form factor is found variationally on a succession of increasing rank submanifolds of the phase space. These reduced systems are interacting parton models: a derivation of parton model from the soliton picture. A rank-1 ansatz leads to a Hartree approximation: a relativistic 2d realization of Witten's proposal. The baryon form factor is used to model x_B dependence of deep inelastic structure functions. In Part II, euclidean large-N multi-matrix models are reformulated as classical systems for U(N) invariants. The configuration space of gluon correlations is a space of non-commutative probability distributions. Classical equations of motion (factorized loop equations) contain an anomaly that leads to a cohomological obstruction to finding an action principle. This is circumvented by expressing the configuration space as a coset space of the automorphism group of the tensor algebra. The action principle is interpreted as the Legendre transform of the entropy of operator-valued random variables. The free energy and correlations in the large-N limit are found variationally. The simplest variational ansatz is an analogue of mean-field theory and compares well with exact and numerical solutions of 1 and 2 matrix models away from phase transitions.