Large-N Limit as a Classical Limit: Baryon in Two-Dimensional QCD and Multi-Matrix Models
Abstract
Description
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.
PhD thesis, defended May 26, 2004 University of Rochester, 157 pages and 22 figures
PhD thesis, defended May 26, 2004 University of Rochester, 157 pages and 22 figures