Limit Of Matrix Models Research Articles

Renormalization group approach to matrix models via noncommutative space

AbstractWe develop a new method for analyzing the large‐N limit of matrix models. We assign the concept of energy to each matrix element and integrate the most highest energy to get a new matrix model which has reduced rank. By regarding this procedure as a renormalization group, we deduce the critical exponents in the large‐N limit by elaborating fixed points of renormalization group transformation. For consistency of our method, we compare our result to that obtained by another method and find nice agreement.

Quantum aspects of GMS solutions of non-commutative field theory and large N limit of matrix models

We investigate quantum aspects of Gopakumar-Minwalla-Strominger (GMS) solutions of noncommutative field theory (NCFT) at large noncommutativity limit, $\theta \to \infty$. Building upon a quantitative map between operator formulation of 2-(respectively, (2+1))-dimensional NCFTs and large $N$ matrix models of $c=0$ (respectively, $c=1$) noncritical strings, we show that GMS solutions are quantum mechanically sensible only if we make appropriate joint scaling of $\theta$ and $N$. For 't Hooft's planar scaling, GMS solutions are replaced by large $N$ saddle-point solutions. GMS solutions are recovered from saddle-point solutions at small 't Hooft coupling regime, but are destabilized at large 'tHooft coupling regime by quantum effects. We make comparisons between these large $N$ effects and recently studied infrared effects in NCFTs. We estimate U(N) symmetry breaking gradient effects and argue that they are suppressed only at small 't Hooft coupling regime.

HALF-OPERATORISATION OF A FUNCTION AND PHYSICAL APPLICATIONS

We define an half-operatorisation of a function by using its Fourier series such that the Fourier series basis changes into exponential phase operators with the Fourier coefficients remaining intact. We define and discuss the half-operator version of various Fourier Series properties. One application is the master field describing the large N limits of matrix models. We also define an half-operatorisation using gauge equivalent exponential phase operators which have a shifted vacuum.

The master field for QCD and q-deformed quantum field theory

The master fields for the large- N limit of matrix models and gauge theory are constructed. The master fields satisfy to standard equations of relativistic field theory but the fields are quantized according to a new rule. To define the master field we use the Yang-Feldman equation with a free field quantized in the Boltzmannian Fock space. The master field for gauge theory does not take values in a finite-dimensional Lie algebra, however, there is a non-Abelian gauge symmetry. For the construction of the master field it is essential to work in Minkowski space-time and to use the Wightman correlation functions. The BRST quantization of the master field for gauge theory and a loop equation are considered.

Mastering the master field

The basic concepts of non-commutative probability theory are reviewed and applied to the large- N limit of matrix models. We argue that this is the appropriate framework for constructing the master field in terms of which large- N theories can be written. We explicitly construct the master field in a number of cases including QCD 2. There we both give an explicit construction of the master gauge field and construct master loop operators as well. Most important we extend these techniques to deal with the general matrix model, in which the matrices do not have independent distributions and are coupled. We can thus construct the master field for any matrix model, in a well defined Hilbert space, generated by a collection of creation and annihilation operators—one for each matrix variable—satisfying the Cuntz algebra. We also discuss the equations of motion obeyed by the master field.

The double-well phase of matrix models: the large-N general solution

The authors provide the general solution of the large-N limit of matrix models with even polynomial potential in the phase with two minimal. The solution is given both in the saddle point and in the orthogonal polynomials approaches.