Quantized Noncommutative Geometry from Multitrace Matrix Models

TL;DR

This paper introduces a novel mechanism for emergent quantum geometry using multitrace scalar matrix models, contrasting with traditional gauge theory approaches, and employs statistical physics tools to analyze phase transitions and geometry emergence.

Contribution

It proposes a new way for noncommutative geometry to emerge from one-matrix multitrace models, expanding understanding of quantum gravity and matrix model phase structures.

Findings

Emergence of geometry linked to phase transitions in matrix models

Critical exponents and Wigner's law used to determine dimension and metric

Monte Carlo and saddle point analyses support the scenario

Abstract

In this article the geometry of quantum gravity is quantized in the sense of being noncommutative (first quantization) but it is also quantized in the sense of being emergent (second quantization). A new mechanism for quantum geometry is proposed in which noncommutative geometry can emerge from "one-matrix multitrace scalar matrix models" by probing the statistical physics of commutative phases of matter. This is in contrast to the usual mechanism in which noncommutative geometry emerges from "many-matrix singletrace Yang-Mills matrix models" by probing the statistical physics of noncommutative phases of gauge theory. In this novel scenario quantized geometry emerges in the form of a transition between the two phase diagrams of the real quartic matrix model and the noncommutative scalar phi-four field theory. More precisely, emergence of the geometry is identified here with the emergence of the uniform-ordered phase and the corresponding commutative (Ising) and noncommutative (stripe) coexistence lines. The critical exponents and the Wigner's semicircle law are used to determine the dimension and the metric respectively. Arguments from the saddle point equation, from Monte Carlo simulation and from the matrix renormalization group equation are provided in support of this scenario.