Matrix generalizations of some dynamic field theories

TL;DR

This paper introduces matrix generalizations of the Navier--Stokes and KPZ equations, analyzing their properties in the large N limit through perturbative methods, and highlighting the complexity of solving these models fully.

Contribution

It proposes matrix extensions of key fluid and interface growth equations, exploring their behavior in the large N limit with a one-loop analysis.

Findings

Perturbative expansions simplify at large N due to planar graphs.

One-loop analysis performed for the matrix models.

Full solutions of the matrix theories remain elusive.

Abstract

We introduce matrix generalizations of the Navier--Stokes (NS) equation for fluid flow, and the Kardar--Parisi--Zhang (KPZ) equation for interface growth. The underlying field, velocity for the NS equation, or the height in the case of KPZ, is promoted to a matrix that transforms as the adjoint representation of $SU(N)$. Perturbative expansions simplify in the $N\to\infty$ limit, dominated by planar graphs. We provide the results of a one--loop analysis, but have not succeeded in finding the full solution of the theory in this limit.