On Field Theories of Loops

TL;DR

This paper develops a field theory of loops using stochastic quantization of matrix models, connecting it to non-critical string theory, dynamical triangulation, and the Virasoro algebra, for both discretized and continuum cases.

Contribution

It introduces a novel approach to loop field theories via stochastic quantization, linking matrix models to string field theory and dynamical triangulation.

Findings

Establishes equivalence between stochastic quantization and transfer-matrix formalism.

Derives the non-critical string field theory from matrix models.

Clarifies the origin of the Virasoro algebra in loop field theories.

Abstract

We apply stochastic quantization method to matrix models for the second quantization of loops in both discretized and continuum levels. The fictitious time evolution described by the Langevin equation is interpreted as the time evolution in a field theory of loops. The corresponding Fokker-Planck hamiltonian defines a non-critical string field theory. We study both orientable and non-orientable interactions of loops in terms of matrix models and take the continuum limit for one-matrix case. As a consequence, we show the equivalence of stochastic quantization of matrix models in loop space to the transfer-matrix formalism in dynamical triangulation of random surfaces. We also clarifies the origin of Virasoro algebra in this context.