skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes

Abstract

We propose a stochastic method for solving Schwinger-Dyson equations in large-N quantum field theories. Expectation values of single-trace operators are sampled by stationary probability distributions of the so-called nonlinear random processes. The set of all the histories of such processes corresponds to the set of all planar diagrams in the perturbative expansions of the expectation values of singlet operators. We illustrate the method on examples of the matrix-valued scalar field theory and the Weingarten model of random planar surfaces on the lattice. For theories with compact field variables, such as sigma models or non-Abelian lattice gauge theories, the method does not converge in the physically most interesting weak-coupling limit. In this case one can absorb the divergences into a self-consistent redefinition of expansion parameters. A stochastic solution of the self-consistency conditions can be implemented as a 'memory' of the random process, so that some parameters of the process are estimated from its previous history. We illustrate this idea on the two-dimensional O(N) sigma model. The extension to non-Abelian lattice gauge theories is discussed.

Authors:
 [1]
  1. ITEP, Bolshaya Cheremushkinskaya 25, 117218 Moscow (Russian Federation)
Publication Date:
OSTI Identifier:
21505027
Resource Type:
Journal Article
Journal Name:
Physical Review. D, Particles Fields
Additional Journal Information:
Journal Volume: 83; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevD.83.045021; (c) 2011 American Institute of Physics; Journal ID: ISSN 0556-2821
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; COUPLING; DISTRIBUTION; EQUATIONS; EXPANSION; GAUGE INVARIANCE; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; PROBABILITY; QUANTUM FIELD THEORY; RANDOMNESS; SCALAR FIELDS; SCHWINGER FUNCTIONAL EQUATIONS; SIGMA MODEL; SIMULATION; STOCHASTIC PROCESSES; BOSON-EXCHANGE MODELS; DIFFERENTIAL EQUATIONS; FIELD THEORIES; INVARIANCE PRINCIPLES; MATHEMATICAL MODELS; PARTICLE MODELS; PERIPHERAL MODELS

Citation Formats

Buividovich, P V, and JINR, Joliot-Curie 6, 141980 Dubna, Moscow region. Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes. United States: N. p., 2011. Web. doi:10.1103/PHYSREVD.83.045021.
Buividovich, P V, & JINR, Joliot-Curie 6, 141980 Dubna, Moscow region. Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes. United States. https://doi.org/10.1103/PHYSREVD.83.045021
Buividovich, P V, and JINR, Joliot-Curie 6, 141980 Dubna, Moscow region. 2011. "Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes". United States. https://doi.org/10.1103/PHYSREVD.83.045021.
@article{osti_21505027,
title = {Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes},
author = {Buividovich, P V and JINR, Joliot-Curie 6, 141980 Dubna, Moscow region},
abstractNote = {We propose a stochastic method for solving Schwinger-Dyson equations in large-N quantum field theories. Expectation values of single-trace operators are sampled by stationary probability distributions of the so-called nonlinear random processes. The set of all the histories of such processes corresponds to the set of all planar diagrams in the perturbative expansions of the expectation values of singlet operators. We illustrate the method on examples of the matrix-valued scalar field theory and the Weingarten model of random planar surfaces on the lattice. For theories with compact field variables, such as sigma models or non-Abelian lattice gauge theories, the method does not converge in the physically most interesting weak-coupling limit. In this case one can absorb the divergences into a self-consistent redefinition of expansion parameters. A stochastic solution of the self-consistency conditions can be implemented as a 'memory' of the random process, so that some parameters of the process are estimated from its previous history. We illustrate this idea on the two-dimensional O(N) sigma model. The extension to non-Abelian lattice gauge theories is discussed.},
doi = {10.1103/PHYSREVD.83.045021},
url = {https://www.osti.gov/biblio/21505027}, journal = {Physical Review. D, Particles Fields},
issn = {0556-2821},
number = 4,
volume = 83,
place = {United States},
year = {Tue Feb 15 00:00:00 EST 2011},
month = {Tue Feb 15 00:00:00 EST 2011}
}