Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes
- ITEP, Bolshaya Cheremushkinskaya 25, 117218 Moscow (Russian Federation)
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.
- OSTI ID:
- 21505027
- Journal Information:
- Physical Review. D, Particles Fields, Vol. 83, Issue 4; Other Information: DOI: 10.1103/PhysRevD.83.045021; (c) 2011 American Institute of Physics; ISSN 0556-2821
- Country of Publication:
- United States
- Language:
- English
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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