Generalized constructive tree weights
Abstract
The Loop Vertex Expansion (LVE) is a quantum field theory (QFT) method which explicitly computes the Borel sum of Feynman perturbation series. This LVE relies in a crucial way on symmetric tree weights which define a measure on the set of spanning trees of any connected graph. In this paper we generalize this method by defining new tree weights. They depend on the choice of a partition of a set of vertices of the graph, and when the partition is non-trivial, they are no longer symmetric under permutation of vertices. Nevertheless we prove they have the required positivity property to lead to a convergent LVE; in fact we formulate this positivity property precisely for the first time. Our generalized tree weights are inspired by the Brydges-Battle-Federbush work on cluster expansions and could be particularly suited to the computation of connected functions in QFT. Several concrete examples are explicitly given.
- Authors:
-
- LPT, CNRS UMR 8627, Univ. Paris 11, 91405 Orsay Cedex, France and Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Ontario N2L 2Y5, Waterloo (Canada)
- Université Paris 13, Sorbonne Paris Cité, 99, Avenue Jean-Baptiste Clément LIPN, Institut Galilée, CNRS UMR 7030, F-93430 Villetaneuse, France and Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O.B. MG-6, 077125 Magurele (Romania)
- Publication Date:
- OSTI Identifier:
- 22250773
- Resource Type:
- Journal Article
- Journal Name:
- Journal of Mathematical Physics
- Additional Journal Information:
- Journal Volume: 55; Journal Issue: 4; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; CALCULATION METHODS; CLUSTER EXPANSION; DIAGRAMS; FUNCTIONS; GRAPH THEORY; PARTITION; PERTURBATION THEORY; QUANTUM FIELD THEORY; SYMMETRY
Citation Formats
Rivasseau, Vincent, and Tanasa, Adrian. Generalized constructive tree weights. United States: N. p., 2014.
Web. doi:10.1063/1.4871176.
Rivasseau, Vincent, & Tanasa, Adrian. Generalized constructive tree weights. United States. https://doi.org/10.1063/1.4871176
Rivasseau, Vincent, and Tanasa, Adrian. 2014.
"Generalized constructive tree weights". United States. https://doi.org/10.1063/1.4871176.
@article{osti_22250773,
title = {Generalized constructive tree weights},
author = {Rivasseau, Vincent and Tanasa, Adrian},
abstractNote = {The Loop Vertex Expansion (LVE) is a quantum field theory (QFT) method which explicitly computes the Borel sum of Feynman perturbation series. This LVE relies in a crucial way on symmetric tree weights which define a measure on the set of spanning trees of any connected graph. In this paper we generalize this method by defining new tree weights. They depend on the choice of a partition of a set of vertices of the graph, and when the partition is non-trivial, they are no longer symmetric under permutation of vertices. Nevertheless we prove they have the required positivity property to lead to a convergent LVE; in fact we formulate this positivity property precisely for the first time. Our generalized tree weights are inspired by the Brydges-Battle-Federbush work on cluster expansions and could be particularly suited to the computation of connected functions in QFT. Several concrete examples are explicitly given.},
doi = {10.1063/1.4871176},
url = {https://www.osti.gov/biblio/22250773},
journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 4,
volume = 55,
place = {United States},
year = {Tue Apr 15 00:00:00 EDT 2014},
month = {Tue Apr 15 00:00:00 EDT 2014}
}