Cluster Convergence Theorem
| dc.creator | Austin, Chris | |
| dc.date | 2005-09-04 | |
| dc.date.accessioned | 2026-07-07T04:18:34Z | |
| dc.date.available | 2026-07-07T04:18:34Z | |
| dc.description | A power-counting theorem is presented, that is designed to play an analogous role, in the proof of a BPHZ convergence theorem, in Euclidean position space, to the role played by Weinberg's power-counting theorem, in Zimmermann's proof of the BPHZ convergence theorem, in momentum space. If $x$ denotes a position space configuration, of the vertices, of a Feynman diagram, and $σ$ is a real number, such that $0 < σ< 1$, a $σ$-cluster, of $x$, is a nonempty subset, $J$, of the vertices of the diagram, such that the maximum distance, between any two vertices, in $J$, is less than $σ$, times the minimum distance, from any vertex, in $J$, to any vertex, not in $J$. The set of all the $σ$-clusters, of $x$, has similar combinatoric properties to a forest, and the configuration space, of the vertices, is cut up into a finite number of sectors, classified by the set of all their $σ$-clusters. It is proved that if, for each such sector, the integrand can be bounded by an expression, that satisfies a certain power-counting requirement, for each $σ$-cluster, then the integral, over the position, of any one vertex, is absolutely convergent, and the result can be bounded by the sum of a finite number of expressions, of the same type, each of which satisfies the corresponding power-counting requirements. | |
| dc.description | LaTeX2e transcription of 1988 paper, with references added. 42 pages. Needs amsfonts. For an application of the theorem to a BPHZ convergence proof, in Euclidean position space, without exponentiating the propagators, see http://web.ukonline.co.uk/chrisaustin/ | |
| dc.identifier | https://arxiv.org/abs/hep-th/0509033 | |
| dc.identifier | http://arxiv.org/abs/hep-th/0509033 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/53229 | |
| dc.subject | High Energy Physics - Theory | |
| dc.title | Cluster Convergence Theorem | |
| dc.type | text |