Non-Archimedean strings and Bruhat-Tits trees
type: Item
instance of: scholarly article
title: Non-Archimedean strings and Bruhat-Tits trees
zbMATH Open document ID: 0676.22006
DOI: 10.1007/BF01238811
published in: Communications in Mathematical Physics
publication date: 1989-01-01T00:00:00Z
review text: In the Nambu-Goto approach the string action is given by the area swept by the world sheet. As it was noticed by Freund and Olson the world sheet by itself is unobservable. Starting from this remark they proposed to choose it to be a manifold over the p-adic number field \(Q_ p\). They also gave a formal recipe for calculating tree bosonic string amplitudes. On the other hand there exists the alternative approach to string theory, due to Polyakov, more convenient for quantization purposes: one considers two-dimensional theory with its fields taking values in a D-dimensional space (space-time) and integrates over that fields as well as the metrics. In the paper under review the author poses and solves the question whether it is possible to obtain the Freund-Olson results within Polyakov's approach, i.e. by constructing the relevant field theory on a p-adic manifold. The starting point is the observation that in the standard approach the field of real numbers plays the role of a boundary of an open string world sheet. It is well known that the field \(Q_ p\) may be identified with the boundary of a discrete homogeneous space \(T_ p\)- the Bruhat-Tits tree which therefore becomes a natural candidate for the interior of the open p-adic string world sheet. To construct the relevant field theory the Laplace operator on B-T tree is defined, the harmonic functions are constructed and the Dirichlet and Neumann problems are solved. Then the field theory on B-T tree with simple quadratic action is proposed and proven to lead to Freund-Olson amplitudes. The relation to the alternative approach based on non-local action is also discussed.
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 83E99
zbMATH DE Number: 4107121
zbMATH Keywords: p-adic number field
zbMATH Keywords: bosonic string amplitudes
zbMATH Keywords: string theory
zbMATH Keywords: p-adic manifold
zbMATH Keywords: string world sheet
zbMATH Keywords: Bruhat-Tits tree
zbMATH Keywords: p-adic string
zbMATH Keywords: harmonic functions
zbMATH Keywords: Dirichlet and Neumann problems
author: A. Zabrodin
reviewed by: Paweł Maślanka
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4107121