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Non-Archimedean strings and Bruhat-Tits trees. (English) Zbl 0676.22006
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.
Reviewer: P.Maślanka

MSC:
22E35 Analysis on \(p\)-adic Lie groups
81T08 Constructive quantum field theory
83E99 Unified, higher-dimensional and super field theories
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