Discrete Strings and Deterministic Cellular Strings

Abstract

A discrete string theory --a theory of embeddings from ${\bf Z}\times {\bf Z}_C\to {\bf R}^D$, where $C$ is the number of components of the string-- is explored. The closure of the algebra of constraints (`${\bf Z}_C$-Virasoro algebra') is exhibited. The ${\bf Z}_C$-Virasoro `algebra' is shown to be anomaly free in arbitrary number of target space dimensions. We prove the existence of a (manifestly unitary) light-cone gauge with anomaly free Lorentz algebra in any dimensions. The analog of vertex operators are introduced and the physical spectrum is analysed. There are an infinite number of higher-level states repeating a certain mass pattern and leading to an infinite degeneracy. The connection with the continuum string theory (in $D=26$) is investigated. Independently, following a method recently introduced by 't Hooft based on Hilbert-space extension of deterministic systems, a particular one-dimensional cellular automaton submitted to a deterministic evolution is shown to reduce to a massless scalar field theory at long distance scales. This automaton is utilized to define a (`cellular') string theory with world-sheet variables evolving under deterministic rules, which in the framework of `first-quantization' corresponds to the ${\bf Z}_C$ string theory mentioned above. We show that in this theory also the target space motion of free strings is governed by deterministic laws. Finally, we discuss a model for (off-shell) interacting strings where space-time determinism is fully restored.