The Operator Quantization of the Open Bosonic String: Field Algebra

Our aim in this paper is to make explicit the operator theory of the heuristic open Bosonic string and to abstract a suitable field algebra for the string. This is done on a Fock-Krein space and we examine integrability and J-unitary implementability of all the defining transformations of the string, i.e. time translations, gauge transformations and Poincare transformations. The results obtained agree partially with those of Bowick and Rajeev, i.e. the gauge transformations do not leave the Fock-Krein complex structure invariant. Once we obtained integrated transformation groups on a suitable symplectic space for the infinitesimal transformations of the string, and proved implementability of these for the FockKrein representation, we are then free to define an abstract C*-algebra carrying all the algebraic information of the string, and to examine different representations.