The quantum algebra of observables of closed bosonic strings moving in 1 3±dimensional Minkowski space is constructed up to degree five. All independent relations of degree four are computed; they involve three as yet undetermined parameters. Definitions and symbols are used as introduced in the above-mentioned article.

We proceed with the investigation of a method of quantization of the observable sector of closed bosonic strings. For the presentation of the quantum algebra of observables the construction cycle concerning elements of order 6 has been carried out. We have computed the quantum corrections to the only generating relation of order 6 . This relation is of spin-parity J P = 0 + . We found that the quantum corrections to this relation break the semidirect splitting of the classical algebra into an abelian, infinitely generated subalgebra a and a non-abelian, finitely generated subalgebra U. We have established that there are no ("truly independent") generating relations of order 7 .

To postulate correspondence for the observables only is a promising approach to a fully satisfying quantization of the Nambu–Goto string. The relationship between the Poisson algebra of observables and the corresponding quantum algebra is established in the language of generators and relations. A very valuable tool is the transformation to the string's rest frame, since a substantial part of the relations are solved. It is the aim of this paper to clarify the relationship between the fully covariant and the rest frame description. Both in the classical and in the quantum case, an efficient method for recovering the covariant algebra from the one in the rest frame is described. Restrictions on the quantum defining relations are obtained, which are not taken into account when one postulates correspondence for the rest frame algebra. For the part of the algebra studied up to now in explicit computations, these further restrictions alone determine the quantum algebra uniquely — in full consistency with the further restrictions found in the rest frame.

The quantum algebra of observables of closed bosonic strings moving in 1 + 3–dimensional Minkowski space is constructed up to degree five. All independent relations of degree four are computed; they involve three as yet undetermined parameters. Definitions and symbols are used as introduced in the above‐mentioned article.

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