THE PRESENTATION OF THE QUANTUM ALGEBRA OF OBSERVABLES OF THE CLOSED BOSONIC STRING IN 1+3 DIMENSIONS: THE EXACT QUANTIZED GENERATING RELATIONS OF ORDERS ℏ⁶ AND ℏ⁷

Abstract

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 .

Algebraic Quantization of the Closed Bosonic String

“…Note added in proof. A failure of the semidirect product structure [a, U] ⊂ U in the intrinsic approach to quantization was reported recently in [18].…”

The Algebra of Observables of the Closed Bosonic String in (1+3)-Dimensional Minkowski Space: Updating the Structural Analysis

“…There is still no general proof that the Poisson algebra h possesses the structure corresponding to the decomposition of the Lie algebra G into a semi-direct sum of the abelian Lie algebra A, generated by the exceptional elements, and the Lie algebra G rem . In fact, in view of the results of [5] on the quantum algebra of string observables, interest has somewhat weakened regarding the existence a Lie algebra g with the following properties:…”

“…The cycles carried out so far are of degrees 2 to 5 corresponding to the powers 4 to 7 in Planck's constant. For the symbolic computations and their results, see [7] concerning the degrees 2 and 3, [8] and [5] concerning the degrees 4 and 5, respectively.…”

“…(b) the structural similarity requirement applied to the semi-direct splitting g = a ⋉ U cannot be upheld [5].…”

Structure Analysis of the Pohlmeyer-Rehren Lie Algebra and Adaptations of the Hall Algorithm to Non-Free Graded Lie Algebras

“…It was however demonstrated by G. Handrich, C. Paufler, J. B. Tausk and M. Walter that the quantum analogon ĥ = â Û of conjecture 3.2.8 is false [HPTW02,p. 4].…”