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Generalized uncertainty principle from quantum geometry. (English) Zbl 0981.83021

The generalized uncertainty principle of string theory \(\Delta x\geq \hbar /2\Delta p+\alpha c^{-3}G\Delta p\) (where \(G\) is the Newton constant and \( \alpha \) is a constant) is derived in the framework of quantum geometry by taking into account the existence of an upper limit on the acceleration of massive particles. Its formulation is based on the fact that the position and momentum operators are represented as covariant derivatives with an appropriate connection in the eight-dimensional manifold and the quantization is geometrically interpreted as curvature of phase space. A consequence of this geometric approach is the existence of maximal acceleration \({\mathcal A}\) defined as the upper limit to the proper acceleration experienced by massive particles along their worldlines, which can be interpreted as mass dependent, \({\mathcal A}_{m}=2mc^{3}/\hbar \) (\(m\) is the mass of particle), or as a universal constant, \({\mathcal A} =m_{P}c^{3}/\hbar \) (\(m_{P}\) is the Planck mass). Since the regime of validity of the generalized uncertainty principle is at Planck scales, in order to derive it from quantum geometry, the maximal acceleration depending on the Planck mass is considered.

MSC:

83C45 Quantization of the gravitational field
83E05 Geometrodynamics and the holographic principle
83E15 Kaluza-Klein and other higher-dimensional theories
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