Quantum Gravity and String Theory

   

On Modified Weyl-Heisenberg Algebras, Noncommutativity Matrix-Valued Planck Constant and QM in Clifford Spaces

Authors: Carlos Castro

A novel Weyl-Heisenberg algebra in Clifford-spaces is constructed that is based on a matrix-valued HAB extension of Planck's constant. As a result of this modifiedWeyl-Heisenberg algebra one will no longer be able to measure, simultaneously, the pairs of variables (x, px); (x, py); (x, pz); (y, px), ... with absolute precision. New Klein-Gordon and Dirac wave equations and dispersion relations in Clifford-spaces are presented. The latter Dirac equation is a generalization of the Dirac-Lanczos-Barut-Hestenes equation. We display the explicit isomorphism between Yang's Noncommutative space-time algebra and the area-coordinates algebra associated with Clifford spaces. The former Yang's algebra involves noncommuting coordinates and momenta with a minimum Planck scale λ (ultraviolet cutoff) and a minimum momentum p = ℏ/R (maximal length R, infrared cutoff ). The double-scaling limit of Yang's algebra λ → 0, R → ∞, in conjunction with the large n → ∞ limit, leads naturally to the area quantization condition λR = L2 = nλ2 ( in Planck area units ) given in terms of the discrete angular-momentum eigenvalues n. It is shown how Modified Newtonian dynamics is also a consequence of Yang's algebra resulting from the modified Poisson brackets. Finally, another noncommutative algebra ( which differs from the Yang's algebra ) and related to the minimal length uncertainty relations is presented . We conclude with a discussion of the implications of Noncommutative QM and QFT's in Clifford-spaces.

Comments: 22 pages, This article appeared in the Journal of Physics A : Math. Gen 39 (2006) 14205-14229.

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[v1] 28 Aug 2009

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