## 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 H^{AB}
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, p_{x}); (x, p_{y}); (x, p_{z}); (y, p_{x}), ... 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 = L^{2} = nλ^{2} ( 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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### Submission history

[v1] 28 Aug 2009

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