## 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.

**Download:** **PDF**

### Submission history

[v1] 28 Aug 2009

**Unique-IP document downloads:** 221 times

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*