On modified Weyl–Heisenberg algebras, noncommutativity, matrix-valued Planck constant and QM in Clifford spaces

Abstract

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 modified Weyl-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 =h/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.

On the noncommutative and nonassociative geometry of octonionic space time, modified dispersion relations and grand unification

“…Next we will review how the minimal length string uncertainty relations can be obtained from polyparticle dynamics in Clifford-spaces (C-spaces) [50]. The truly C-space invariant norm of a momentum poly-vector is defined (after introducing suitable powers of the Planck mass that is set to unity in order to match units)…”

“…The mass-shell condition in C-space, after imposing the constraints among the poly-vector valued components, yields an effective mass M = mf (Λm/h). The generalized de Broglie relations, which are no longer linear, are [50] |P ef f ective | = |p| f (Λm/h) =h ef f ective (k 2 ) |k|.…”

“…QM in Clifford-spaces (C-spaces) is very rich with many novelties [50]. A novel Weyl-Heisenberg algebra in Clifford-spaces was constructed that is based on a matrix-valued H AB extension of Planck's constant [50].…”

“…A novel Weyl-Heisenberg algebra in Clifford-spaces was constructed that is based on a matrix-valued H AB extension of Planck's constant [50]. As a result of this modified Weyl-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.…”

The Euclidean gravitational action as black hole entropy, singularities, and spacetime voids

“…The most general p-brane uncertainty relations based on a unified treatment of p-branes, for all values of p, in Clifford spaces was derived in [61]. The physical interpretation of the phase space null horizon at r = 0, null from the perspective of the full fledged phase space metric g µν (x, p), or Finsler metric g µν (x, v) is that it is the "attractor" region where a test particle (of mass m ) approaches asymptotically as it moves in the gravitational background produced by the point mass M located at r = 0 (when m <<< M ) .…”