Abstract
Stabilization, by deformation, of the Poincaré-Heisenberg algebra requires both the introduction of a fundamental lentgh and the noncommutativity of translations which is associated to the gravitational field. The noncommutative geometry structure that follows from the deformed algebra is studied both for the non-commutative tangent space and the full space with gravity. The contact points of this approach with the work of David Finkelstein are emphasized.
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Notes
\(\overline {p}^{\mu }, \overline {\Im } \) denote the tangent space (R→∞) limits of the operators, not be confused with the physical p μ,I operators. According to the deformation-stability principle they are stable physical operators only when R is finite, that is, when gravity is turned on.
2 Notice that additional components are not necessarily required for spinors because the Clifford algebras C(3,2) or C(4,1) both have four-dimensional representations.
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Appendix: Irreducible Representations of the Space-Time Algebra
Appendix: Irreducible Representations of the Space-Time Algebra
For S O(3,2)(𝜖 4=+1) a way to characterize the irreducible representations of this groups is to consider its action on functions on a V 3,2 cone, with coordinates
Then, on this cone, consider a space S σ,ε of functions satisfying the homogeneity conditions [23]
\(\sigma \in \mathbb {R}\) and ε={0,1}. In M σ,ε the group operators act as follows
Because of (35) the functions are uniquely characterized by their values in the (s=0)Γ1 contour. This contour is topologically S 2×S 1. The spaces of homogeneous functions on this contour will be denoted \(S^{{\Gamma }_{1}}\). Denote by g i j (𝜃) a rotation in the plane ij and by \(g_{ij}^{\prime } \left (t\right ) \) a hyperbolic rotations in the plane ij.
Given \(f\in S^{{\Gamma }_{1}}\), using (35) and (36) one obtains for an hyperbolic rotation in the 1, 4 plane
Similar expressions are obtained for the other elementary operations. From these one obtains, as infinitesimal generators, a representation for the generators of the algebra {X μ ,M μ ν } as operators in \(S^{{\Gamma }_{1}}\)
These representations are irreducible for non-integer σ. There are also conditions for unitary of the representations, but this is not so important because only the M μ ν (μ,ν=0,1,2,3) are generators of symmetry operations.
A similar construction is possible for S O(4,1)(𝜖 4=−1) with functions on a V 4,1(𝜖 5=−1) cone, with coordinates
the contour Γ2(s=0) in this case being topologically S 3.
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Mendes, R.V. The Geometry of Noncommutative Space-Time. Int J Theor Phys 56, 259–269 (2017). https://doi.org/10.1007/s10773-016-3166-9
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DOI: https://doi.org/10.1007/s10773-016-3166-9