Heisenberg relation(s) for spectral non commutative spacetime?

The first Heisenberg-like commutation relation for a spectral non commutative spacetime / La première relation de commutation de type Heisenberg pour un espacetemps spectral non commutatif


In this letter we shall take equation (2), and its two sided refinement (4) below using the real structure, as a geometric analogue of the Heisenberg commutation relations [p,q]=i where D plays the role of p (momentum) and Y the role of q (coordinate) and use it as a starting point of quantization of geometry with quanta corresponding to irreducible representations of the operator relations. The above integrality result on the volume is a hint of quantization of geometry. We first use the one-sided (2) as the equations of motion of some field theory on M and describe the solutions as follows. (For details and proofs see [arxiv.org/abs/1411.0977])... 
Each geometric quantum is a sphere of arbitrary shape and unit volume (in Planck units).
It would seem at this point that only disconnected geometries fit in this framework but this is ignoring an es-sential piece of structure of the NCG framework, which allows one to refine (2). It is the real structure J, an antilinear isometry in the Hilbert space H which is the algebraic counterpart of charge conjugation. This leads to refine the quantization condition by taking J into account as the two-sided equation


where E is the spectral projection for {1,i}⊂C of the double slash Y=Y+⊕Y-∈ C(M,C+⊕C-). It is the classification of finite geometries of [4] which suggested to use the direct sum C+⊕C-  of two Clifford algebras and the algebra C(M,C+⊕C-). It turns out moreover that in dimension 4 one has C+=M2(H) and C-=M2(C) which is in perfect agreement with the algebraic constituents of the Standard Model
(last revised 9 Sep 2014 (this version, v2))


Verstehen Sie nur Bahnhof? Vergessen Sie die Bahn und halten nur den Hof! / Vous n'y pigez que dalle? Oubliez la trajectoire et ne gardez que le halo 

Über den anschaulichen Inhalt der quantentheoretischen Kinematik und MechanikHeisenberg, 23. März 1927
//ajout d'une référence le 9 novembre 2014


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