### 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 sumC_{+}⊕C_{- }of two Cliﬀord algebras and the algebraC^{∞}(M,C_{+}⊕C_{-}).It turns out moreover that in dimension 4 one hasC_{+}=M_{2}(H) and C_{-}=M_{2}(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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