The Geometry of Noncommutative Space-Time

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

  1. 1.

    \(\overline {p}^{\mu }, \overline {\Im } \) 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. 2.

    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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Correspondence to R. Vilela Mendes.

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

$$\begin{array}{@{}rcl@{}} y_{1} & = & e^{s}\cos \varphi_{2} \\ y_{2} & = & e^{s}\sin \varphi_{2}\cos \varphi_{1} \\ y_{3} & = & e^{s}\sin\varphi_{2}\sin \varphi_{1} \\ y_{4} & = & e^{s}\sin \theta_{1} \\ y_{0} & = & e^{s}\cos \theta_{1} \end{array} $$
(34)

Then, on this cone, consider a space S σ,ε of functions satisfying the homogeneity conditions [23]

$$ f\left( ax\right) =\left\vert a\right\vert^{\sigma} sign^{\varepsilon} af\left( x\right) $$
(35)

\(\sigma \in \mathbb {R}\) and ε={0,1}. In M σ,ε the group operators act as follows

$$ T\left( g\right) f\left( x\right) =f\left( g^{-1}x\right) $$
(36)

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

$$\begin{array}{@{}rcl@{}} T^{\sigma} \left( g_{14}^{\prime} \left( t\right) \right) f\left( \varphi_{1},\varphi_{2},\theta_{1}\right) &=&\left\vert a\right\vert^{\sigma /2}f\left( \varphi_{1},\varphi_{2}^{\prime} ,\theta_{1}^{\prime} \right) \\ \left\vert a\right\vert &=&\left\{ \sin^{2}\theta_{1}+\left( \cos \theta_{1}\cosh t-\cos \varphi_{2}\sinh t\right)^{2}\right\}^{1/2} \\ \cos \varphi_{2}^{\prime} &=&\frac{\cos \varphi_{2}\cosh t-\cos \theta_{1}\sinh t}{\left\vert a\right\vert} \\ \cos \theta_{1}^{\prime} &=&\frac{\cos \theta_{1}\cosh t-\cos \varphi_{2}\sinh t}{\left\vert a\right\vert} \end{array} $$
(37)

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 \(S^{{\Gamma }_{1}}\)

$$\begin{array}{@{}rcl@{}} iX_{1} &=&iM_{14}=\sigma \cos \theta_{1}\cos \varphi_{2}-\sin \theta_{1}\cos \varphi_{2}\frac{\partial} {\partial\theta_{1}}-\cos \theta_{1}\sin \varphi_{2}\frac{\partial} {\partial \varphi_{2}}\\ iX_{2} &=& iM_{24}=\sigma\cos \theta_{1}\sin \varphi_{2}\cos \varphi_{1}-\sin \theta_{1}\sin \varphi_{2}\cos \varphi_{1}\frac{\partial}{\partial \theta_{1}}\\ &&+\cos \theta_{1}\cos \varphi_{2}\cos \varphi_{1}\frac{\partial}{\partial \varphi_{2}}-\frac{\cos \theta_{1}\sin \varphi_{1}}{\sin \varphi_{2}}\frac{\partial} {\partial \varphi_{1}}\\ iX_{3} &=& iM_{34}=\sigma \cos \theta_{1}\sin \varphi_{2}\sin \varphi_{1}-\sin \theta_{1}\sin \varphi_{2}\sin \varphi_{1}\frac{\partial}{\partial \theta_{1}}\\ && + \cos \theta_{1}\cos \varphi_{2}\sin \varphi_{1}\frac{\partial}{\partial \varphi_{2}}+\frac{\cos \theta_{1}\cos \varphi_{1}}{\sin \varphi_{2}} \frac{\partial}{\partial \varphi_{1}}\\ iX_{0} &=& iM_{04}=\frac{\partial}{\partial \theta_{1}} \end{array} $$
$$\begin{array}{@{}rcl@{}} iM_{12} &=& -\cos \varphi_{1}\frac{\partial}{\partial \varphi_{2}}+\frac{\cos \varphi_{2}\sin \varphi_{1}}{\sin \varphi_{2}}\frac{\partial}{\partial \varphi_{1}}\\ iM_{13} &=& -\sin \varphi_{1}\frac{\partial} {\partial \varphi_{2}}-\frac{\cos \varphi_{2}\cos \varphi_{1}}{\sin \varphi_{2}}\frac{\partial}{\partial \varphi_{1}}\\ iM_{23} &=& -\frac{\partial}{\partial \varphi_{1}} \\ iM_{10} &=& \sigma \sin \theta_{1}\cos \varphi_{2}+\cos \theta_{1}\cos\varphi_{2}\frac{\partial}{\partial \theta_{1}}-\sin \theta_{1}\sin \varphi_{2}\frac{\partial}{\partial \varphi_{2}}\\ iM_{20} &=&\sigma \sin \theta_{1}\sin \varphi_{2}\cos \varphi_{1}+\cos \theta_{1}\sin \varphi_{2}\cos\varphi_{1}\frac{\partial}{\partial \theta_{1}}\\ &&+\sin \theta_{1}\cos \varphi_{2}\cos \varphi_{1}\frac{\partial}{\partial \varphi_{2}} + \frac{\sin \theta_{1}\sin \varphi_{1}}{\sin \varphi_{2}}\frac{\partial}{\partial \varphi_{1}}\\ iM_{30} &=&\sigma \sin \theta_{1}\sin \varphi_{2}\sin \varphi_{1}+\cos \theta_{1}\sin \varphi_{2}\sin \varphi_{1}\frac{\partial}{\partial \theta_{1}}\\ &&+\sin \theta_{1}\cos \varphi_{2}\sin \varphi_{1}\frac{\partial}{\partial \varphi_{2}}+\frac{\sin \theta_{1}\cos \varphi_{1}}{\sin \varphi_{2}}\frac{\partial}{\partial \varphi_{1}} \end{array} $$
(38)

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

$$\begin{array}{@{}rcl@{}} y_{1} & = & e^{s}\cos \varphi_{3} \\ y_{2} & = & e^{s}\sin \varphi_{3}\cos \varphi_{2} \\ y_{3} & = & e^{s}\sin \varphi_{3}\sin \varphi_{2}\cos \varphi_{1} \\ y_{4} & = & e^{s}\sin \varphi_{3}\sin \varphi_{2}\sin \varphi_{1}\\ y_{0} & = & e^{s} \end{array} $$
(39)

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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Keywords

  • Spacetime
  • Noncommutative geometry
  • Gravity