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.

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.