Quantization of the G-Connections via the Tangent Groupoid

Abstract

A description of the space of G-connections using the tangent groupoid is given. As the tangent groupoid parameter is away from zero, the G-connections act as convolution operators on a Hilbert space. The gauge action is examined in the tangent groupoid description of the G-connections. Tetrads are formulated as Dirac type operators. The connection variables and tetrad variables in Ashtekar's gravity are presented as operators on a Hilbert space. IntroductionIn Ashtekar's theory of gravity, a SU (2)-connection captures the extrinsic curvature of a space-like leaf in a time-transversal foliation, and the intrinsic geometry of the leaf is given by a tetrad [2]. It is shown that the Einstein-Hilbert functional and Einstein equations can be written in terms of SU (2)-connections and tetrads, thus these variables together recast Einstein's theory of gravity. By rewriting Einstein's gravity with the connection variables and tetrad variables, one could attempt to quantize gravity via the Hamiltonian formalism, and obtain a theory of quantum gravity [3]. The reader may refer to Thiemann's introductory [9]. This article offers an alternative view of the space of connections to ones that appear in other loop quantum gravity literatures.To take the connections as dynamic variables in a quantum theory, one studies wave functions on the space of connections. Unfortunately, the lack of a measure on such a space already poses the first challenge. One typical solution to obtain a measure on the connection space comes from spaces of progressively refined cylindrical functions. In another description, one uses a finite set of curves to probe the space of G-connections to obtain a finite dimensional manifold that depend on the sets of curves. By successively refining the finite sets of curves, one obtains a pro-manifold that extends the original space of connections to the so-called space of generalized connections [4].As a step to quantizing gravity in the Ashtekar framework, there are recent developments of describing such an extended space of connections using a spectral triple in noncommutative geometry [1,8], which captures the geometry of the space of generalized connections as operators on a Hilbert space. While the geometries of the base manifold and the space of G-connections on it are in theory retained, the construction of the spectral triple is considered too discrete to practically allow one to reconstruct the base manifold and its G-connections. To rid the discrete description of using finite sets of embedded curves, this article proposes an alternative way to smoothly probe the space of connections using the tangent groupoid. The purposes of this article are to present an alternative idea to the studies *

Strict Deformation Quantisation of the 𝐺-connections via Lie Groupoid

“…Since connection variables are only half of the variables in gravity in Arnowitt-Deser-Misner formulation, one still has to look into the other half of the variables that the connections are conjugate dual to, tetrads or metrics. It is known that tetrads are quantised to degree one differential operators [1,9], and its semi-classical limit can be obtained from the → 0 limit of integral kernel of the differential operator multiplied by , which gives nothing but the symbol of the differential operator [10]. In the case of the manifold M being three dimensional, the symbol is an su(2)-valued function the Poisson manifold T * M .…”