Gravity as an SU(1,1) gauge theory in four dimensions

Abstract

We start with the Hamiltonian formulation of the first order action of pure gravity with a full $\mathfrak{sl}(2,\mathbb{C})$ internal gauge symmetry. We make a partial gauge-fixing which reduces $\mathfrak{sl}(2,\mathbb{C})$ to its sub-algebra $\mathfrak{su}(1,1)$. This case corresponds to a splitting of the space-time $\mathcal{M}=\Sigma \times \mathbb{R}$ where $\Sigma$ inherits an arbitrary Lorentzian metric of signature $(-,+,+)$. Then, we find a parametrization of the phase space in terms of an $\mathfrak{su}(1,1)$ commutative connection and its associated conjugate electric field. Following the techniques of Loop Quantum Gravity, we start the quantization of the theory and we consider the kinematical Hilbert space on a given fixed graph $\Gamma$ whose edges are colored with unitary representations of $\mathfrak{su}(1,1)$. We compute the spectrum of area operators acting of the kinematical Hilbert space: we show that space-like areas have discrete spectra, in agreement with usual $\mathfrak{su}(2)$ Loop Quantum Gravity, whereas time-like areas have continuous spectra. We conclude on the possibility to make use of this formulation of gravity to construct a holographic description of black holes in the framework of Loop Quantum Gravity.

19 pages, 2 figures

References in corpus (5)

• Black hole state counting in Loop Quantum Gravity: A number theoretical approach
• Towards Loop Quantum Gravity without the time gauge
• First order gravity on the light front
• Timelike twisted geometries
• Semi-classical analysis of black holes in Loop Quantum Gravity: Modelling Hawking radiation with volume fluctuations

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