Gravitational SL(2, ℝ) algebra on the light cone

In his recent paper, Wolfgang Wieland, researcher at IQOQI-Vienna, looks at the gravitational phase space in a region. The region is defined operationally as a laboratory bounded by light rays. At the boundary, otherwise unphysical gauge degrees of freedom turn into physical boundary (edge) modes. The paper explains how parity-violating terms in the action deform the SL(2,R) symmetries at the boundary. In quantum gravity, this deformation has no effect on the radiative modes, but the effect on the boundary modes is dramatic: the area spectrum, which would otherwise be continuous, becomes discrete.

In a region with a boundary, the gravitational phase space consists of radiative modes in the interior and edge modes at the boundary. Such edge modes are necessary to explain how the region couples to its environment. In this paper, we characterise the edge modes and radiative modes on a null surface for the tetradic Palatini-Holst action. Our starting point is the definition of the action and its boundary terms. We choose the least restrictive boundary conditions possible. The fixed boundary data consists of the radiative modes alone (two degrees of freedom per point). All other boundary fields are dynamical. We introduce the covariant phase space and explain how the Holst term alters the boundary symmetries. To infer the Poisson brackets among Dirac observables, we define an auxiliary phase space, where the SL(2, ℝ) symmetries of the boundary fields are manifest. We identify the gauge generators and second-class constraints that remove the auxiliary variables. All gauge generators are at most quadratic in the fundamental SL(2, ℝ) variables on phase space. We compute the Dirac bracket and identify the Dirac observables on the light cone. Finally, we discuss various truncations to quantise the system in an effective way.


Journal reference:

Journal of High Energy Physics volume 2021, Article number: 57 (2021)