This talk describes how the Barbero–Immirzi parameter, a couling constant akin to the theta parameter in QCD, deforms the SL(2,R) symmetries of the gravitational boundary data on a null surface. Our starting point is the definition of the action and its boundary terms. We introduce the covariant phase space and explain how the Holst term alters the symmetries on a null surface. We show that this alteration only affects the algebra of the edge modes, whereas the algebra of the radiative modes is unchanged. To compute the Poisson brackets explicitly, we work on an auxiliary phase space, where the SL(2,R) symmetries of the boundary fields are manifest. The physical phase space is obtained by imposing both first-class and second-class constraints. All gauge generators are at most quadratic in terms of the fundamental SL(2,R) variables. Finally, we discuss various strategies to quantise the system.
A summary will be given of the papers arXiv:2104.08377, 2104.05803 and 2012.01889. The main point of this seminar is to discuss how the ADM phase space for general relativity splits into edge modes and radiative modes and how these two different components can each be understood as genuine phase spaces for themselves. Each of these smaller phase spaces is equipped with a symplectic structure. The symplectic structure on the radiative phase space defines the familiar Poisson commutation relations for the asymptotic shear. Following the argument given in arXiv:2104.08377, it will be shown that the Poisson bracket for the edge modes (obtained by formally imposing second-class constraints on the ADM phase space and calculating the resulting Dirac bracket) is the Barnich–Troessaert bracket on a cross section of null infinity.
I will consider the phase space at null-infinity from the \(r\rightarrow\infty\) limit of a quasi-local phase space for a finite box with a boundary that is null. This box will serve as a natural IR regulator. To remove the IR regulator, I will consider a double null foliation together with an adapted Newman–Penrose null tetrad. The limit to null infinity (on phase space) is obtained in the limit where the boundary is sent to infinity. I will introduce various charges and explain the role of the corresponding balance laws. The talk is based on the paper: arXiv:2012.01889.