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Quantum gravity at the corner

Laurent Freidel, Alejandro Perez

TL;DR

The paper reveals that the boundary of a 4d gravitational region carries a natural 2d symplectic structure that encapsulates would-be boundary gauge degrees of freedom, enabling a boundary 2+1 theory described by an $\mathrm{SU}(2)\times\mathrm{SL}(2,\mathbb{R})$ structure at each boundary point. By discretizing the boundary, the authors construct a discrete quantum representation in terms of three harmonic oscillators, yielding a rich spectrum including degenerate boundary geometries labeled by a quantum number $k$. They analyze diffeomorphism symmetry on the boundary, show how to realize and restrict these boundary degrees of freedom, and discuss connections to 2d conformal field theories and horizon modeling, thereby extending the quantum description of horizons beyond Chern-Simons-based isolated-horizon approaches. The work highlights new boundary degrees of freedom tied to boundary diffeomorphisms and proposes concrete mechanisms to relate them to black hole entropy and horizon dynamics.

Abstract

We investigate the quantum geometry of $2d$ surface $S$ bounding the Cauchy slices of 4d gravitational system. We investigate in detail and for the first time the symplectic current that naturally arises boundary term in the first order formulation of general relativity in terms of the Ashtekar-Barbero connection. This current is proportional to the simplest quadratic form constructed out of the triad field, pulled back on $S$. We show that the would-be-gauge degrees of freedom---arising from $SU(2)$ gauge transformations plus diffeomorphisms tangent to the boundary, are entirely described by the boundary $2$-dimensional symplectic form and give rise to a representation at each point of $S$ of $SL(2,\mathbb{R}) \times SU(2)$. Independently of the connection with gravity, this system is very simple and rich at the quantum level with possible connections with conformal field theory in 2d. A direct application of the quantum theory is modelling of the black horizons in quantum gravity.

Quantum gravity at the corner

TL;DR

The paper reveals that the boundary of a 4d gravitational region carries a natural 2d symplectic structure that encapsulates would-be boundary gauge degrees of freedom, enabling a boundary 2+1 theory described by an structure at each boundary point. By discretizing the boundary, the authors construct a discrete quantum representation in terms of three harmonic oscillators, yielding a rich spectrum including degenerate boundary geometries labeled by a quantum number . They analyze diffeomorphism symmetry on the boundary, show how to realize and restrict these boundary degrees of freedom, and discuss connections to 2d conformal field theories and horizon modeling, thereby extending the quantum description of horizons beyond Chern-Simons-based isolated-horizon approaches. The work highlights new boundary degrees of freedom tied to boundary diffeomorphisms and proposes concrete mechanisms to relate them to black hole entropy and horizon dynamics.

Abstract

We investigate the quantum geometry of surface bounding the Cauchy slices of 4d gravitational system. We investigate in detail and for the first time the symplectic current that naturally arises boundary term in the first order formulation of general relativity in terms of the Ashtekar-Barbero connection. This current is proportional to the simplest quadratic form constructed out of the triad field, pulled back on . We show that the would-be-gauge degrees of freedom---arising from gauge transformations plus diffeomorphisms tangent to the boundary, are entirely described by the boundary -dimensional symplectic form and give rise to a representation at each point of of . Independently of the connection with gravity, this system is very simple and rich at the quantum level with possible connections with conformal field theory in 2d. A direct application of the quantum theory is modelling of the black horizons in quantum gravity.

Paper Structure

This paper contains 11 sections, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The origin of the 2d symplectic structure
  3. Symmetries
  4. Hamiltonian generators
  5. Boundary symplectic structure
  6. The associated boundary $2+1$ dynamical theory
  7. Quantisation: the discrete representation
  8. Diffeomorphism symmetry
  9. Representation
  10. The geometry of the $k$ quantum number
  11. Conclusion

Figures (3)

  • Figure 1: Spacetime region obtained from the time flow that is allowed in our analysis. Lapse and shift are constrained on the corners $\partial \Sigma$ in order to preserve the boundary fixed up to tangent diffeomorphisms and gauge transformations.
  • Figure 2: Deformation of a ball of geodesics normal to $\Sigma$. On the left panel $h_{\xi}=0$: the principal axis of the shear are tangent to $\partial \Sigma$ and normal to $\partial \Sigma$. On the right panel $h_{\xi}\not=0$, the boundary "moves".
  • Figure 3: The thick segments represent the paths $L_x$ and $L_y$ used in the regularization of the basic observables used in (\ref{['una']}). The square oriented path represents the contour $C$ used in (\ref{['emi']}) defined by four oriented segments $\{u, d, r, l\}$. The diagram should be thought of as embedded inside a coordinate ball $x^2+y^2\le \epsilon^2$. The regularisation is removed in the limit $\epsilon \to 0$.