Covariant μˉ{\bar{\mu}}-scheme effective dynamics, mimetic gravity, and non-singular black holes: Applications to spherical symmetric quantum gravity and CGHS model

We propose a new $\bar{\mu}$-scheme Hamiltonian effective dynamics in the spherical symmetric sector of Loop Quantum Gravity (LQG). The effective dynamics is generally covariant as derived from a covariant Lagrangian. The Lagrangian belongs to the class of extended mimetic gravity Lagrangians in 4 dimensions. We apply the effective dynamics to both cosmology and black hole. The effective dynamics reproduces the non-singular Loop-Quantum-Cosmology (LQC) effective dynamics. From the effective dynamics, we obtain the non-singular black hole solution, which has a killing symmetry in addition to the spherical symmetry and reduces to the Schwarzschild geometry asymptotically near the infinity. The black hole spacetime resolves the classical singularity and approaches asymptotically the Nariai geometry $\mathrm{dS}_2\times S^2$ at the future infinity in the interior of the black hole. The resulting black hole spacetime has the complete future null infinity $\mathscr{I}^+$. Thanks to the general covariance, the effective dynamics can be reformulated in the light-cone gauge. We generalize the covariant $\bar{\mu}$-scheme effective dynamics to the Callan-Giddings-Harvey-Strominger (CGHS) model and apply the light-cone formulation to the CGHS black hole solution with the null-shell collapse. We focus on the effective dynamics projected along the null shell. The result shows that both the 2d scalar curvature and the derivative of dilaton field are finite, in contrast to the divergence in the CGHS model.Comment: 49 pages, 23 figure

Fermions in Loop Quantum Gravity and Resolution of Doubling Problem

The fermion propagator is derived in detail from the model of fermion coupled to loop quantum gravity. As an ingredient of the propagator, the vacuum state is defined as the ground state of some effective fermion Hamiltonian under the background geometry given by a coherent state resembling the classical Minkowski spacetime. Moreover, as a critical feature of loop quantum gravity, the superposition over graphs is employed to define the vacuum state. It turns out that the graph superposition leads to the propagator being the average of the propagators of the lattice field theory over various graphs so that all fermion doubler modes are suppressed in the propagator. This resolves the doubling problem in loop quantum gravity. Our result suggests that the superposition nature of quantum geometry should, on the one hand, resolve the tension between fermion and the fundamental discreteness and, on the other hand, relate to the continuum limit of quantum gravity.Comment: 25+9 pages, 2 figure

Spinfoams and high performance computing

Numerical methods are a powerful tool for doing calculations in spinfoam theory. We review the major frameworks available, their definition, and various applications. We start from $\texttt{sl2cfoam-next}$, the state-of-the-art library to efficiently compute EPRL spin foam amplitudes based on the booster decomposition. We also review two alternative approaches based on the integration representation of the spinfoam amplitude: Firstly, the numerical computations of the complex critical points discover the curved geometries from the spinfoam amplitude and provides important evidence of resolving the flatness problem in the spinfoam theory. Lastly, we review the numerical estimation of observable expectation values based on the Lefschetz thimble and Markov-Chain Monte Carlo method, with the EPRL spinfoam propagator as an example.Comment: 33 pages, 11 figures. Invited chapter for the book "Handbook of Quantum Gravity" (Eds. C. Bambi, L. Modesto and I.L. Shapiro, Springer Singapore, expected in 2023

Improved effective dynamics of loop-quantum-gravity black hole and Nariai limit

AbstractWe propose a new model of the spherical symmetric quantum black hole in the reduced phase space formulation. We deparametrize gravity by coupling to the Gaussian dust which provides the material coordinates. The foliation by dust coordinates covers both the interior and exterior of the black hole. After the spherical symmetry reduction, our model is a 1 + 1 dimensional field theory containing infinitely many degrees of freedom. The effective dynamics of the quantum black hole is generated by an improved physical Hamiltonian HΔ. The holonomy correction in HΔ is implemented by the μ-scheme regularization with a Planckian area scale Δ (which often chosen as the minimal area gap in loop quantum gravity). The effective dynamics recovers the semiclassical Schwarzschild geometry at low curvature regime and resolves the black hole singularity with Planckian curvature, e.g. RμνρσRμνρσ∼ 1/Δ2. Our model predicts that the evolution of the black hole at late time reaches the charged Nariai geometry dS2 × S2 with Planckian radii∼Δ. The Nariai geometry is stable under linear perturbations but may be unstable by nonperturbative quantum effects. Our model suggests the existence of quantum tunneling of the Nariai geometry and a scenario of black-hole-to-white-hole transition