The four-dimensional metric f(R) theories of gravity are cast into connection-dynamical formalism with real su(2) connections as configuration variables. Through this formalism, the classical metric f(R) theories are quantized by extending the loop quantization scheme of general relativity. Our results imply that the nonperturbative quantization procedure of loop quantum gravity is valid not only for general relativity but also for a rather general class of four-dimensional metric theories of gravity.

The Hamiltonian formulation of scalar-tensor theories of gravity is derived from their Lagrangian formulation by Hamiltonian analysis. The Hamiltonian formalism marks off two sectors of the theories by the coupling parameter $\omega${}($\phi${}). In the sector of $\omega${}($\phi${})=-32, the feasible theories are restricted and a new primary constraint generating conformal transformations of spacetime is obtained, while in the other sector of $\omega${}($\phi${})$\ne${}-32, the canonical structure and constraint algebra of the theories are similar to those of general relativity coupled with a scalar field. By canonical transformations, we further obtain the connection-dynamical formalism of the scalar-tensor theories with real su(2) connections as configuration variables in both sectors. This formalism enables us to extend the scheme of nonperturbative loop quantum gravity to the scalar-tensor theories. The quantum kinematical framework for the scalar-tensor theories is rigorously constructed. Both the Hamiltonian constraint operator and master constraint operator are well defined and proposed to represent quantum dynamics. Thus the loop quantum gravity method is also valid for general scalar-tensor theories.

The interior of a Schwarzschild black hole is quantized by the method of loop quantum gravity. The Hamiltonian constraint is solved and the physical Hilbert space is obtained in the model. The properties of a Dirac observable corresponding to the Arnowitt-Deser-Misner mass of the Schwarzschild black hole are studied by both analytical and numerical techniques. It turns out that zero is not in the discrete spectrum of this Dirac observable. This supports the existence of a stable remnant after the evaporation of a black hole. Our conclusion is valid for a general class of schemes adopted for loop quantization of the model.

As modified gravity theories, the four-dimensional metric f(R) theories are cast into connection-dynamical formalism with real su(2) connections as configuration variables. This formalism enables us to extend the nonperturbative loop quantization scheme of general relativity to any metric f(R) theories. The quantum kinematical framework of f(R) gravity is rigorously constructed, where the quantum dynamics can be launched. Both Hamiltonian constraint operator and master constraint operator for f(R) theories are well defined. Our results show that the nonperturbative quantization procedure of loop quantum gravity are valid not only for general relativity but also for a rather general class of four-dimensional metric theories of gravity.

In this paper, we investigate the topological number of de-Sitter black hole solutions with different charges (q) and rotational (a) parameters. By using generalized free energy and Duan’s $$\phi $$ϕ -mapping topological current theory, we find that the topological numbers of black holes can still be classified into three types. In addition, we interestingly found the topological classes for de-Sitter (dS) spacetime with distinct horizons, i.e., black hole event horizon and cosmological horizon, will be different. Moreover, we also investigate topological classifications of dS black hole solutions in higher dimensions with or without the Gauss-Bonnet term.

We present the detailed analyses of a model of loop quantum Schwarzschild interior coupled to a massless scalar field and extend the results in our previous rapid communication [C. Zhang, Y. Ma, S. Song, and X. Zhang, Loop quantum Schwarzschild interior and black hole remnant, Phys. Rev. D 102, 041502 (2020)] to more general schemes. It is shown that the spectrum of the black hole mass is discrete and does not contain zero. This supports the existence of a black hole remnant after Hawking evaporation due to loop quantum gravity effects. Besides to show the existence of a nonvanishing minimal black hole mass in the vacuum case, the quantum dynamics for the nonvacuum case is also solved and compared with the effective one.