Selected Topics in Theoretical Physics from Quantum Null Energy Condition to Black Hole Thermodynamics

The quantum null energy condition (QNEC) is a quantum generalization of the null energy condition which gives a lower bound on the null energy in terms of the second derivative of the von Neumann entropy or entanglement entropy of some region with respect to a null direction. The QNEC states that $\langle T_{kk}\rangle _{p}\geq lim_{A\rightarrow 0}\left(\frac{\hbar}{2\pi A}S_{out}^{\prime\prime}\right)$ where $S_{out}$ is the entanglement entropy restricted to one side of a codimension-2 surface $\Sigma$ which is deformed in the null direction about a neighborhood of point $p$ with area $A$. A proof of QNEC has been given which applies to free and super-renormalizable bosonic field theories, and to any points that lie on a stationary null surface. Using similar assumptions and method, we prove QNEC for fermionic field theories in Chapter \ref{C:QNEC}. It has been argued that using the Weyl tensor, $C_{\mu\nu\lambda \rho}$, a dimensionless integral can be constructed on a spacelike hypersurface in 5-dimensional spacetime, $\int C_{\mu\nu\lambda \rho}C^{\mu\nu\lambda \rho} dV_4$ and using the Penrose Weyl curvature hypothesis as guide, $C_{\mu\nu\lambda \rho}C^{\mu\nu\lambda \rho}$ was tested as an entropy density by comparing the Bekeinstein-Hawking entropy of a black hole to above integral. In chapter \ref{C:Weyl}, we test $C_{\mu\nu\lambda \rho}C^{\mu\nu\lambda \rho}$ as an entropy density using similar methods but in Gauss-Bonnet gravity. Typically, the entropy of an isolated system in equilibrium is calculated by counting the number of accessible microstates, or in more general cases by using the Gibbs formula. In irreversible processes entropy spontaneously increases and this is understood from statistical arguments. In Chapter \ref{C:RR}, we propose a new definition of entropy directly based on the level of irreversibility of a process. This formulation agrees in first approximation with the usual methods of calculating entropy and can be readily applied in the case of a black hole in the semiclassical regime