This monograph introduces the basic concepts of the theory of causal fermion systems, a recent approach to the description of fundamental physics. The theory yields quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for describing and analyzing non-smooth geometries and ``quantum geometries.'' The dynamics is described by a novel variational principle, called the causal action principle. In addition to the basics, the book provides all the necessary mathematical background and explains how the causal action principle gives rise to the interactions of the standard model plus gravity on the level of second-quantized fermionic fields coupled to classical bosonic fields. The focus is on getting a mathematically sound connection between causal fermion systems and physical systems in Minkowski space. The book is intended for graduate students entering the field, and is furthermore a valuable reference work for researchers in quantum field theory and quantum gravity.

We apply our model of quantum gravity to an AdS black hole resulting in a wave equation in a quantum spacetime which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur.

Three dimensional topologically massive gravity (TMG) with a negative cosmological constant -\ell^{-2} and positive Newton constant G admits an AdS_3 vacuum solution for any value of the graviton mass \mu. These are all known to be perturbatively unstable except at the recently explored chiral point \mu\ell=1. However we show herein that for every value of \mu\ell< 3 there are two other (potentially stable) vacuum solutions given by SL(2,R)x U(1)-invariant warped AdS_3 geometries, with a timelike or spacelike U(1) isometry. Critical behavior occurs at \mu\ell=3, where the warping transitions from a stretching to a squashing, and there are a pair of warped solutions with a null U(1) isometry. For \mu\ell>3, there are known warped black hole solutions which are asymptotic to warped AdS_3. We show that these black holes are discrete quotients of warped AdS_3 just as BTZ black holes are discrete quotients of ordinary AdS_3. Moreover new solutions of this type, relevant to any theory with warped AdS_3 solutions, are exhibited. Finally we note that the black hole thermodynamics is consistent with the hypothesis that, for \mu\ell>3, the warped AdS_3 ground state of TMG is holographically dual to a 2D boundary CFT with central charges c_R={15(\mu\ell)^2+81\over G\mu((\mu\ell)^2+27)} and c_L={12 \mu\ell^2\over G((\mu\ell)^2+27)}.

Three dimensional Einstein gravity with negative cosmological constant -1/\ell^2 deformed by a gravitational Chern-Simons action with coefficient 1/\mu is studied in an asymptotically AdS_3 spacetime. It is argued to violate unitary or positivity for generic \mu due to negative-energy massive gravitons. However at the critical value \mu\ell=1, the massive gravitons disappear and BTZ black holes all have mass and angular momentum related by \ell M=J. The corresponding chiral quantum theory of gravity is conjectured to exist and be dual to a purely right-moving boundary CFT with central charges (c_L,c_R)=(0,3\ell /G).

We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in Wang and Yau (Commun Math Phys 288(3):919–942, 2009) to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group is assigned. Quasi-local angular momentum and quasi-local center of mass correspond to pairing this element with rotation Killing fields and boost Killing fields, respectively. They obey classical transformation laws under the action of the Poincaré group. We further justify these definitions by considering their limits as the total angular momentum and the total center of mass of an isolated system. These expressions were derived from the Hamilton–Jacobi analysis of the gravitational action and thus satisfy conservation laws. As a result, we obtained an invariant total angular momentum theorem in the Kerr spacetime. For a vacuum asymptotically flat initial data set of order 1, it is shown that the limits are always finite without any extra assumptions. We also study these total conserved quantities on a family of asymptotically flat initial data sets evolving by the vacuum Einstein evolution equation. It is shown that the total angular momentum is conserved under the evolution. For the total center of mass, the classical dynamical formula relating the center of mass, energy, and linear momentum is recovered, in the nonlinear context of initial data sets evolving by the vacuum Einstein evolution equation. The definition of quasi-local angular momentum provides an answer to the second problem in classical general relativity on Penrose’s list (Proc R Soc Lond Ser A 381(1780):53–63, 1982).