Based on a family of indefinite unitary representations of the diffeomorphism group of an oriented smooth $4$-manifold, a manifestly covariant $4$ dimensional and non-perturbative algebraic quantum field theory formulation of gravity is exhibited. More precisely among the bounded linear operators acting on these representation spaces we identify algebraic curvature tensors hence a net of local quantum observables can be constructed from $C^*$-algebras generated by local curvature tensors and vector fields. This algebraic quantum field theory is extracted from structures provided by an oriented smooth $4$-manifold only hence possesses a diffeomorphism symmetry. In this way classical general relativity exactly in $4$ dimensions naturally embeds into a quantum framework.
Several Hilbert space representations of the theory are found. First a "tautological representation" of the limiting global $C^*$-algebra is constructed allowing to associate to any oriented smooth $4$-manifold a von Neumann algebra in a canonical fashion. Secondly, influenced by the Dougan--Mason approach to gravitational quasilocal energy-momentum, we construct certain representations what we call "positive mass representations" with unbroken diffeomorphism symmetry. Thirdly, we also obtain "classical representaions" with spontaneously broken diffeomorphism symmetry corresponding to the classical limit of the theory which turns out to be general relativity.
Finally we observe that the whole family of "positive mass representations" comprise a $2$ dimensional conformal field theory in the sense of G. Segal.
Common features of known quantum gravity theories may hint at the general nature of quantum gravity. The absence of continuous global symmetries is one such feature. This inspired the Weak Gravity Conjecture, which bounds masses of charged particles. We propose the Lattice Weak Gravity Conjecture, which further requires the existence of an infinite tower of particles of all possible charges under both abelian and nonabelian gauge groups and directly implies a cutoff for quantum field theory. It holds in a wide variety of string theory examples and has testable consequences for the real world a…
We describe the freedom of choosing a square root in massive gravity from algebraic point of view. This contribution is based on the talk given by one of us (AG) at the Geometric Foundations of Gravity conference in Tartu in August 2017.
An algebraic analysis of the Hamiltonian formulation of the model two-dimensional gravity is performed. The crucial fact is an exact coincidence of the Poisson brackets algebra of the secondary constraints of this Hamiltonian formulation with the SO(2,1)-algebra. The eigenvectors of the canonical Hamiltonian $H_{c}$ are obtained and explicitly written in closed form.
The quadratic gravity constraints are reformulated in terms of the Newman-Penrose-like quantities. In such a frame language, the field equations represent a linear algebraic system for the Ricci tensor components. In principle, a procedure for the combination of the Ricci components with standard geometric identities can be applied in a similar way as in the case of general relativity. These results could serve in various subsequent analyses and physical interpretations of admitted solutions to quadratic gravity. Here, we demonstrate the utility of such an approach by explicitly proving genera…
We propose a model with quantum bosons on the fcc lattice, which has a stable algebraic Bose liquid phase at low energy. We show that this phase is described by emergent quantum gravity at the Gaussian z = 3 Lifshitz fixed point in 3+1 dimensions. The stability of this algebraic Bose liquid phase is guaranteed by the gauge symmetry of gravitons and self-duality of the low energy field theory. By tuning one parameter in the lattice boson model we can drive a phase transition between the z = 3 Lifshitz gravity and another algebraic Bose liquid phase, described by gravity at the z = 2 Lifshitz po…
We investigate the ghostfree scalar-tensor theory with a timelike scalar field, with derivatives of the scalar field up to the third order and with the Riemann tensor up to the quadratic order. We build two types of linear spaces. One is the set of linearly independent generally covariant scalar-tensor monomials, the other is the set of linearly independent spatially covariant gravity monomials. We argue that these two types of linear space are isomorphic to each other in the sense of gauge fixing/recovering procedures. We then identify the subspaces in the spatially covariant gravity, which a…
We study the Kawai-Lewellen-Tye (KLT) relations for quantum field theory by reformulating it as an isomorphism between two Lie algebras. We also show how explicit formulas for KLT relations arise when studying rational functions on ${\mathcal M}_{0,n}$, and prove identities that allow for arbitrary rational functions to be expanded in any given basis. Via the Cachazo-He-Yuan formulas for, these identities also lead to new formulas for gauge and gravity tree amplitudes, including formulas for so-called Bern-Carrasco-Johansson numerators, in the case of non-linear sigma model and maximal-helicit…
Four principles are proposed to underlie the quantum theory of gravity. We show that these suffice to recover the Einstein equations. We also suggest that MOND results from a modification of the classical equivalence principle, due to quantum gravity effects.
We study exact solutions of three-dimensional gravity with a cosmological constant and a gravitational Chern-Simons term: the theory known as topologically massive gravity. After reviewing the algebraic classification, we show that if a solution has curvature of algebraic type D, then it is biaxially squashed AdS_3. Applying the classification, we provide a comprehensive review of the literature, showing that most known solutions are locally equivalent to biaxially squashed AdS_3 or to AdS pp-waves.
To ensure the existence of a well defined linearized gravitational wave equation, we show that the spacetimes in the so-called "Einstein-Gauss-Bonnet gravity in four dimension" have to be locally conformally flat.
We reexamine the concept of the attractive gravity probe surface recently proposed as an indicator for strength of gravity. Then, we propose three new variant concepts and show refined inequalities for the four types of the AGPSs by taking account of the angular momentum, gravitational waves and matters.
This article is an extension of the author's second master thesis [1]. It aims to introduce to the theory of perturbatively quantized General Relativity coupled to Spinor Electrodynamics, provide the results thereof and set the notation to serve as a starting point for further research in this direction. It includes the differential geometric and Hopf algebraic background, as well as the corresponding Lagrange density and some renormalization theory. Then, a particular problem in the renormalization of Quantum General Relativity coupled to Quantum Electrodynamics is addressed and solved by a g…
Fermions in (anti) de Sitter gravity theory in four dimensions are considered. Especially we propose new fermion actions to derive a Weyl or Majorana fermion action if we break the AdS (dS) group to the Lorentz group in curved spacetime.
We study the algebraic structure of the configuration space of the Kodaira-Spencer gravity theory on a Calabi-Yau threefold. We then investigate the deformation problem of the Kodaira-Spencer gravity at the classical level using the algebraic tools obtained here.
We construct a generalized massive gravity by combining quadratic curvature gravity with the Chern-Simons term in four dimensions. This may be a candidate for the parity-odd tricritical gravity theory. Considering the AdS$_4$ vacuum solution, we derive the linearized Einstein equation, which is not similar to that of the three dimensional (3D) generalized massive gravity. When a perturbed metric tensor is chosen to be the Kerr-Schild form, the linearized equation reduces to a single massive scalar equation. At the tricritical points where two masses are equal to -1 and 2, we obtain a log-squar…
If quantum gravity respects the principles of quantum mechanics, suitably generalized, it may be that a more viable approach to the theory is through identifying the relevant quantum structures rather than by quantizing classical spacetime. This viewpoint is supported by difficulties of such quantization, and by the apparent lack of a fundamental role for locality. In finite or discrete quantum systems, important structure is provided by tensor factorizations of the Hilbert space. However, even in local quantum field theory properties of the generic type III von Neumann algebras and of long ra…
In this work we present a non-relativistic gravity theory defined in four spacetime dimensions using the MacDowell-Mansouri geometrical formulation. We obtain a Newtonian gravity action which is constructed from the curvature of a Newton-Hooke version of the so-called Newtonian algebra. We show that the non-relativistic gravity theory presented here contains the Poisson equation in presence of a cosmological constant. Moreover we make contact with the Modified Newtonian Dynamics (MOND) approach for gravity by considering a particular ansatz for a given gauge field. We extend our results to a g…
We consider conformal gravity boundary conditions and outline the highest dimensional non-trivial asymptotic symmetry algebras of conformal gravity. The highest among them is five dimensional and leads to a global geon solution.
We study four-dimensional gravity theories that are rendered renormalisable by the inclusion of curvature-squared terms to the usual Einstein action with cosmological constant. By choosing the parameters appropriately, the massive scalar mode can be eliminated and the massive spin-2 mode can become massless. This "critical" theory may be viewed as a four-dimensional analogue of chiral topologically massive gravity, or of critical "New Massive Gravity" with a cosmological constant, in three dimensions. We find that the on-shell energy for the remaining massless gravitons vanishes. There are als…
We comment on structural properties of the algebras $\mathfrak{A}_{LQG/LQC}$ underlying loop quantum gravity and loop quantum cosmology, especially the representation theory, relating the appearance of the (dynamically induced) superselection structure ($θ$-sectors) in loop quantum cosmology to recently proposed representations with non-degenerate background geometries in loop quantum gravity with Abelian structure group. To this end, we review and employ the concept of extending a given (observable) algebra with possibly non-trivial centre to a (charged) field algebra with (global) gauge grou…
We investigate the localization of four-dimensional metastable gravity in supersymmetric dilatonic domain walls through massive modes by considering several scenarios in the model. We compute corrections to the Newtonian potential for small and long distances compared with a crossover scale given in terms of the dilatonic coupling. 4D gravity behavior is developed on the brane for distance very much below the crossover scale, while for distance much larger, the 5D gravity is recovered. Whereas in the former regime gravity is always attractive, in the latter regime due to non-normalizable unsta…
A new family of analytical solutions in a four dimensional static spacetime is presented for $f\left( R\right) $-gravity. In contrast to General Relativity, we find that a non trivial black string/ring solution is supported in vacuum power law $f\left( R\right) $-gravity for appropriate values of the parameters characterizing the model and when axisymmetry is introduced in the line element. For the aforementioned solution, we perform a brief investigation over its basic thermodynamic quantities.
We calculate canonical charges in four dimensional conformal gravity using the generalised boundary conditions presented in \cite{Grumiller:2013mxa}. We show that the charges are finite and conserved. The asymptotic symmetry algebras generated by these charges are classified according to the conditions imposed on the first subleading term in the generalised Fefferman-Graham expansion.
We investigate string-like solutions in four dimensions based on Hořava-Lifshitz gravity. For a restricted class of solutions where the Cotton tensor vanishes, we find that the string-like solutions in Einstein gravity including the BTZ black strings are solutions in Hořava-Lifshitz gravity as well. The geometry is warped in the same way as in Einstein gravity, but the "conformal" lapse function is not constrained in Hořava-Lifshitz gravity. It turns out that if $λ\ne 1$, there exist no other solutions. For the value of model parameter with which Einstein gravity recovers in IR limit (i.e., $λ…
We develop a proposal for a theory of simplicial gravity with spinors as the fundamental configuration variables. The underlying action describes a mechanical system with finitely many degrees of freedom, the system has a Hamiltonian and local gauge symmetries. We will close with some comments on the resulting quantum theory, and explain the relation to loop quantum gravity and twisted geometries. The paper appears in parallel with an article by Cortês and Smolin, who study the relevance of the model for energetic causal sets and various other approaches to quantum gravity.
At the horizon of a black hole, the action of (3+1)-dimensional loop quantum gravity acquires a boundary term that is formally identical to an action for three-dimensional gravity. I show how to use this correspondence to obtain the entropy of the (3+1)-dimensional black hole from well-understood conformal field theory computations of the entropy in (2+1)-dimensional de Sitter space.
Noncritical Conformal Gravity and Four-Dimensional Liouville Theory
National Taiwan University（台湾大学）；National Center for Theoretical Sciences（国家理论科学中心）；Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University（大阪公立大学南阳一郎理论与实验物理研究所（NITEP））；Institute of Fundamental Physics and Quantum Technology, Department of Physics, School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院物理学与量子技术基础物理研究所）
The aim of this thesis is to explore the quantum aspects of Higher Spin Gravities (HSGRAs) and their underlining algebraic structures. We give a concise review of HSGRAs followed by three chapters with original results. The first chapter is dedicated to the study of the vacuum one-loop correction of holographic HSGRAs in Anti-de Sitter space. We show that there is a remarkable agreement between the $F$-energy of HSGRAs in the bulk and the predictions coming from the dual CFTs in integer dimensions. We extend this result to continuous dimension and show that vacuum one-loop corrections in HSGRA…
