Emilio Elizalde, Pedro J. Silva
The ${\bf f}(R)$ gravity field equations are derived as an equation of state of local space-time thermodynamics. Jacobson's arguments are non-trivially extended, by means of a more general definition of local entropy, for which Wald's definition of dynamic black hole entropy is used, as well as the concept of an effective Newton constant for graviton exchange, recently appeared in the literature.
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The universality of the non-perturbative definition of Hermitian one-matrix models following the quantum, stochastic, or $d=1$-like stabilization is discussed in comparison with other procedures. We also present another alternative definition, which illustrates the need of new physical input for $d=0$ matrix models to make contact with 2D quantum gravity at the non-perturbative level.
A note on the definition of gravitational energy for quadratic curvature gravity via topological regularization
本文证明在四维二次曲率引力中通过拓扑正则化定义的引力能量可被文献[3]的定义涵盖，并将该结果推广至任意偶数维的通用引力理论。
We show that the partition function of the Ponzano-Regge quantum gravity model can be written as a sum over surfaces in a $(2+1)$ dimensional space-time. We suggest a geometrical meaning, in terms of surfaces, for the (regulated) divergences that appear in the partition function.
We study dynamical surface gravity in a general spherically symmetric setting using Painlevé-Gullstrand (PG) coordinates. Our analysis includes several definitions that have been proposed in the past as well as two new definitions adapted to PG coordinates. Various properties are considered, including general covariance, value at extremality, locality and static limit. We illustrate with specific examples of "dirty" black holes that even for spacetimes possessing a global timelike Killing vector, local definitions of surface gravity can differ substantially from "non-local" ones that require a…
We provide a simple method to compute the energy in higher curvature gravity in asymptotically AdS spacetimes in even dimensions. It follows from the combined use of topological terms added to the gravity action, and the Wald charges derived from the augmented action functional. No additional boundary terms are needed. As a consistency check, we show that the formula for the conserved quantities derived in this way yields the correct result for the mass of asymptotically AdS black holes.
We present a recent definition of symmetry generating vector fields on manifolds equipped with a first-order reductive Cartan geometry. We apply this definition to a number of spacetime geometries used in gravity theories and show that it agrees with the usual notions of symmetry of affine, Riemann-Cartan, Riemannian, Weizenböck and Finsler spacetimes.
Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energy-momentum naturally leads to the gauge theory of volume-preserving diffeormorphisms of an inner Minkowski space. The generalized asymptotic free scalar, Dirac and gauge fields in that theory are canonically quantized, the Fock spaces of stationary states are constructed and the gravitational limit - mapping the gravitational energy-momentum onto the inertial energy-momentum to account for their observed equality - is introduced. Next the S-matrix in quantum gravity is defined as the grav…
I would claim that we do not have a suitably general definition of what a topological phase is, or more importantly, any robust understanding of how to enter one even in the world of mathematical models. The latter is, of course, the more important issue and the main subject of this note. But a good definition can sharpen our thinking and a poor definition can misdirect us. I will not attempt a final answer here but merely comment on the strengths and weaknesses of possible definitions and argue for some flexibility. In particular, I describe a rather simple class of ``quantum gravity'' models…
In this work it is introduced an extension of the study of the first law of thermodynamics of black holes based on the geometry of the extended phase space for AdS Lovelock gravities which includes changes of scales. As expected, the result obtained coincides with the previously known four dimensional case. For higher dimensions, the result is the arise of two new contributions to the first law of thermodynamics. The first term corresponds to corrections of the usual definition of thermodynamics volume at the horizon, due to the presence of the higher curvature terms. The second term arises in…
A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of `gravity $=$ gauge $\times$ gauge'. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the `gauge $\times$ gauge' convolution product on, for example, the static Einstein universe, and more generally for ultrastatic …
We give a geometrical definition of the asymptotic flatness at null infinity in spacetimes of even dimension $d$ greater than 4 within the framework of conformal infinity. Our definition is shown to be stable against perturbations to linear order. We also show that our definition is stringent enough to allow one to define the total energy of the system viewed from null infinity as the generator conjugate to an asymptotic time translation. We derive an expression for the generator conjugate within the Hamiltonian framework, and propose to take this notion of energy as the natural generalisation…
We argue that a certain twisted supersymmetric Yang-Mills theory in three dimensions with gauge group SU(2) possesses a set of topological observables whose expectation values can be computed in a related Chern Simons theory. This Chern Simons theory has been proposed as a definition of three dimensional Euclidean quantum gravity. Since the YM theory admits a discretization which preserves the values of topological observables we conjecture that it can be used as a non-perturbative definition of the quantum gravity theory.
A proper definition of the path integral of quantum gravity has been a long-standing puzzle because the Weyl factor of the Euclidean metric has a wrong-sign kinetic term. We propose a definition of two-dimensional Liouville quantum gravity with cosmological constant using conformal bootstrap for the timelike Liouville theory coupled to supercritical matter. We prove a no-ghost theorem for the states in the BRST cohomology. We show that the four-point function constructed by gluing the timelike Liouville three-point functions is well defined and crossing symmetric (numerically) for external Lio…
We review the covariant canonical formalism initiated by D'Adda, Nelson and Regge in 1985, and extend it to include a definition of form-Poisson brackets (FPB) for geometric theories coupled to $p$-forms, gauging free differential algebras. The form-Legendre transformation and the form-Hamilton equations are derived from a $d$-form Lagrangian with $p$-form dynamical fields $ϕ$. Momenta are defined as derivatives of the Lagrangian with respect to the "velocities" $dϕ$ and no preferred time direction is used. Action invariance under infinitesimal form-canonical transformations can be studied in …
Based on the definition of the apparent horizon in a general two-dimensional dilaton gravity theory, we analyze the tunnelling phenomenon of the apparent horizon by using Hamilton-Jacobi method. In this theory the definition of the horizon is very different from those in higher-dimensional gravity theories. The spectrum of the radiation is obtained and the temperature of the radiation is read out from this spectrum and it satisfies the usual relationship with the surface gravity. Besides, the calculation with Parikh's null geodesic method for a simple example conforms to our result in general …
Causality in quantum field theory is defined by the vanishing of field commutators for space-like separations. However, this does not imply a direction for causal effects. Hidden in our conventions for quantization is a connection to the definition of an arrow of causality, i.e. what is the past and what is the future. If we mix quantization conventions within the same theory, we get a violation of microcausality. In such a theory with mixed conventions the dominant definition of the arrow of causality is determined by the stable states. In some quantum gravity theories, such as quadratic grav…
I review the definition of n-point functions in loop quantum gravity, discussing what has been done and what are the main open issues. Particular attention is dedicated to gauge aspects and renormalization.
In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry. However, such an approach cannot be employed to define `conserved quantities' in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) becau…
We construct a 2nd quantized reformulation of canonical Loop Quantum Gravity at both kinematical and dynamical level, in terms of a Fock space of spin networks, and show in full generality that it leads directly to the Group Field Theory formalism. In particular, we show the correspondence between canonical LQG dynamics and GFT dynamics leading to a specific GFT model from any definition of quantum canonical dynamics of spin networks. We exemplify the correspondence of dynamics in the specific example of 3d quantum gravity. The correspondence between canonical LQG and covariant spin foam model…
We give a general geometric definition of asymptotic flatness at null infinity in $d$-dimensional general relativity ($d$ even) within the framework of conformal infinity. Our definition is arrived at via an analysis of linear perturbations near null infinity and shown to be stable under such perturbations. The detailed fall off properties of the perturbations, as well as the gauge conditions that need to be imposed to make the perturbations regular at infinity, are qualitatively different in higher dimensions; in particular, the decay rate of a radiating solution at null infinity differs from…
In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice. This geodesic distance is used to numerically determine the Hausdorff dimension associated to shortest cycles of 2d quantum gravity on the torus coupled to conformal matter fields, showing agreement with a conjectured formula by Y. Watabiki. Finally, the numerical tools are put to test by quantitatively comparing the distribution of lengths of shortest cycles to the corresponding distribution in large random trian…
First we contemplate the operational definition of space-time in four dimensions in light of basic principles of quantum mechanics and general relativity and consider some of its phenomenological consequences. The quantum gravitational fluctuations of the background metric that comes through the operational definition of space-time are controlled by the Planck scale and are therefore strongly suppressed. Then we extend our analysis to the braneworld setup with low fundamental scale of gravity. It is observed that in this case the quantum gravitational fluctuations on the brane may become unacc…
Euclidean quantum gravity might be defined by stochastic quantisation that is governed by a higher order Langevin equation rather than a first order stochastic equation. In a transitory phase where the Lorentz time cannot be defined, the parameter that orders the evolution of quantum gravity phenomena is the stochastic time. This changes the definition of causality in the period of primordial cosmology. The prediction of stochastically quantised gravity is that there will a transition from an oscillating quantum phase to a semi-classical one, when the Lorentz time emerges. The end of the trans…
Following Mach's definition of mass and force in terms of space time measurement we build the conceptual structure of classical mechanics and show that it is remarkably similar to the results of Carnot and Clausius in thermodynamics. The analogy is extended to equality to derive the laws of gravity from the laws of thermodynamics.
These notes are a didactic overview of the non perturbative and background independent approach to a quantum theory of gravity known as loop quantum gravity. The definition of real connection variables for general relativity, used as a starting point in the program, is described in a simple manner. The main ideas leading to the definition of the quantum theory are naturally introduced and the basic mathematics involved is described. The main predictions of the theory such as the discovery of Planck scale discreteness of geometry and the computation of black hole entropy are reviewed. The quant…
We generalize the idea of Vassiliev invariants to the spin network context, with the aim of using these invariants as a kinematical arena for a canonical quantization of gravity. This paper presents a detailed construction of these invariants (both ambient and regular isotopic) requiring a significant elaboration based on the use of Chern-Simons perturbation theory which extends the work of Kauffman, Martin and Witten to four-valent networks. We show that this space of knot invariants has the crucial property -from the point of view of the quantization of gravity- of being loop differentiable …
Asymptotic symmetry algebra of Einstein gravity and Lorentz generators
本文证明在渐近平坦四维 Einstein 引力中，基于 Hamiltonian 形式主义和类光无穷远处分别提出的两种消除超平移歧义的 Lorentz 生成元重新定义方法是等价的。
In this paper, we address the problem of the dynamics in three dimensional loop quantum gravity with zero cosmological constant. We construct a rigorous definition of Rovelli's generalized projection operator from the kinematical Hilbert space--corresponding to the quantization of the infinite dimensional kinematical configuration space of the theory--to the physical Hilbert space. In particular, we provide the definition of the physical scalar product which can be represented in terms of a sum over (finite) spin-foam amplitudes. Therefore, we establish a clear-cut connection between the canon…
A new generalization of the Hawking-Hayward quasilocal energy to scalar-tensor gravity is proposed without assuming symmetries, asymptotic flatness, or special spacetime metrics. The procedure followed is simple but powerful and consists of writing the scalar-tensor field equations as effective Einstein equations and then applying the standard definition of quasilocal mass.
What are the conditions on a field theoretic model leading to a finite entanglement entropy density? We prove two very general results: 1) Ultraviolet finiteness of a theory does not guarantee finiteness of the entropy density; 2) If the spectral dimension of the spatial boundary across which the entropy is calculated is non-negative at all scales, then the entanglement entropy cannot be finite. These conclusions, which we verify in several examples, negatively affect all quantum-gravity models, since their spectral dimension is always positive. Possible ways out are considered, including aban…
We explicitly show that the equations of motion for modified gravity theories of $F(R)$-gravity, the scalar-Gauss-Bonnet gravity, $F(\mathcal{G})$-gravity and the non-local gravity are equivalent to the Clausius relation in thermodynamics. In addition, we discuss the relation between the expression of the entropy and the contribution from the modified gravity as well as the matter to the definition of the energy flux (heat).
Quantum complexity in gravity, quantum field theory, and quantum information science
本文梳理跨量子信息、多体物理、量子场论与引力领域的量子复杂度定义、性质及应用，打通不同领域的复杂度研究框架，阐释全息AdS/CFT对偶下复杂度与引力可观测量的关联。
Starting from the definition of entropy used in statistical mechanics we show that it is proportional to the gravity action. For a stationary black hole this entropy is expressed as $S = E/ 2T$, where $T$ is the Hawking temperature and $E$ is shown to be the Komar energy. This relation is also compatible with the generalised Smarr formula for mass.
The framework of slowly evolving horizons is generalized to the case of black branes in asymptotically anti-de Sitter spaces in arbitrary dimensions. The results are used to analyze the behavior of both event and apparent horizons in the gravity dual to boost-invariant flow. These considerations are motivated by the fact that at second order in the gradient expansion the hydrodynamic entropy current in the dual Yang-Mills theory appears to contain an ambiguity. This ambiguity, in the case of boost-invariant flow, is linked with a similar freedom on the gravity side. This leads to a phenomenolo…
Tidal response beyond vacuum General Relativity with a canonical definition
本文指出微扰框架下潮汐响应函数定义的模糊性，提出规范潮汐响应函数（CTRF）以统一比较超越真空广义相对论的理论，并首次推导多种非爱因斯坦理论中黑洞的潮汐耗散洛夫数。
I investigate the relationship between the phase space path integral in (2+1)-dimensional gravity and the canonical quantization of the corresponding reduced phase space in the York time slicing. I demonstrate the equivalence of these two approaches, and discuss some subtleties in the definition of the path integral necessary to prove this equivalence.
I formulate several statements demonstrating that the local metric redefinition can be used to reduce the UV divergences present in the quantum action for the Einstein gravity in $d=4$ dimensions. In its most general form, the proposal is that any UV divergences in the quantum action can be removed by an appropriate field re-definition and a renormalization of cosmological constant.
Entanglement entropy is a valuable tool for characterizing the correlation structure of quantum field theories. When applied to gauge theories, subtleties arise which prevent the factorization of the Hilbert space underlying the notion of entanglement entropy. Borrowing techniques from extended topological field theories, we introduce a new definition of entanglement entropy for both Abelian and non-Abelian gauge theories. Being based on the notion of excitations, it provides a completely relational way of defining regions. Therefore, it naturally applies to background independent theories, e.…
We show that the ``time'' t_s defined via spin clusters in the Ising model coupled to 2d gravity leads to a fractal dimension d_h(s) = 6 of space-time at the critical point, as advocated by Ishibashi and Kawai. In the unmagnetized phase, however, this definition of Hausdorff dimension breaks down. Numerical measurements are consistent with these results. The same definition leads to d_h(s)=16 at the critical point when applied to flat space. The fractal dimension d_h(s) is in disagreement with both analytical prediction and numerical determination of the fractal dimension d_h(g), which is base…
Following the formalism of enveloping algebras and star product calculus we formulate and analyze a model of gauge gravity on noncommutative spaces and examine the conditions of its equivalence to general relativity. The corresponding Seiberg-Witten maps are established which allow the definition of respective dynamics for a finite number of gravitational gauge field components on noncommutative spaces.
We consider defining a fuzzy space by a specific state in a fermionic field theory in terms of which all the observables for the space can be evaluated. This allows for a definition of entanglement for a fuzzy space by direct integration of the fields over a certain region. Even though the resulting formula for the entanglement entropy (EE) is similar to what has been used in the quantum Hall effect, our derivation provides a novel perspective. We also review and strengthen the arguments for the EE to be described by a generalized Chern-Simons form.
New progress in loop gravity has lead to a simple model of `general-covariant quantum field theory'. I sum up the definition of the model in self-contained form, in terms accessible to those outside the subfield. I emphasize its formulation as a generalized topological quantum field theory with an infinite number of degrees of freedom, and its relation to lattice theory. I list the indications supporting the conjecture that the model is related to general relativity and UV finite.
We determine the most general form of the interaction between the gravitational field and an arbitrary Yang-Mills system of fields (massless and massive). We work in the perturbative quantum framework of the causal approach (of Epstein and Glaser) and use a cohomological definition of gauge invariance for both gauge fields. We also consider the case of massive gravity. We discuss the question whether gravity couples to the unphysical degrees of freedom in the Yang-Mills fields.
It is shown that the surface gravity and temperature of a stationary black hole are invariant under conformal transformations of the metric that are the identity at infinity. More precisely, we find a conformal invariant definition of the surface gravity of a conformal Killing horizon that agrees with the usual definition(s) for a true Killing horizon and is proportional to the temperature as defined by Hawking radiation. This result is reconciled with the intimate relation between the trace anomaly and the Hawking effect, despite the {\it non}invariance of the trace anomaly under conformal tr…
Basis and limitations of singularity theorems for Gravity are examined. As singularity is a critical situation in course of time, study of time paths, in full generality of Equivalence principle, provides two mechanisms to prevent singularity. Resolution of singular Time translation generators into space of its orbits, and essential higher dimensions for Relativistic particle interactions has facets to resolve any real singularity problem. Conceptually, these varied viewpoints have a common denominator: arbitrariness in the definition of `energy' intrinsic to the space of operation in each cas…
We study further the rôle of the boundary operator $Ø_B$ for macroscopic loop length in the stable definition of 2D quantum gravity provided by the $[{\tilde P},Q]=Q$ formulation. The KdV flows are supplemented by an additional flow with respect to the boundary cosmological constant $σ$. We numerically study these flows for the $m=1$, $2$ and $3$ models, solving for the string susceptibility in the presence of $Ø_B$ for arbitrary coupling $σ$. The spectrum of the Hamiltonian of the loop quantum mechanics is continuous and bounded from below by $σ$. For large positive $σ$, the theory is dominat…
We introduce the key ideas behind the group field theory approach to quantum gravity, and the basic elements of its formalism. We also briefly report on some recent results obtained in this approach, concerning both the mathematical definition of these models, and possible avenues towards extracting interesting physics from them.
In this paper we discuss massive gravity in de Sitter space via gravitational Higgs mechanism, which provides a nonlinear definition thereof. The Higgs scalars are described by a nonlinear sigma model, which includes higher derivative terms required to obtain the Fierz-Pauli mass term. Using the aforesaid non-perturbative definition, we address appearance of an enhanced local symmetry and a null norm state in the linearized massive gravity in de Sitter space at the special value of the graviton mass to the Hubble parameter ratio. By studying full non-perturbative equations of motion, we argue …
A general homogeneous two dimensional dilaton gravity model considered recently by Lemos and S\` a, is given quantum matter Polyakov corrections and is solved numerically for several static, equilibrium scenarii. Classically the dilaton field ranges the whole real line, whereas in the semi-classical theory, with the usual definition, it is always below a certain critical value at which a singularity appears. We give solutions for both sub- and super-critical dilaton field. The pasting together of the spacetime on both sides of a singularity in semi-classical planar general relativity is discus…
Recent results in Local Regge Calculus are confronted with Spin Foam Formalism. Introducing Barrett-Crane Quantization in Local Regge Calculus makes it possible to associate a unique Spin $j_{h}$ with an hinge $h$, fulfilling one of the requirements of Spin Foam definition. It is shown that inter-twiner terms of Spin Foam can follow from the closure constraint in Local Regge Calculus. Dedicated to Beppe Marmo for his 65th Birthday
Without imposing the trapping boundary conditions and only from within the very definition of area it is shown that the loop quantization of area manifests an unexpected degeneracy in area eigenvalues. This could lead to a deeper understanding of the microscopic description of a quantum black hole. If a certain number of semi-classically expected properties of black holes are imposed on a quantum surface its entropy coincides with the Bekenstein-Hawking entropy.
This paper proposes a definition of gravitational observables and of their path integral formula within the framework of ADM foliation and the choice of unimodular gauge classes. The method enforces a BRST invariant gauge fixing of the lapse and shift fields. It yields the quantum level extension of the known classical property that the conformal classes of internal metrics of constant Lorentz time leafs define the gravitational physical degrees of freedom.
It is shown that every Feynman integral can be interpreted as Green function of some linear differential operator with constant coefficients. This definition is equivalent to usual one but needs no regularization and application of $R$-operation. It is argued that presented formalism is convenient for practical calculations of Feynman integrals.
String theory tells us that quantum gravity has a dual description as a field theory (without gravity). We use the field theory dual to ask what happens to an object as it falls into the simplest black hole: the 2-charge extremal hole. In the field theory description the wavefunction of a particle is spread over a large number of `loops', and the particle has a well-defined position in space only if it has the same `position' on each loop. For the infalling particle we find one definition of `same position' on each loop, but there is a different definition for outgoing particles and no canonic…
In the light of recent interest in quantum gravity in de Sitter space, we investigate semi-classical aspects of 4-dimensional Schwarzschild-de Sitter space-time using the method of complex paths. The standard semi-classical techniques (such as Bogoliubov coefficients and Euclidean field theory) have been useful to study quantum effects in space-times with single horizons; however, none of these approaches seem to work for Schwarzschild-de Sitter or, in general, for space-times with multiple horizons. We extend the method of complex paths to space-times with multiple horizons and obtain the spe…
We investigate theories in which gravity arises as a consequence of entropy. We distinguish between two approaches to this idea: holographic gravity, in which Einstein's equation arises from keeping entropy stationary in equilibrium under variations of the geometry and quantum state of a small region, and thermodynamic gravity, in which Einstein's equation emerges as a local equation of state from constraints on the area of a dynamical lightsheet in a fixed spacetime background. Examining holographic gravity, we argue that its underlying assumptions can be justified in part using recent result…
We present a systematic definition and analysis of the thermo-electric linear response in gauge/gravity systems focusing especially on models with massive gravity in the bulk and therefore momentum dissipation in the dual field theory. A precise treatment of finite counter-terms proves to be essential to yield a consistent physical picture whose hydrodynamic and beyond-hydrodynamics behaviors noticeably match with field theoretical expectations. The model furnishes a possible gauge/gravity description of the crossover from the quantum-critical to the disorder-dominated Fermi-liquid behaviors, …
We present a method for the first principles calculation of tachyon one-point amplitudes in $(2,2p+1)$ minimal Liouville gravity defined on a torus. The method is based on the higher equations of motion in the Liouville CFT. These equations were earlier successfully applied for analytic calculations of the amplitudes in the spherical topology. We show that this approach allows to reduce the moduli integrals entering the definition of the torus amplitudes to certain boundary contributions, which can be calculated explicitly. The results agree with the calculations performed in the matrix models…
We give a brief overview of the string landscape and techniques used to construct string compactifications. We then explain how this motivates the notion of the swampland and review a number of conjectures that attempt to characterize theories in the swampland. We also compare holography in the context of superstrings with the similar, but much simpler case of topological string theory. For topological strings, there is a direct definition of topological gravity based on a sum over a "quantum gravitational foam." In this context, holography is the statement of an identification between a gravi…
A Background Independent Algebra in Quantum Gravity
本文提出沿观测者世界线的算子代数作为量子引力中的背景无关代数，并以哈特尔-霍金无边界态为最大熵普适态，用相对熵定义熵，在德西特真空情形下得到合理结果。
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