Loop quantum cosmology has achieved great successes, in which the polymerization plays a crucial role. In particular, the phase-space-variable dependent polymerization turns out to be the unique one that leads to consistent quantization of the homogeneous and isotropic universe. However, when applying the same scheme to the quantization of black holes, it meets resistances, when the Kantowski-Sachs (KS) gauge is adopted. In this paper, we continue to study the quantum effects of the polymerization near the location that a classical black hole horizon used to be, from the point of view of effective loop quantum gravity in the KS gauge. In particular, we find a phase-space-variable dependent polymerization scheme that leads to negligible quantum effects near the location of the classical black hole horizon, but significantly alters the spacetime structure near the origin, so that the classical singularity is finally replaced by a finite and regular transition surface. The final geodesically-complete spacetime consists of the regular transition surface that connects a black hole in one side and an anti-trapped region in the other side. In the anti-trapped region, no white hole horizons are found and the spacetime is extended to infinity, at which the geometric radius of the two-spheres becomes infinitely large.
We discuss the quantization of pure string--inspired dilaton--gravity in $(1+1)$--dimensions, and of the same theory coupled to scalar matter. We perform the quantization using the functional Schroedinger and BRST formalisms. We find, both for pure gravity and the matter--coupled theory, that the two quantization procedures give inequivalent ``physical'' results.
We compare three approaches to the quantization of (2+1)-dimensional gravity with a negative cosmological constant: reduced phase space quantization with the York time slicing, quantization of the algebra of holonomies, and quantization of the space of classical solutions. The relationships among these quantum theories allow us to define and interpret time-dependent operators in the ``frozen time'' holonomy formulation.
We discuss string theory relations between gravity and gauge theory tree amplitudes. Together with $D$-dimensional unitarity, these relations can be used to perturbatively quantize gravity theories, i.e. they contain the necessary information for calculating complete gravity $S$-matrices to any loop orders. This leads to a practical method for computing non-trivial gravity $S$-matrix elements by relating them to much simpler gauge theory ones. We also describe arguments that N=8 D=4 supergravity is less divergent in the ultraviolet than previously thought.
It was recently observed in \cite{Park:2014tia} that the holographic nature of gravity may hold a key to quantization of gravity. The so-called "holographic quantization" has been carried out in \cite{Park:2014noa,Park:2015ota} for Einstein gravity in a flat background. Generalizing the procedure to a curved background is the main goal of the present work. In particular, we consider the Einstein action expanded around a Schwarzschild background
We perform canonical quantization of gravity in the background of a Schwarzschild black hole in the generalized Regge-Wheeler gauge proposed in \cite{Kallosh:2021ors}. We find that the Hamiltonian at the quadratic level is unitary and ghost-free. Two canonical degrees of freedom are associated with Zerilli-Moncrief and Cunningham-Price-Moncrief functions of the metric perturbations. The ${\ell }<2$ part of the Hamiltonian vanishes. This quantization with the unitary Hamiltonian for gravity is valid also in Minkowski space in spherical coordinates.
The spinfoam approach to quantum gravity rests on a "quantization" of BF theory using 2-complexes and group representations. We explain why, in dimension three and higher, this "spinfoam quantization" must be amended to be made consistent with the gauge symmetries of discrete BF theory. We discuss a suitable generalization, called "cellular quantization", which (1) is finite, (2) produces a topological invariant, (3) matches with the properties of the continuum BF theory, (4) corresponds to its loop quantization. These results significantly clarify the foundations - and limitations - of the sp…
While many aspects of general relativity have been tested, and general principles of quantum dynamics demand its quantization, there is no direct evidence for that. It has been argued that development of detectors sensitive to individual gravitons is unlikely, and perhaps impossible. We argue here, however, that measurement of polarization of the Cosmic Microwave Background due to a long wavelength stochastic background of gravitational waves from Inflation in the Early Universe would firmly establish the quantization of gravity.
It is pointed out that the action recently proposed by Bañados et al. for gravitation in odd dimensions higher (and lower) than four, provides a natural quantization for the gravitational constant. These theories possess no dimensionful parameters and hence they may be power counting renormalizable.
Fundamental quantum gravity theories are known to be notoriously difficult to extract viable testable predictions out of. In this paper, we aim to incorporate putative quantum corrections coming from loop quantum gravity in deriving modified dispersion relations for particles on a deformed Minkowski spacetime. We show how different choices of the Immirzi parameter can, in some cases, serendipitously lead to different outcomes for such modifications, depending on the quantization scheme chosen. This allows one to differentiate between these quantization choices via testable phenomenological pre…
A previously used quantization mechanism is applied to the continuous states of the shielded strong gravity scenario (hep-th/0602183), yielding two types of spectra for uncharged black hole scalars. Each yields the general morphology for states expected in this scenario at LHC and at arbitrarily higher energies, once the parameters are determined by the two lowest-lying scalar states. A particularized example for the preferred type of quantization is numerically evaluated.
We discuss the relation between canonical and Schrodinger quantization of the CGHS model. We also discuss the situation when background charges are added to cancel the Virasoro anomaly. New physical states are found when the square of the background charges vanishes.
A method for consistent quantization of conformal gravity treating conformal symmetry in a very controllable way is presented. First, we discuss local conformal symmetry in the framework of gravitational interactions, where we view it as an example of a general gauge theory. We also present some early attempts at quantization of conformal gravity and use the generalized framework of covariant quantization due to Faddeev and Popov. Some salient issues such as the need for conformal gauge-fixing, an issue with conformal third ghosts, and discontinuities in conformal gravity are studied as well. …
We consider the loop quantization of Maxwell theory. A quantization of this type leads to a quantum theory in which the fundamental excitations are loop-like rather than particle-like. Each such loop plays the role of a quantized Faraday's line of electric flux. We find that the quantization depends on an arbitrary choice of a parameter e that carries the dimension of electric charge. For each value of e an electric charge that can be contained inside a bounded spatial region is automatically quantized in units of hbar/4*pi*e. The requirement of consistency with the quantization of electric ch…
De Sitter Chern-Simons gravity in D = 1 + 2 spacetime is known to possess an extension with a Barbero-Immirzi like parameter. We find a partial gauge fixing which leaves a compact residual gauge group, namely SU(2). The compacticity of the residual gauge group opens the way to the usual LQG quantization techniques. We recall the exemple of the LQG quantization of SU(2) CS theory with cylindrical space topology, which thus provides a complete LQG of a Lorentzian gravity model in 3-dimensional space-time.
A generalization of the recently formulated nonlinear quantization of a parameterized theory is presented in the context of quantum gravity. The parametric quantization of a Friedmann universe with a massless scalar field is then considered in terms of analytic solutions of the resulting evolution equations.
In this paper we carry out a quantization of gravity coupled to massless spin 1/2 fermions through the instanton representation of Plebanski gravity. We have constructed a Hilbert space of states for this model, and we have computed the Hamilton's equations of motion. The classical equations appear, at a superficial level, to be consistent with the quantum dynamics of the theory.
We show that the noncritical string field theory developed from two-dimensional quantum gravity in the framework of causal dynamical triangulations can be viewed as arising through a stochastic quantization. This requires that the proper time appearing in the string field theory be identified with the stochastic time of the stochastic formulation. The framework of stochastic quantization gives rise to a natural nonperturbative quantum Hamiltonian, which incorporates a sum over all spacetime topologies. We point out that the external character of stochastic time is a feature that pertains more …
We study the quantization of two versions of unimodular gravity, namely, fully diffeomorphism-invariant unimodular gravity and unimodular gravity with fixed metric determinant utilizing standard path integral approach. We derive the BRST symmetry of effective actions corresponding to several relevant gauge conditions. We observe that for some gauge conditions, the restricted gauge structure may complicate the formulation and effective actions, in particular, if the chosen gauge conditions involve the canonical momentum conjugate to the induced metric on the spatial hypersurface. The BRST symme…
Gravity does not naturally fit well with canonical quantization. Affine quantization is an alternative procedure that is similar to canonical quantization but may offer a positive result when canonical quantization fails to offer a positive result. Two simple examples given initially illustrate the power of affine quantization. These examples clearly point toward an affine quantization procedure that vastly simplifies a successful quantization of general relativity.
We study quantum corrections to projectable Horava gravity with $z = 2$ scaling in 2+1 dimensions. Using the background field method, we utilize a non-singular gauge to compute the anomalous dimension of the cosmological constant at one loop, in a normalization adapted to the spatial curvature term.
Based on certain assumptions for the expectation value of a product of the quantum fluctuating metric at two points, the gravitational and scalar field Lagrangians are evaluated. Assuming a vanishing expectation value of the first order terms of the metric, the calculations are performed with an accuracy of second order. It is shown that such quantum corrections give rise to modified gravity.
It is well known that Einstein gravity is non-renormalizable; however a generalized approach is proposed that leads to Einstein gravity {\it after} renormalization. This them implies that at least one candidate for quantum gravity treats all matter on an equal footing with regard to the gravitational behaviour.
This is a review article on quantum gravity. In section 1, the Penrose singularity theorem is proven. In section 2, the covariant quantization approach of gravity is reviewed. In section 3, an article by Hawking is reviewed that shows the gravitational path integral at one loop level to be dominated by contributions from some kind of virtual gravitational instantons. In section 4, the canonical, non-perturbative quantization approach is reviewed. In section 5, arguments from Hawking are mentioned which show the gravitational path integral to be an approximate solution of the Wheeler deWitt equ…
We re-examine the process of loop quantization for flat isotropic models in cosmology. In particular, we contrast different inequivalent `loop quantizations' of these simple models through their respective successes and limitations and assess whether they can lead to any viable physical description. We propose three simple requirements which any such admissible quantum model should satisfy: i) independence from any auxiliary structure, such as a fiducial interval/cell introduced to define the phase space when integrating over non-compact manifolds; ii) existence of a well defined classical lim…
Local path integral quantization of generic 2D dilaton gravity is considered. Locality means that we assume asymptotic fall off conditions for all fields. We demonstrate that in the absence of `matter' fields to all orders of perturbation theory and for all 2D dilaton theories the quantum effective action coincides with the classical one. This resolves the apparent contradiction between the well established results of Dirac quantization and perturbative (path-integral) approaches which seemed to yield non-trivial quantum corrections. For a particular case, the Jackiw--Teitelboim model, our res…
An abelian version of standard general relativity in the Cartan-Palatini gauge-like formulation in four dimensions has been introduced. Traditional canonical analysis utilizing similarities to the akin Husain-Kuchar SU(2) version of gravity has been performed. The model has been next quantized in the canonical path-integral Faddeev-Popov formalism yielding abelian BF theory.
The quantum theory of gravity is considered based on the assumption that gravitational interaction occurs by means of the vector field of the Planck mass. Gravitational emission is considered as a process of the decay of proton into some matter fields at the Planck scale. Within the framework of grand unification vector field of the Planck mass may be thought of as those which realize the interaction between leptons and quarks.
We consider the quantization of the midi-superspace associated with a class of spacetimes with toroidal isometries, but without the compact spatial hypersurfaces of the well-known Gowdy models. By a symmetry reduction, the phase space for the system at the classical level can be identified with that of a free massless scalar field on a fixed background spacetime, thereby providing a simple route to quantization. We are then able to study certain non-perturbative features of the quantum gravitational system. In particular, we examine the quantum geometry of the asymptotic regions of the spaceti…
In this paper is considered a generalized quantization principle for the gravitational field in canonical quantum gravity, especially with respect to quantum geometrodynamics. This assumption can be interpreted as a transfer from the generalized uncertainty principle in quantum mechanics, which is postulated as generalization of the Heisenberg algebra to introduce a minimal length, to a corresponding quantization principle concerning the quantities of quantum gravity. According to this presupposition there have to be given generalized representations of the operators referring to the observabl…
Canonical quantization (CQ) is built around $[Q,P]=i\hbar1\!\!1$, while affine quantization (AQ) is built around $[Q,D]=i\hbar\,Q$, where $D\equiv(PQ+QP)/2$. The basic CQ operators must fit $-\infty< P, Q <\infty$, while the basic AQ operators can fit $-\infty<P<\infty$ and $ 0<Q<\infty$, $-\infty <Q<0$, or even $-\infty<Q\neq0<\infty$. AQ can also be the key to quantum gravity, as our simple outline demonstrates.
A canonical quantization of two-dimensional gravity minimally coupled to real scalar and spinor Majorana fields is presented. The physical state space of the theory is completely described and calculations are also made of the average value of the metric tensor relative to states close to the ground state
We present an interpretation of loop quantization in the framework of lattice gauge theory. Within this context the lack of appropriate notions of effective theories and renormalization group flow exhibit loop quantization as an incomplete framework. This interpretation includes a construction of embedded spin foam models which does not rely on the choice of any auxiliary structure (e.g. triangulation) and has the following straightforward consequences: (1) The values of the coupling constants need to be those of an UV-attractive fixed point (2) The kinematics of canonical loop quantization an…
Nonholonomic distributions and adapted fame structures on (pseudo) Riemannian manifolds of even dimension are employed to build structures equivalent to almost Kahler geometry and which allows to perform a Fedosov-like quantization of gravity. The nonlinear connection formalism that was formally elaborated for Lagrange and Finsler geometry is implemented in classical and quantum Einstein gravity.
The vacuum Gowdy models provide much studied, non-trivial midi-superspace examples. Various technical issues within Loop Quantum Gravity can be studied in these models as well as one can hope to understand singularities and their resolution in the loop quantization. The first step in this program is to reformulate the model in real connection variables in a manner that is amenable to loop quantization. We begin with the unpolarized model and carry out a consistent reduction to the polarized case. Carrying out complete gauge fixing, the known solutions are recovered.
We perform the Hamiltonian analysis of unimodular gravity in terms of the connection representation. The unimodular condition is imposed straightforwardly into the action with a Lagrange multiplier. After classifying constraints into first class and second class, the canonical quantization is carried out. We consider the difference of the corresponding physical states between unimodular gravity and general relativity.
We construct phase-space structure of a typical higher-order theory of gravity, in the background of anisotropic Bianchi-1 mini-superspace, following `Modified Horowitz Formalism' as well as applying `Dirac Algorithm' (after taking care of the divergent terms), and establish equivalence. Canonical quantization, and semiclassical approximation are performed to expatiate the fact that such a quantum theory transits successfully to a classical de-Sitter universe. Inflation has thereafter been studied. The numerical values of the inflationary parameters show excellent agreement with the latest rel…
Model of noncommutative gravity is constructed by means of Fedosov deformation quantization of endomorphism bundle. The fields describing noncommutativity -- symplectic form and symplectic connection -- are dynamical, and the resulting theory is completely coordinate covariant and background independent. Its interpretation in terms of Seiberg-Witten map is provided. Also, new action for ordinary (commutative) general relativity is given, which in the present context appears as a commutative limit of noncommutative theory.
There is no conclusive theoretical argument that would require the quantization of the gravitational field. A paradox presented by Mari et al. [Sci. Rep. 6, 22777 (2016)] and its resolution by Belenchia et al. [Phys. Rev. D 98, 126009 (2018)] did not change anything about this assertion. Despite being interesting in other respects, efforts of a recent work by Ryding, Aurell, and Pikovski [arXiv:2107.07514] to further establish the inconclusiveness of the question of necessity of quantization are futile.
We give a brief introduction to matrix models and the group field theory (GFT) formalism as realizations of the idea of a third quantization of gravity, and present in some more detail the idea and basic features of a continuum third quantization formalism in terms of a field theory on the space of connections, building up on the results of loop quantum gravity that allow to make the idea slightly more concrete. We explore to what extent one can rigorously define such a field theory. Concrete examples are given for the simple case of Riemannian GR in 3 spacetime dimensions. We discuss the rela…
The present article summarizes the work of the papers \cite{1} dealing with the quantization of pure gravity and gravity coupled to a Maxwell field and a cosmological constant in presence of spherical symmetry. The class of models presented is intended as an interesting testing ground for the quantization of full 3+1 gravity. We are working in Ashtekar's self-dual representation.
Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with non-renormalizable scalar models as well as quantum gravity. The basic applications of this approach lead to the common goals of any quantization, such as Schroedinger's representation and Schroedinger's equation. Careful attention is paid toward seeking favored classical variables, which are those that should be promoted to the principal quantum operators. This effort leads toward classical variables that have a constant positive, zero, or negative curvature, which typically characteriz…
The derivation of the recently proposed nonlinear quantum evolution of gravity from an action principle is considered in this brief note. It is shown to be possible if a set of consistency conditions are satisfied that are analogous to the Dirac relations for the super-Hamiltonian and momenta in classical canonical gravity.
In loop quantum gravity, modifications to the geometrical density cause a self-interacting scalar field to accelerate away from a minimum of its potential. In principle, this mechanism can generate the conditions that subsequently lead to slow-roll inflation. The consequences for this mechanism of various quantization ambiguities arising within loop quantum cosmology are considered. For the case of a quadratic potential, it is found that some quantization procedures are more likely to generate a phase of slow--roll inflation. In general, however, loop quantum cosmology is robust to ambiguities…
In this work we explore ideas in quantizing AdS$_3$ Einstein gravity. We start with the most general solution to the 3d gravity theory which respects Brown-Henneaux boundary conditions. These solutions are specified by two holomorphic functions and satisfy simple superposition rule. These geometries generically have a bifurcate Killing horizon (with a noncompact bifurcation curve) which is not an event horizon and are hence not black holes. Nonetheless, there are superpositions of these geometries which have event horizon. We propose to view these geometries as "semiclassical fuzzball microsta…
We develop a simple scheme of quantization for the dilaton CGHS model without scalar fields, that uses the Gupta-Bleuler approach for the string fields. This is possible because the constraints can be linearized classically, due to positivity conditions that are present in the model (and not in the general string case). There is no ambiguity nor anomalies in the quantization. The expectation values of the metric and dilaton fields obey the classical requirements, thus exhibiting at the quantum level the Birkhoff theorem.
In our previous paper, we have presented a covariant BRST quantization of unimodular gravity which may account for the smallness of the cosmological constant, and have shown that the physical degrees of freedom in the theory are the same as general relativity. The formulation has been given by using rank-2 antisymmetric tensor fields for both ghosts and antighosts. Here we give an alternative formulation using a vector field for the antighost but keeping the same structure for the ghosts. This gives a significantly simpler covariant quantization with less ghosts and no tripole modes in the gho…
In this paper we quantize the Klein--Gordon scalar field coupled to gravity in the instanton representation for spatially homogeneous variables. The construction provides a well-defined Hilbert space of states for generic self-interaction potential V, with a direct link to the semiclassical limit below the Planck scale. Additionally, we compute the Hamilton's equations of motion for this model, performing various consistency checks on the quantum theory.
We have recently proposed in \cite{Park:2014tia} the quantization of pure 4D Einstein gravity through hypersurface foliation, and observed that the 4D Einstein gravity becomes renormalizable once all (or most) of the unphysical degrees of freedom are removed. In this work, we confirm this observation from a more mathematical angle. In particular, we show that the physical state condition arising from the shift vector constraint connects with the requirement that the manifold admit "totally geodesic (TG) foliation". The TG foliation, in turn, makes it possible to view the 4D manifold as abelian…
In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant $Λ>0$ in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2+1 dimensions is coordinatized by an SU(2) connection $A$ and the canonically conjugate triad field $e$. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of $A+=A + \sqrtΛe$. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection $A_λ=A+λe$ on the kinematical Hilbert spa…
Loop Quantum Gravity is widely developed using canonical quantization in an effort to find the correct quantization for gravity. Affine quantization, which is like canonical quantization augmented bounded in one orientation, e.g., a strictly positive coordinate. We open discussion using canonical and affine quantizations for two simple problems so each procedure can be understood. That analysis opens a modest treatment of quantum gravity gleaned from some typical features that exhibit the profound differences between aspects of seeking the quantum treatment of Einstein's gravity.
We develop a formalism to calculate the response of a model gravitational wave detector to a quantized gravitational field. Coupling a detector to a quantum field induces stochastic fluctuations ("noise") in the length of the detector arm. The statistical properties of this noise depend on the choice of quantum state of the gravitational field. We characterize the noise for vacuum, coherent, thermal, and squeezed states. For coherent states, corresponding to classical gravitational configurations, we find that the effect of gravitational field quantization is small. However, the standard devia…
We present a first attempt to apply the approach of deformation quantization to linearized Einstein's equations. We use the analogy with Maxwell equations to derive the field equations of linearized gravity from a modified Maxwell Lagrangian which allows the construction of a Hamiltonian in the standard way. The deformation quantization procedure for free fields is applied to this Hamiltonian. As a result we obtain the complete set of quantum states and its discrete spectrum.
The main goal of this work is to explore the symmetries and develop the dynamics associated to a 3D Abelian BF model coupled to scalar fields submitted to a sigma model like constraint, at the classical and quantum levels. We apply to the present model the techniques of Loop Quantum Gravity, construct its physical Hilbert space and its observables.
On the basis of dynamic quantization method we build in this paper a new mathematically correct quantization scheme of gravity. In the frame of this scheme we develop a canonical formalism in tetrad-connection variables in 4-D theory of pure gravity. In this formalism the regularized quantized fields corresponding to the classical tetrad and connection fields are constructed. It is shown, that the regularized fields satisfy to general covariant equations of motion, which have the classical form. In order to solve these equations the iterative procedure is offered.
We investigate a minisuperspace model of Einstein gravity plus dilaton that describes a static spherically symmetric configuration or a Kantowski - Sachs like universe. We develop the canonical formalism and identify canonical quantities that generate rigid symmetries of the Hamiltonian. Quantization is performed by the Dirac and the reduced methods. Both approaches lead to the same positive definite Hilbert space.
Unimodular gravity (UG) is an interesting theory that may explain why the cosmological constant is extremely small, in contrast to general relativity (GR). The theory has only the transverse diffeomorphism invariance and this causes lots of debate as to the equivalence of UG to GR in the covariant quantization. We study the covariant BRST quantization of UG by gauge fixing only the transverse diffeomorphism and show that the remaining physical degrees of freedom are two, the same number as GR. This is achieved by using antisymmetric tensor ghost fields which automatically satisfy the transvers…
We present a complete quantization of Lorentzian D=1+2 gravity with cosmological constant, coupled to a set of topological matter fields. The approach of Loop Quantum Gravity is used thanks to a partial gauge fixing leaving a residual gauge invariance under a compact semi-simple gauge group, namely Spin(4) = SU(2) x SU(2). A pair of quantum observables is constructed, which are non-trivial despite of being null at the classical level.
We perform a covariant (Lagrangian) quantization of perturbative gravity in the background of a Schwarzschild black hole. The key tool is a decomposition of the field into spherical harmonics. We fix Regge-Wheeler gauge for modes with angular momentum quantum number $l \geq 2$, while for low multipole modes with $l$ $=$ $0$ or $1$ -- for which Regge-Wheeler gauge is inapplicable -- we propose a set of gauge fixing conditions which are 2D background covariant and perturbatively well-defined. We find that the corresponding Faddeev-Popov ghosts are non-propagating for the $l\geq2$ modes, but are …
