Conformal sector of Quantum Einstein Gravity in the local potential approximation: non-Gaussian fixed point and a phase of unbroken diffeomorphism invariance
We explore the nonperturbative renormalization group flow of Quantum Einstein Gravity (QEG) on an infinite dimensional theory space. We consider "conformally reduced" gravity where only fluctuations of the conformal factor are quantized and employ the Local Potential Approximation for its effective average action. The requirement of "background independence" in quantum gravity entails a partial differential equation governing the scale dependence of the potential for the conformal factor which differs significantly from that of a scalar matter field. In the infinite dimensional space of potential functions we find a Gaussian as well as a non-Gaussian fixed point which provides further evidence for the viability of the asymptotic safety scenario. The analog of the invariant cubic in the curvature which spoils perturbative renormalizability is seen to be unproblematic for the asymptotic safety of the conformally reduced theory. The scaling fields and dimensions of both fixed points are obtained explicitly and possible implications for the predictivity of the theory are discussed. Spacetime manifolds with $R^d$ as well as $S^d$ topology are considered. Solving the flow equation for the potential numerically we obtain examples of renormalization group trajectories inside the ultraviolet critical surface of the non-Gaussian fixed point. The quantum theories based upon some of them show a phase transition from the familiar (low energy) phase of gravity with spontaneously broken diffeomorphism invariance to a new phase of unbroken diffeomorphism invariance; the latter phase is characterized by a vanishing expectation value of the metric.
This paper is intended to study diffeomorphism invariance and diffeomorphism generation in the modified theory of gravity proposed by Horava. Firstly, we demonstrate that the theory does not lose diffeomorphism invariance due to the parameter $λ$, as it was previously believed. However, we show that the presence of terms containing the Levi-Civita symbol in the original proposal of Horava makes the theory diffeomorphism dependent. By neglecting such terms, what returns fully diffeomorphism invariance to the action, we obtain the equations of motion. Secondly, in the Hamiltonian formalism, we c…
In this paper, we explicitly prove the presymplectic forms of the Palatini and Ashtekar gravity to be zero along gauge orbits of the Lorentz and diffeomorphism groups, which ensures the diffeomorphism invariance of these theories.
A comprehensive account of a new structured algorithm for obtaining nonrelativistic diffeomorphism invariances in both space and spacetime by gauging the Galilean symmetry in a generic nonrelativistic field theoretical model is provided. Various applications like the obtention of nonrelativistic diffeomorphism invariance, the introduction of Chern-Simons term and its role in fractional quantum Hall effect, induction of diffeomorphism in irrotational fluid model, abstraction of Newton-Cartan geometry and the emergence of Horava-Lifshitz gravity are discussed in details.
A scalar field can be inserted in Maxwell and/or Einstein theory to effect symmetry breaking. Consequences of such a modification are discussed. Possible dynamics for the scalar field are presented.
The regularities at large distances of complete gauge invariance in QCD are shown to bear nontrivial consequences for the selection among inequivalent representations of canonical commutation (anticommutation) rules for gauge boson (quark) fields. The trace anomaly forces a modification of the gauge boson Lagrangean and by this of the entire associated canonical structure.
An algorithmic approach towards the formulation of non-relativistic diffeomorphism invariance has been developed which involves both matter and gauge fields. A step by step procedure has been provided which can accommodate all types of (abelian) gauge interaction. The algorithm is applied to the problem of a two dimensional electron moving under an external field and also under the Chern-Simons dynamics.
In this paper we consider generalised Proca theories coupled to any background field and with time-time and time-space components of Hessian of the vector sector are zero, whereas the space-space part is non-degenerate. By using Faddeev-Jackiw analysis, we derive the conditions that these theories have to satisfy in order for the vector sector to have three propagating degrees of freedom. Most of these conditions are trivialised due to diffeomorphism invariance requirements. This leaves only a condition that a complicated combination of terms should not be trivially zero. This condition is the…
The interplay between the diffeomorphism and conformal symmetries (a feature common in quantum field theories) is shown to be exhibited for the case of black holes in two dimensional classical Liouville theory. We show that although the theory is conformally invariant in the near horizon limit, there is a breaking of the diffeomorphism symmetry at the classical level. On the other hand, in the region away from the horizon, the conformal symmetry of the theory gets broken with the diffeomorphism symmetry remaining intact.
In the $torsion \otimes curvature$ approach of gravity Chern-Simons modification has been considered here. It has been found that Chern-Simons contribution to the bianchi identity has become cancelled from that of the scalar field part. But "homogeneity and isotropy" consideration of present day cosmology is a consequence of the "strong equivalence principle" and vice-versa.
We complement investigations of the gauge structure of general relativity with an analysis of the physical consequences of the spontaneous breaking of diffeomorphism invariance which manifests itself in e.g. the socalled Einstein hole problem. We analyze the nature of the gravitons as Goldstone excitations both in the classical and the quantum case. We show that the metrical field and the classical space-time manifold play the role of an order parameter field and order parameter manifold as macroscopic super structures living in an underlying presumed quantum space-time. We furthermore relate …
It is shown that when the Einstein-Hilbert Lagrangian is considered without any non-covariant modifications or change of variables, its Hamiltonian formulation leads to results consistent with principles of General Relativity. The first-class constraints of such a Hamiltonian formulation, with the metric tensor taken as a canonical variable, allow one to derive the generator of gauge transformations, which directly leads to diffeomorphism invariance. The given Hamiltonian formulation preserves general covariance of the transformations derivable from it. This characteristic should be used as th…
We propose a new theory of gravitation on noncommutative space-time which is invariant under the general coordinate transformations, while the local Lorentz invariance is realized as twisted gauge symmetry. Our theory is remarkably simpler compared to the existing formulations of noncommutative gravity.
We provide a new formulation of nonrelativistic diffeomorphism invariance. It is generated by localising the usual global Galilean Symmetry. The correspondence with the type of diffeomorphism invariant models currently in vogue in the theory of fractional quantum Hall effect has been discussed. Our construction is shown to open up a general approach of model building in theoretical condensed matter physics. Also, this formulation has the capacity of obtaining Newton - Cartan geometry from the gauge procedure.
A new analysis of the gauge invariances and their unity with diffeomorphism invariances in second order metric gravity is presented which strictly follows Dirac's constrained Hamiltonian approach.
We show that the Wilsonian renormalization group (RG) provides a natural regularisation of the Quantum Master Equation such that to first order the BRST algebra closes on local functionals spanned by the eigenoperators with constant couplings. We then apply this to quantum gravity. Around the Gaussian fixed point, RG properties of the conformal factor of the metric allow the construction of a Hilbert space $\mathfrak{L}$ of renormalizable interactions, non-perturbative in $\hbar$, and involving arbitrarily high powers of the gravitational fluctuations. We show that diffeomorphism invariance is…
If general relativity is an emergent phenomenon, there may be small violations of diffeomorphism invariance. We propose a phenomenology of perturbatively small violations of general relativity by the inclusion of terms which break general covariance. These can be tested by matching to the Parametrized Post Newtonian (PPN) formalism. The most sensitive tests involve pulsar timing and provide an extremely strong bound, with a dimensionless constraint of order 10^{-20} relative to gravitational strength.
We study certain aspects of the recently proposed notion of nonrelativistic diffeomorphism invariance. In particular, we consider specific examples of invariant actions, extended gauge symmetry as well as an application to the theory of quantum Hall effect. We also discuss an alternative approach based on Horava-Lifshitz gravity.
We present an investigation on the invariance properties of noncommutative Yang-Mills theory in two dimensions under area preserving diffeomorphisms. Stimulated by recent remarks by Ambjorn, Dubin and Makeenko who found a breaking of such an invariance, we confirm both on a fairly general ground and by means of perturbative analytical and numerical calculations that indeed invariance under area preserving diffeomorphisms is lost. However a remnant survives, namely invariance under linear unimodular tranformations.
We argue that the resolution to the black hole information paradox lies in a proper accounting of the implications of diffeomorphism invariance for the Hilbert space and observables of quantum gravity. The setting of asymptotically Anti-de Sitter spacetime is adopted for most of the paper, but in the framework of canonical quantum gravity, without invoking AdS/CFT duality. We present Marolf's argument that boundary unitarity is a consequence of diffeomorphism invariance, and show that its failure to apply in the classical limit results from a lack of analyticity that has no quantum counterpart…
We propose a lattice counterpart of diffeomorphism symmetry in the continuum. A functional integral for quantum gravity is regularized on a discrete set of space-time points, with fermionic or bosonic lattice fields. When the space-time points are positioned as discrete points of a continuous manifold, the lattice action can be reformulated in terms of average fields within local cells and lattice derivatives. Lattice diffeomorphism invariance is realized if the action is independent of the positioning of the space-time points. Regular as well as rather irregular lattices are then described by…
The key ingredient for lattice regularized quantum gravity is diffeomorphism symmetry. We formulate a lattice functional integral for quantum gravity in terms of fermions. This allows for a diffeomorphism invariant functional measure and avoids problems of boundedness of the action. We discuss the concept of lattice diffeomorphism invariance. This is realized if the action does not depend on the positioning of abstract lattice points on a continuous manifold. Our formulation of lattice spinor gravity also realizes local Lorentz symmetry. Furthermore, the Lorentz transformations are generalized…
We study a model for quantum gravity on a circle in which the notion of a classical metric tensor is replaced by a quantum metric with an inhomogeneous transformation law under diffeomorphisms. This transformation law corresponds to the co--adjoint action of the Virasoro algebra, and resembles that of the connection in Yang--Mills theory. The transformation property is motivated by the diffeomorphism invariance of the one dimensional Schrödinger equation. The quantum distance measured by the metric corresponds to the phase of a quantum mechanical wavefunction. The dynamics of the quantum gravi…
The problem of observables in classical and quantum gravity is a long-standing one. It is sometimes argued that observable quantities should be diffeomorphsm invariant, following the philosophy of Dirac. We argue that diffeomorphism invariance in classical and quantum gravity is not primarily a case of gauge invariance but rather an example of spontaneous symmetry breaking of the diffeomorphism group. As a consequence, observables fall into two classes, Dirac observables which are invariant under the full diffeomorphism group and more general observables which take on different (expectation) v…
Standard inflationary models yield a characteristic signature of a primordial power spectrum with a red tensor and scalar tilt. Nevertheless, Cannone et al recently suggested that, by breaking the assumption of spatial diffeomorphism invariance in the context of the effective field theory of inflation, a blue tensor spectrum can be achieved without violating the Null Energy Condition. In this context, we explore in which cases a blue tensor tilt can be obtained along with a red tilt in the scalar spectrum. Ultimately, we analyze under which conditions this model can reproduce the specific cons…
The constraint structure of 2D-gravity with the Weyl and area-preserving diffeomorphism invariances is analysed in the ADM formulation. It is found that when the area-preserving diffeomorphism constraints are kept, the usual conformal gauge does not exist, whereas there is the possibility to choose the so-called ``quasi-light-cone'' gauge, in which besides the area-preserving diffeomorphism invariance, the reduced Lagrangian also possesses the SL(2,R) residual symmetry. The string-like approach is applied to quantise this model, but a fictitious non-zero central charge in the Virasoro algebra …
Certain aspects of nonrelativistic diffeomorphisms in 2+1 dimensions are investigated. These include a nonrelativistic limit of some relativistic actions in 3 dimensions, the Seiberg-Witten map, a modification of the viscosity tensor in particular due to a non-uniform magnetic field, a redefinition of background fields, and 1/R terms on Riemann surfaces of constant curvature.
To guarantee the stability of the cosmological constant sector against radiative corrections coming from quantum matter fields, one of the most natural ingredients to invoke is the symmetry under scale transformations of the gravitational field. Previous attempts to follow this path have nevertheless failed in providing a consistent picture. Here we point out that this failure is intimately tied up to an assumption which is typically embedded in modern studies of the gravitational interaction: invariance under the full group of diffeomorphisms. We base the discussion on the gravitational theor…
We propose a scenario for particle-mass generation, assuming the existence of a physical regime where, firstly, physical particles can be considered as point-like objects moving in a background space-time and, secondly, their mere presence spoils the invariance under the local diffeomorphism group, resulting in an anomalous realization of the latter. Under these hypotheses, we describe mass generation starting from the massless free theory. The mechanism is not sensitive to the detailed description of the underlying theory at higher energies, leaning only on general structural features of it, …
We study a series of the Wess-Zumino actions obtained by repeatedly integrating conformal anomalies with respect to the conformal-factor field that appear at higher loops. We show that they arise as physical quantities required to make nonlocal loop correction terms diffeomorphism invariant. Specifically, in a conformally flat spacetime $ds^2=e^{2ϕ}(-dη^2 + d{\bf x}^2)$, we find that effective actions are described in terms of momentum squared expressed as a physical $Q^2 = q^2/e^{2ϕ}$ for $q^2$ measured by the flat metric, which recalls the relationship between physical momentum and comoving …
We analyze asymptotic symmetries on the Killing horizon of the four-dimensional Kerr--Newman black hole. We first derive the asymptotic Killing vectors on the Killing horizon, which describe the asymptotic symmetries, and find that the general form of these asymptotic Killing vectors is the universal one possessed by arbitrary Killing horizons. We then construct the phase space associated with the asymptotic symmetries. It is shown that the phase space of an extreme black hole either has the size comparable with a non-extreme black hole, or is small enough to exclude degeneracy, depending on w…
We complete the four-dimensional ${\cal N}=1$ superfield description of six-dimensional supergravity. The missing ingredients in the previous works are the superfields that contain the sechsbein $e_4^{\;\;\underlineν}$, $e_5^{\;\;\underlineν}$, $e_μ^{\;\;\underline{4}}$, $e_μ^{\;\;\underline{5}}$ and the second gravitino. They are necessary to make the action invariant under the diffeomorphisms and the Lorentz transformations involving the extra dimensions. We find the corresponding superfield transformation laws, and show the invariance of the action under them. We also check that the resulta…
We use the requirement of diffeomorphism invariance in the Bianchi I context to derive the form of the quantum Hamiltonian constraint. After imposing the correct classical behavior and making a certain minimality assumption, together with a certain restriction to "planar loops", we then obtain a unique expression for the quantum Hamiltonian operator for Bianchi I to both leading and subleading orders in $\hbar$. Specifically, this expression is found to exactly match the form proposed by Ashtekar and Wilson-Ewing in the loop quantum cosmology (LQC) literature. Furthermore, by using the project…
We present a matrix action based on the unitary group U(N) whose large N ground states are conjectured to be in precise correspondence with the weak-strong dual effective field theory limits of M theory preserving sixteen supersymmetries. We identify a finite N matrix algebra that corresponds to the spacetime and internal symmetry algebra of the Lorentz invariant field theories obtained in the different large N limits. The manifest diffeomorphism invariance of matrix theory is spontaneously broken upon specification of the large N ground state. We verify that there exist planar limits which yi…
We study the extent to which diffeomorphism invariance restricts the properties of the primordial perturbations in single scalar field models. We derive a set of identities that constrain the connected correlators of the cosmological perturbations, as well as the one-particle-irreducible vertices of the theory in any gauge. These identities are the analogues of Slavnov-Taylor identities in gauge theories, and follow essentially from diffeomorphism invariance alone. Yet because quantization requires diffeomorphism invariance to be broken, they not only reflect invariance under diffeomorphisms, …
Four-dimensional Einstein gravity in the Palatini first order formalism is shown to possess a vector supersymmetry of the same type as found in the topological theories for Yang-Mills fields. A peculiar feature of the gravitational theory, characterized by diffeomorphism invariance, is a direct link of vector supersymmetry with the field equation of motion for the Faddeev-Popov ghost of diffeomorphisms.
Einstein gravity in the Palatini first order formalism is shown to possess a vector supersymmetry of the type encountered in the topological gauge theories. A peculiar feature of the gravitationel theory is the link of this vector supersymmetry with the field equation of motion of the Faddeev-Popov ghost associated to diffeomorphism invariance.
This series of lectures gives a simple and self-contained introduction to the non-perturbative and background independent loop approach of canonical quantum gravity. The Hilbert space of kinematical quantum states is constructed and a complete basis of spin network states is introduced. An application of the formalism is provided by the spectral analysis of the area operator, which is the quantum analogue of the classical area function. This leads to one of the key results of loop quantum gravity: the derivation of the discreteness of the geometry and the computation of the quanta of area. Fin…
We revisit the dynamical action, developed in earlier studies [1-3], of the gravitational analog of Yang-Mills field, called the diffeomorphism field. We show an inconsistency in the construction of this action and solve it by a modification. The modified action becomes structurally similar to the Yang-Mills action. Then we explain the problems that arise from interpreting the diffemorphism field as a tensor. We leave out this interpretation, thereby also covariantization as a method to reach diffeomorphism invariance. By introducing corrections that involve affine connection coefficients we r…
In arXiv:hep-th/0011245, in the context of domain wall backgrounds, it was shown that spontaneous breaking of diffeomorphism invariance results in gravitational Higgs mechanism. Recently in arXiv:0708.3184 't Hooft discussed gravitational Higgs mechanism in the context of obtaining massive gravity directly in four dimensions, and pointed out some subtleties with unitarity. We attribute these subtleties to breaking time-like diffeomorphisms, and discuss gravitational Higgs mechanism with all but time-like diffeomorphisms spontaneously broken. The resulting background is no longer flat but exhib…
Gravitational theories with fixed background fields break local Lorentz and diffeomorphism invariance either explicitly or spontaneously. In the case of explicit breaking it is known that conflicts can arise between the dynamics and geometrical constraints, while spontaneous breaking evades this problem. It is for this reason that in the gravity sector of the Standard-Model Extension (SME) it is assumed that the background fields (SME coefficients) originate from spontaneous symmetry breaking. However, in other examples, such as Chern-Simons gravity and massive gravity, diffeomorphism invarian…
A Lorentz covariant quantization of membrane dynamics is defined, which also leaves unbroken the full three dimensional diffeomorphism invariance of the membrane. Among the applications studied are the reduction to string theory, which may be understood in terms of the phase space and constraints, and the interpretation of physical,zero-energy states. A matrix regularization is defined as in the light cone gauged fixed theory but there are difficulties implementing all the gauge symmetries. The problem involves the non-area-preserving diffeomorphisms which are realized non-linearly in the clas…
We use twist deformation techniques to analyse the behaviour under area-preserving diffeomorphisms of quantum averages of Wilson loops in Yang-Mills theory on the noncommutative plane. We find that while the classical gauge theory is manifestly twist covariant, the holonomy operators break the quantum implementation of the twisted symmetry in the usual formal definition of the twisted quantum field theory. These results are deduced by analysing general criteria which guarantee twist invariance of noncommutative quantum field theories. From this a number of general results are also obtained, su…
Recently, using a local action satisfying the Wess-Zumino condition as a kinetic term of the conformal mode, we formulated a four-dimensional quantum geometry (4DQG). The conformal mode can be treated exactly, and it was shown that the part of the effective action related to this mode is given by the scale-invariant non-local Riegert action. As for the traceless mode, we introduce dimensionless coupling, which is a unique gravitational coupling of this theory satisfying the conditions of renormalizability and asymptotic freedom. Although this theory is asymptotically free, the physical states …
We argue that when conformal symmetry is spontaneously broken the trace anomalies in the broken and unbroken phases are matched. This puts strong constraints on the various couplings of the dilaton. Using the uniqueness of the effective action for the Goldstone supermultiplet for broken ${\cal N}=1$ superconformal symmetry the dilaton effective action is calculated.
We construct a gauge-fixing procedure in the path integral for gravitational models with branes and boundaries. This procedure incorporates a set of gauge conditions which gauge away effectively decoupled diffeomorphisms acting in the $(d+1)$-dimensional bulk and on the $d$-dimensional brane. The corresponding gauge-fixing factor in the path integral factorizes as a product of the bulk and brane (surface-theory) factors. This factorization underlies a special bulk wavefunction representation of the brane effective action. We develop the semiclassical expansion for this action and explicitly de…
The technique for constructing conformally invariant theories within the coset space construction is developed. It reproduces all consequences of the conformal invariance and Lagrangians of widely-known conformal field theories. The method of induced representations, which plays the key role in the construction, allows to reveal a special role of the "Nambu-Goldstone fields" for special conformal transformations. Namely, their dependence on the coordinates turns out to be fixed by the symmetries. This results in the appearance of the constraints on possible forms of Lagrangians, which ensure t…
We try to use scale-invariance and the 1/N expansion to construct a non-trivial 4d O(N) scalar field model with controlled UV behavior and naturally light scalar excitations. The principle is to fix interactions at each order in 1/N by requiring the effective action for arbitrary background fields to be scale-invariant. We find a line of non-trivial UV fixed-points in the large-N limit, parameterized by a dimensionless coupling. Nether action nor measure is scale invariant, but the effective action is. Scale invariance makes it natural to set a mass deformation to zero. The model has phases wh…
We study the issue of diffeomorphism symmetry in group field theories (GFT), using the recently introduced noncommutative metric representation. In the colored Boulatov model for 3d gravity, we identify a field (quantum) symmetry which ties together the vertex translation invariance of discrete gravity, the flatness constraint of canonical quantum gravity, and the topological (coarse-graining) identities for the 6j-symbols. We also show how, for the GFT graphs dual to manifolds, the invariance of the Feynman amplitudes encodes the discrete residual action of diffeomorphisms in simplicial gravi…
Conventional quantization of two-dimensional diffeomorphism and Weyl invariant theories sacrifices the latter symmetry to anomalies, while maintaining the former. When alternatively Weyl invariance is preserved by abandoning diffeomorphism invariance, we find that some invariance against coordinate redefinition remains: one can still perform at will transformations possessing a constant Jacobian. The alternate approach enjoys as much ``gauge'' symmetry as the conventional formulation.
We analyze the symmetries that are realized on the massive Kaluza-Klein modes in generic D-dimensional backgrounds with three non-compact directions. For this we construct the unbroken phase given by the decompactification limit, in which the higher Kaluza-Klein modes are massless. The latter admits an infinite-dimensional extension of the three-dimensional diffeomorphism group as local symmetry and, moreover, a current algebra associated to SL(D-2,R) together with the diffeomorphism algebra of the internal manifold as global symmetries. It is shown that the `broken phase' can be reconstructed…
A brief recap is given of issues that can occur when fixed nondynamical backgrounds that explicitly break diffeomorphism invariance are included in gravity theories. Implications for the Standard-Model Extension (SME) are summarized, including recent results showing how the SME can be modified to accommodate such fixed nondynamical backgrounds.
We study the renormalizability of quantum gravity near two dimensions. Our formalism starts with the tree action which is invariant under the volume preserving diffeomorphism. We identify the BRS invariance which originates from the full diffeomorphism invariance. We study the Ward-Takahashi identities to determine the general structure of the counter terms. We prove to all orders that the counter terms can be supplied by the coupling and the wave function renormalization of the tree action. The bare action can be constructed to be the Einstein action form which ensures the full diffeomorphism…
