Conformally Coupled General Relativity

Gravity model developed in the series of papers \cite{Arbuzov:2009zza,Arbuzov:2010fz,Pervushin:2011gz} is revisited. Model is based on Ogievetsky theorem that specifies structure of general coordinate transformation group. The theorem is implemented in the context of Noether theorem with the use of nonlinear representation technique. Canonical quantization is performed with the use of reparametrization-invariant time and ADM foliation techniques. Basic quantum features of the models are discussed. Mistakes occurred in the previous papers are corrected.


Introduction
General relativity forms our understanding of spacetime.It is verified by the Solar system and cosmological tests [4,5].The recent discovery of gravitational waves provided another evidence supporting theory viability in the classical regime [6][7][8][9].Despite these successes there are reasons to believe that general relativity is unable to provide an adequate description of gravitational phenomena in the high energy regime and should be either modified or replaced by a new theory of gravity [10][11][12].
One of the main issues is the phenomenon of inflation.It appears that an inflationary phase of a universe expansion is necessary for a self-consistent cosmological model [13][14][15].Inflation takes place at the early stage of a universe expansion, in the high energy regime, and should be driven either by a new scalar field [16] or by quantum effects associated with high order gravitational terms [17].In that regime gravity should be considered in a framework of quantum theory, but attempts to construct a proper description within a perturbative approach failed.It is possible to construct renormalizable gravity theory with higher derivatives [18], but general relativity is nonrenormalizable in the second order of perturbation theory [19].Multiple attempts to construct a proper quantum gravity model gave birth to several major research programs such as Asymptotic Safety [20], Loop Quantum Gravity [21], and String Theory [22,23].
In this paper we revisit another approach to quantum gravity that we are going to call Conformally Coupled General Relativity (CCGR).CCGR is founded on a series of techniques that may resolve some of general relativity issues appearing in the quantum regime.These are techniques of nonlinear symmetry representation (used to implement the Ogievetsky theorem [24]) [25][26][27], Hamiltonian formalism [28,29], conformal coupling of matter to gravity [30][31][32], and parameterization-invariant time [33][34][35][36].This approach was presented in a broad series of papers [1][2][3] and our main aim is to present it in a self-consistent comprehensive form.We present a brief discussion of all aforementioned techniques in order to provide a necessary theoretical background required for CCGR.The main focus of this paper are quantum feature of CCGR because they were discussed very briefly and obscure in the previous papers.
This paper organized as follows.In the Section 2 we provide a detailed motivation for usage of these techniques, their physical relevance, and applications.In the Section 3 we recall results of nonlinear symmetry realization theory.In the Section 4 we demonstrate nonlinear realization theory implementation in a context of the Ogievetsky theorem.Finally, in the Section 5 we apply all aforementioned techniques to general relativity to construct CCGR and discuss its quantum features in the Section 6.

Motivation
The Ogievetsky theorem [24] is crucial for gravity research, as it specifies a structure of general coordinate transformations.Transformations are given by the following expression: where x µ are original coordinates, x ′ µ are new coordinates, and Greek indices take their values from zero to three.One can expand (2.1) in the Taylor series and obtain an infinite number of generators: 2) The theorem states that every generator (2.2) is a linear combination of generator commutators from special linear SL(4, R) and conformal C(1, 3) groups.Corresponding algebras are presented in the Appendix 7.1.Thus transformation features of a quantity with respect to a coordinate frame change can be expressed with a finite number of generators.
A gravity model must be stated in a coordinate independent way, thus one should be able to identify structures of SL(4, R) and C (1,3) groups in any gravity model.At the same time Noether theorems establish a connection between model symmetries and conservation laws and there are no conservation laws corresponding to the whole SL(4, R) and C(1, 3) groups in nature.Energy-momentum and angular momentum conservation laws correspond only to the Lorentz subgroup.
That apparent contradiction was resolved in the paper by Borisov and Ogievetsky [37].Their idea was to use the nonlinear symmetry representation theory [25][26][27].Because of a nonlinear structure of a representation one does not simply imply Noether theorems.Despite C(1, 3) and SL(4, R) symmetries are presented, only the Lorentz subgroup receives a linear representation and theorems result in the Lorentzian currents but provide no conservation laws for the rest of the symmetry.In the paper vierbein (tetrades) formalism was used to construct the representation and such a method can be applied to any gravity model which considers physical spacetime as a Riemannian manifold.The paper thereby provides a general ground to incorporate the Ogievetsky theorem within any gravity model in a four dimensional (Riemannian) spacetime.
Benefits of this method appear at the level of quantum theory.It relates the original model symmetry with dynamical variables and the choice of dynamical variables affects renormalization features of a model.Moreover, the method introduces new symmetry in a model which may influence model behavior.New symmetry may exclude certain terms from a Lagrangian thereby improve general relativity renormalization behavior.
Another important gravity models issue is definition of a gravity coupling to matter.The coupling should be driven by some fundamental physical principle, as this allows one to narrow a number of possible gravity models and to define interacting states in quantum theory in a unique way.An example of a similar approach is given by the Lovelock theorem [38].It proves that the left hand side of Einstein equations is fixed uniquely, if one considers it to be a second differential order rank-2 divergenceless tensor.Another example is the Horndeski's paper [39].In that paper it was proven that scalar-tensor gravity models with a nontrivial kinetic coupling can result in second differential order field equations.Therefore one cannot define a scalar-tensor model uniquely by a requirement of second differential order field equations.In such a way it appears to be fruitful to find a guiding physical principle constraining a number of possible interactions between matter and gravity.
A proper principle was proposed in Dirac and Deser papers [30][31][32].The paper by Dirac follows the Large Numbers Hypothesis [40] which states that large dimensionless physical constants are connected to a cosmological epoch.Dirac proposed to consider matter coupled to a metric g which is differs from the full metric g governed by Einstein equations [31].Therefore the full metric g cannot be measured directly in any experiment, but it defines particles motion; the metric g is measured by our apparatus, but does not define geodesic motion.The paper by Deser [32] is devoted to study of massless scalar fields which are, unlike massless vector and spinor fields, coupled to gravity not in a conformally invariant way.He proved that one can introduce additional terms to the standard massless scalar field Lagrangian which restore the invariance.The resulted action matches the standard Hilbert action up to a metric conformal transformation.Thus such a construction separates a gravitational scalar degree of freedom from the full metric, but does not make a conformally invariant coupling of a scalar field to gravity.
These results can be used in the following way.First, following Dirac's idea one constructs a coupling of matter to gravity in a conformally invariant way.Second, following Deser's paper one separates the conformal metric at the level of an action via conformal transformations.Thus an action turns up to be separated into two parts: one drives a cosmological evolution and the other drives local gravitational interaction.For the best of our knowledge Penrose, Chernikov, and Tagirov [41,42] were first who wrote the Hilbert action separating the conformal degree of freedom.Usage of a conformal coupling of matter to gravity gave birth to conformal cosmology [43][44][45][46].
We would like to highlight that the conformal symmetry plays an important role in physics.The best example illustrating its role is the standard model of particle physics, as it appears to be almost conformally invariant.Without the Higgs sector the model admits the conformal invariance because it consists of massless vector bosons and fermions.Higgs boson is coupled with all other particles in a conformally invariant way with the use of dimensionless couplings.The conformal invariance is violated when one introduces the scale of spontaneous symmetry breaking.Moreover, as one introduces a single scale it cannot be evaluated from the first principles within the model and one must refer to experimental data.Thus one may expect that conformal symmetry plays a key role in physical theories.
We imply Hamiltonian formalism, as it is considered to be a perspective approach to quantum theory.The formalism was developed by Arnowitt, Deser, and Misner [28] (and founded in Dirac's paper [29]).It treats a four-dimensional spacetime as a series of three-surfaces with dynamical geometry.Constraints play a key role in the formalism, for instance, the Hamiltonian of general relativity vanishes because of constraints.We discuss constraints influence on quantum theory in the Section 6. However the main implication of the Hamiltonian formalism in this paper is reduced to usage of ADM foliation technique for the sake of simplicity.
Implication of ADM foliation gives a rise to the time parameterization problem that was addressed in papers [34][35][36].Within quantum theory time is considered as an external parameter mapping system evolution.In the realm of gravity models time is merely a coordinate and can be chosen arbitrary.In the realm of cosmology one expects to associate time with a cosmological epoch which can be defined via cosmological parameters measurement.Therefore one expects to connect the time with an observable quantity and should find a way to remove an ambiguity from a definition of time.In the series of paper [34][35][36] a method was proposed that allows one to define time in a reparametrization-independent way and to relate time to a cosmological epoch according to the Einstein cosmological principle [47].
Therefore the motivation should be summarized as follows.First, we use nonlinear representation of SL(4, R) and C(1, 4) groups to incorporate the Ogievetsky theorem into the model.Second, we introduce a conformal coupling of matter to gravity and separate the conformal metric at the level of an action.Next we implement ADM foliation to define time in a reparametrization invariant way and obtain CCGR in the classical regime.Techniques used to obtain the classical model affect quantum version of CCGR, namely, additional symmetry allows us to introduce a notion of (conformal) gravitons which Lagrangian is bilinear.Nonetheless gravity preserves its nonlinear nature, as gravitational interaction cannot be reduced to conformal gravitons only.However implication of all aforementioned techniques significantly improves study of CCGR quantum properties.

Nonlinear Symmetry Representation
Nonlinear symmetry representation theory is stated in papers [25][26][27].One can refer to papers [48][49][50] for a detailed review; a detailed example of a model with a nonlinear representation of SO(N) on SO(N)/SO(N −1) coset in a context of nonlinear sigma model is given in the textbook [51].
One starts with a d-dimensional Lie group G and its n-dimensional subgroup H. Lie algebra of H is formed by generators V l , l = 1, • • • , n; Lie algebra of G is formed by generators from algebra H and A l , l = 1, • • • , d − n.An arbitrary element g from a neighborhood of the identity of G can be presented in the following form: Group G is equipped with a natural left group action on itself.An element g ∈ G acting on an element g ′ ∈ G by a right multiplication as gg ′ .One can narrow this group action down to G/H coset.The action of an arbitrary element g from G on an arbitrary element exp[ζ l A l ] from G/H is given by the following: where ζ l are coordinates of the initial coset element, ζ ′ l (ζ, g) are coordinates of the element after the group action, and u l (ζ, g) are coordinates on H, that form an associated nonlinear representation of G on H.
In such a way To relate these representations with physical fields one should, first, treat coset coordinates ζ l as physical fields subjected to the group transformation defined by (3.2): Second, one should take physical fields ψ i subjected to a linear representation of H: ψ → D e φ l V l ψ, and use u l (ζ, g) defined by (3.2) except transformation parameters φ l .Finally, one should introduce a covariant derivative ∇ that transforms in agreement with the nonlinear representation.Covariant derivatives are given by the following: where values of p µ and v µ are defined by the form: One should use fields ζ l , ψ i , and their covariant derivatives to construct an invariant Lagrangian.
A nonlinear symmetry representation, as we mentioned before, can be treated as spontaneous symmetry breaking, thus a correspondent field theory is constraint by the Goldstone theorem.Therefore d − n physical fields associated with coset coordinates can only be massless Goldstone particles.As a nonlinear representation of G on H founded on a linear representation of H, one can introduce an arbitrary number of additional (massive or massless) degrees of freedom subjected to various linear realization of H.
We would like to highlight that nonlinear realization implementation appears to be fruitful in the quantum regime.On the classical level one can treat a nonlinear symmetry realization merely as a new degrees of freedom parameterization.It is impossible to distinguished a model with a linear representation of H and non-minimal interaction from a model with a nonlinear representation of G and minimal interaction solely by a form of the classical Lagrangian.In the quantum regime symmetry may prevent some interactions from appearance which is crucial for renormalization features of the model and for its physical content.This is the main reason why we are interested in usage of that technique and we discuss this issue in detail in the Section 6.

Nonlinear Representation of SL(4, R) and C(1, 3) on Vierbein
Technique used to describe gravity with a nonlinear representation of SL(4, R) and C(1, 3) was proposed in the paper [37].More detailed review can be found in [48].In accordance with the Section 3, one should use variables subjected to a linear Lorentz transformation to construct a nonlinear representation.Vierbein are objects one should use for that purpose [52][53][54].We recall vierbein formalism and then turn to a nonlinear SL(4, R) and C(1, 3) groups representation.
There are several ways to construct vierbein formalism.One can introduce vierbein to obtain a proper generalization of Dirac's equation.In the realm of flat spacetime all physical quantities, such as i ψγ µ ∂ µ ψ or A µ , transform by Lorentz group representations.In the realm of curved spacetime an arbitrary tensor transforms in accordance with the general linear transformation group SL(4, R).But there is no transformation in SL(4, R) that corresponds to the spinor Lorentz group representation.One introduces vierbein e µa , which transform in accordance with SL(4, R) by the Greek index and in accordance with the Lorentz group by the Latin index (both Latin and Greek indices take values from zero to three in this section).These objects connect the spinor Lorentz group representation with the SL(4, R) representation.To generalize Dirac's equation one simply replace the index of standard γ-matrices: In the same manner one uses vierbein to replace SL(4, R) indices with Lorentz ones in an arbitrary tensor.
An alternative approach to vierbein considers vector fields on a spacetime.The set of all vector field forms a linear space and one is free to choose its basis.The standard choice is the holonomic basis, these are vector fields co-directed with coordinate lines in every point of a spacetime.This basis canonically noted as ∂ µ and dx µ because these derivatives transform as vectors [52,55].The manifold metric can be treated as a bilinear form on the linear space of vector fields: One can choose a vector field basis ω m which makes the metric to take the Minkowski form: where η mn is the Minkowski metric and ω m is a vierbein basis (of a vector field space).We should highlight that the choice of a basis is not related with a coordinate frame on a spacetime.It corresponds to gauge transformations or to a coordinate frame change in the tangent bundle.Coefficients that relate the holonomic basis and forms ω m are called vierbein: In vierbein basis the metric tensor coincides with the Minkowski one and as the form of the metric should be preserved, vector fields ω m can only be subjected to Lorentz transformations.Thus vierbein transform by the Lorentz group by the Latin index and in accordance with SL(4, R) by the Greek index.Differential geometry formalism can be stated in terms of vierbein [52,55], in the Appendix 7.3 we present a set of formulae that relates standard and vierbein formalisms.Now we turn to the discussion of a nonlinear representation of SL(4, R) and C(1, 3).For the sake of simplicity we consider the affine group A(4) instead of SL(4, R).This choice has no influence on the representation because we did not include neither new generators nor new structures.The affine group consists of all linear transformations in a four dimensional spacetime x ′ µ = a ν µ x ν + c µ ; it is a semidirect product of SL(4, R) and a group of all shifts.Following the paper [37], first, we consider a nonlinear representation of A(4) on A(4)/L coset.Group A(4) consists of shift generators P (µ) , Lorentz group (asymmetric) generators L (µ)(ν) , and distortion (symmetric) generators R (µ)(ν) .We use indices in brackets to numerate generators of the algebra, as they are not (yet) related with any representation of Lorentz group.Greek indices with and without brackets take their values from zero to three.The correspondent commutation relations are given in the Appendix 7.1.
An arbitrary element G from A(4)/L coset is parameterized as follows: The following forms define transformation features of all objects in the representation: ) ) As we wrote before, L (µ)(ν) is asymmetric and R (µ)(ν) is symmetric therefore the form ω L is also asymmetric and the form ω R is symmetric.One should treat forms ω (µ) as a vector field basis on a spacetime and ω (µ)ν as vierbein: The same thing should be done with a conformal group representation on C(1, 3)/L coset.For the sake of simplicity we do not present the procedure here, as it can be found in the original paper [37].These nonlinear representations should be agreed with one another to provide the following definition of a covariant derivative (applied to a field of an arbitrary spin): Here L Ψ (α)(β) is the Lorentz group representation operator acting on Ψ, forms ω L and ω R are given by (4.9) and (4.8), and (4.12) is the spin connection.The Riemann tensor R (µ)(ν)(α)(β) is defined in the following way: and given by the following formula: We would like to highlight that the Riemann tensor (4.14) is bilinear in forms ω R , ω L and linear in their derivatives because of the form of the spin connection (4.12).
Vierbein ω (µ)ν and the form ω R thereby should be treated as dynamical variables of the theory as they carry system symmetry.Despite both forms ω R and ω L enter a covariant derivative (4.11),only the form ω R can change during system evolution.Definitions (4.8) and (4.9) result in the following identity: As vierbein are constrained by orthogonality identities (see (7.17) in the Appendix 7.3), the following expression for a vierbein differential holds: This identity should be used to obtain the following expression for a metric differential: The forepart ω (ρ)µ ω (σ)ν + ω (σ)ν ω (ρ)µ is symmetric with respect to indices (µ) and (ν), while the former part contains an asymmetric form ω L (µ)(ν) .Hence the asymmetric form ω L cannot enter the metric differential expression and it is given by the following: As the form ω L does not enter the metric differential it does not evolve and cannot be considered as a dynamical variable.We would like to highlight the following feature of the representation.One can construct a nonlinear representation of A(4) on A(4)/L coset, but in that case a covariant derivative cannot be determine in a unique way [37].One can expand the minimal expression for the derivative in a self-consistent way making it dependent on three arbitrary constants.One excludes such an ambiguity by combing representations of A(4) and C(1, 3).On the classical level the technique is merely a way to parameterize the metric and the connection used to agree Ogievetsky and Noether theorems.No additional degrees of freedom are introduced and, in full accordance with a nonlinear representation technique, at the level of a classical Lagrangian the model is indistinguishable from general relativity.At the same time the construction presented in this section is not agreed with the conformal coupling condition, the technique merely connects a nonlinear representation to an arbitrary vierbein.We introduce a conformally invariant coupling in the next section.

Conformally Coupled General Relativity
Following the motivation presented in the Section 2 we apply discussed techniques to general relativity.First, we separate the conformal degree of freedom from the full metric: (5.1) Here g µν in the full (Einstein) metric, g µν in the conformal metric, x µ are arbitrary coordinates on a spacetime, χ µ are correspondent conformally invariant coordinates, and D is the conformal degree of freedom also known as the dilaton.At this level we also implement the conformal coupling of matter to gravity.The metric parameterization (5.1) results in the Penrose-Chernikov-Tagirov action with the conformal coupling to matter [41,42]: 2) Here Λ is the cosmological constant, Λ = e −2D Λ is the conformal cosmological constant, M P is the Planck mass, and M P = M P e −D is the conformal Planck mass.We would like to note that we introduce the cosmological constant in the model for the sake of generality and so far we treat it as a free model parameter.Verification by type Ia supernovae data showed that a conformal cosmological model based on action (5.3) without the cosmological constant can be treated alongside with the standard cosmological model [46].In order to use a nonlinear representation technique from the section (4) we define (conformal) vierbein by the following: (5.4) Here ω (µ) is the (conformal) vierbein basis subjected to the nonlinear symmetry representation (4.7).One can define vierbein at the level of the full metric g µν , but it would be equivalent to the one used by us.One is free to associate the nonlinear representation (4.7) with any vierbein because they all related by conformal transformations.Therefore the choice of vierbein corresponds to different frames in (4.5) and does not affect the structure of (4.5) and the structure of the representation.The conformal metric is conformally invariant in the following sense: any conformal transformation g µν → e 2Ω g µν can be equipped with a dilaton transformation D → D + Ω to preserve the form of metric (5.1).In terms of (conformal) vierbein this symmetry is given by the following: (5.5) One cannot consider the conformal metric and the dilaton as independent variables because the number of degrees of freedom should be equal both in (5.2) and (5.3).Moreover, these variables are related with a symmetry (5.5).We define the dilaton through the three-metric (3) g mn defined by the full metric g µν as follows: ln det (3) g mn . (5.6) However one cannot express the dilaton through the conformal metric because of the symmetry (5.5).Instead one can fix a gauge of variables g and D in full analogy with the standard gauge field theory.
In such a way we obtain CCGR, as we implement a nonlinear symmetry realization and conformal coupling of matter to gravity.As we highlighted in Sections 2 and 4, CCGR is indistinguishable from general relativity on the classical level because of usage of vierbein and nonlinear symmetry representation theory merely introduces a different variable definitions.At the quantum level CCGR has extremely different features, as its dynamics depends on the choice of variables and changes as we had introduced an additional symmetry.
To obtain a quantum version of CCGR one requires to implement the Hamiltonian formalism with the use of the ADM foliation [28].We splits the conformal four-dimensional spacetime in a series of three-spaces equipped with a metric γ ab .The conformal metric g µν is parameterized by the three-metric, the shift three-vector N a , and by the laps function N (see the Appendix 7.4 for notations and definitions).
Implication of the Hamiltonian formalism cannot be constructed in a straightforward way in CCGR because of two reasons.First, canonical implication uses three-space metric as a dynamical variable, but we desire to use (conformal) vierbein.As we constructed nonlinear representations of SL(4, R) and C(1, 3) on (conformal) vierbein, we obligated them to be true dynamical variables carrying the model symmetry.Thus one should write action (5.3) in terms of vierbein and spin connection to obtain the correct Hamiltonian.A similar construction is used in Loop Quantum Gravity research and we discuss it in details in the Section 6.Second, as we highlighted in the Section 2, one has to resolve the time parameterization issue.In the Penrose-Chernikov-Tagirov action (5.3) the time coordinate can be chosen arbitrary, but in cosmology time is no longer treated as a coordinate and becomes a measurable quantity.Thus one needs to define the time in a parameterization-invariant way before the quantization of a model.
In papers [34][35][36] it was proposed to associate physical time with the zero dilaton mode according to the Einstein cosmological principle [47].We introduce the following definitions to separate the zero dilaton mode D which depends only on the time coordinate.We use the following definition of a three-space mean value of a physical quantity X: where V is three-volume of a region over which the averaging is carried out.In such a way any physical quantity X is separated into two parts: the mean part X and the fluctuation part X.They are connected by the following expression: and the following orthogonality condition holds: The same definition should be applied to the dilaton: It allows one to exclude all interferences between D and D from the action (5.3) and to reduce it to the following form: The action for the zero dilaton mode reads (5.12) It gives a way to introduce the parameterization-invariant time.One defines the following factorization of the lapse function N: where N 0 is defined by the following and N is subjected to the orthogonality condition: N −1 = 1.In these terms the dilaton action (5.12) takes the desirable form: (5.15) In the action (5.15) any time reparametrization χ 0 → χ ′0 can be absorbed into the definition of N 0 .Thus we define the parameterization invariant time by the following expression: And the four-volume element can be expressed as follows: (5.17) On the physical level one can associate cosmological epoch with the dilaton zero mode D , as it is related with the parameterization-invariant time t by a Jacobian matrix d D /N 0 dχ 0 taking place in the dilaton action (5.15).Therefore zero dilaton mode can also be considered as a time variable alongside with the parameterizationinvariant time (5.16).
The construction results in two direct corollaries.First, the canonical momentum of the zero dilaton mode P D with respect to D should be related with the evolution operator and must be non-zero: Second, because of the orthogonality condition (5.9), D does not depend on the D and the correspondent canonical momentum is zero: Therefore the action (5.11) decomposes in three parts: ) ) (5.23) Action ( 5.3) decomposed into three terms (5.20) is the main result of CCGR in the classical regime.Such action parameterization illustrates that the cosmological evolution is driven by a mean (global) structure of a gravitational field described by the D , while local gravitational interaction is driven by a fluctuation part of a gravitational field D. These degrees of freedom interact with each other in a non-trivial way and the interaction is given by the action (5.22).Corollaries of that model that take place in the classical regime were studies in papers [1][2][3].A detailed discussion of its cosmological implication including perturbation theory presents in the paper [56].Quantum features rising within the model are discussed in the following chapter.

Quantum Features of CCGR
We should start with a brief discussion of various approaches to quantum gravity.The discussion is due, as we attribute our results to one particular quantum gravity approach and briefly discuss features that may appear in others.Nowadays a number of different approaches is huge, but one can distinguish three most perspective directions within that landscape, namely: String Theory, Loop Quantum Gravity (LQG), and the standard perturbative approach.
We do not discuss string theory because it introduces a new fundamental notion of a (super) string.Therefore one is obliged to found gravity theory on that notion and may adopt results of this paper only within a low energy regime.Put it otherwise, our results may serve as a way to verify low energy string theory limit, but cannot be implemented directly in the theory foundation.
The framework of LQG is valuable in quantum gravity study, as it is an attempt to construct a self-consistent framework of canonical quantum gravity (detailed reviews can be found in [21,57]).The approach shares many similarities with CCGR because LQG is founded on the Hamiltonian vierbein formalism.In the LQG framework one introduces dreibein (triade) connecting spatial indices with SU(2) indices to cast gravity in terms of Ashtekar variables (spin-connection variables).This allows one to use the standard Willson loop to solve Hamiltonian constraints (Wheller-DeWitt equation) and build a space of states basis in the spin network formalism.In our paper, in full analogy with LQG, we adopt vierbein that is reduced to dreibein at the level of ADM foliation to connect spatial indices with a nonlinear representation of SL(4, R) and C(1, 3) groups.In other words, we introduce new symmetry in the model which may affect both Wheller-DeWitt equation and structure of spin network.However this issue requires a separate treatment that lies beyond the scope of this paper.
The standard perturbative approach was developed in papers [58][59][60] and it treats gravity as small perturbations over the Minkoswki background.Although it is possible to construct a renormalizable gravity model [18], general relativity appears to be nonrenormalizable in the second order of perturbation theory [19].Thus it is usually used in the low energy regime within the effective field theory framework [61].Implication of the perturbative approach in the context of CCGR is complicated by following issues.First, the perturbative approach uses the full metric g µν as a dynamical variable.We use vierbein as dynamical variables carrying a nonlinear symmetry representation, thus the perturbative approach should be stated in terms of vierbein variables.Second, in CCGR one should not use Minkowski spacetime as a background because this leads to a contradiction.As we related a mean dilaton value, which is related with the metric, to the cosmological epoch, one should be able to use perturbative approach in the cosmological context.But this is impossible with the Minkowski background, as it does not admit cosmological expansion.Therefore in order to implement the perturbative approach one should define graviton states in terms of vierbein with respect to a cosmological background.Introduction of new dynamical variables alongside with the new symmetry may influence model dynamics and prevent high order gravitational terms from appearance.Further study of perturbative approach implementation lies beyond the scope of this paper.
An alternative way to study quantum features (at least in the quasiclassical regime) was proposed in the aforementioned series of papers [1][2][3].It follows the standard program that was first used in quantum electrodynamics development.First, one finds a single nonlinear plane wave solution within our model.Then we associate a single graviton state with a nonlinear plane wave solution (in a quasiclassical regime) and decompose gravitational field in a series of (nonlinear) plane waves.This allows us to obtain a gravitons Lagrangian and to draw some conclusions on model quantum features.Unlike in the standard perturbative approach, we use ADM split to treat the Lagrangian because we defined time in a reparametrization-invariant way thereby violate model covariance.We also consider the cosmological constant to be zero, as a conformal cosmological model without the cosmological constant provides a good fit for cosmological data [46].Moreover, the cosmological constant problem alone [62,63] lies beyond the scope of this paper.
We start with the choice of a proper background for perturbation theory.As we connected time with the cosmological epoch, we can only adopt the Friedman-Robertson-Walker (FRW) spacetime as a background.In a contrast to the standard approach, we do not consider the scale factor to be a classical (non-quantum) welldefined function.We consider the mean part of the dilaton D to be related with the scale factor a in the following way: In other words, we consider quantum cosmology in the following way.We treat gravity as small perturbations over a conformal metric, while the dilaton which is also a quantum object drives the cosmological expansion.Therefore, as we highlighted before, we consider local gravitational interaction associated with gravitons separately from the global interaction associated with the cosmological expansion and the dilaton.This allows us to treat graviton at the level of the conformal metric thus to imply the standard perturbation approach.Second, we need to define a classical nonlinear plane wave solution.We would like to remind that in the classical theory (Section 5) the conformal metric and the dilaton cannot be treated as an independent variables and they are connected with the symmetry (5.5).Thus we can fix a gauge of the conformal metric by the following: where γ is the conformal three-metric determinant and the gauge is known as the Lichnerowicz gauge [64][65][66].Next, we set the shift vector to be zero N a = 0 by a proper coordinate redefinition ( see equation (7.22) from the Appendix 7.4 that relates four-coordinate frame with the shift vector).We also consider a time parameterization resulting in N = 1.Then we imply a theorem stating that any two dimensional space can be equipped with conformal coordinates [52] in order to foliate three-space into a series of two-spaces.Put it otherwise, any two dimensional metric can be transferred to the conformal form: Therefore one can cast any three-metric in the following form: Metric function g 33 can be put equal to 1 by proper x 3 rescaling.Metric functions g 13 , g 23 are similar to the shift vector in ADM foliation, as they describe two-coordinates shift from layer to layer.Therefore we require g 13 = g 23 = 0, as in a plane-like wave an observer should experience gravitational interaction between two-layers (wavefront), but not within a two-layer.With the use of that factorization we define a nonlinear plane wave solution as a solution given by the following metric ansatz: where σ depends on χ 3 and time and Σ depends only on χ 1 and χ 2 .This metric describes a three-space foliated in a series of two-surfaces with the same topology but their geometry varies with time and from surface to surface.Each two-surface serves as a wavefront, while a normal vector, which co-directed with χ 3 coordinate lines, serve as a wave-vector.In such a metric parameterization two-surfaces can have non-flat geometry and we use function σ and Σ to distinguish two conformal factors.The factor Σ maps two-surface geometry, while σ maps geometry variations with layers and time.Within the ADM foliation the single graviton action (5.22) reads (see the Appendix 7.4 for notations): For the metric (6.5) it is given by the following: This proofs that the metric (6.5) can be treated as a nonlinear plane wave solution because the first action term matches the standard harmonic oscillator Lagrangian.The former term in (6.7) contains no derivatives of σ and appears due to wavefront non-flat geometry.In such a way the factor Σ defines a nonlinear dynamic of a nonlinear plain wave, but it does not required to obtain or support a nonlinear plain wave solution.Therefore we consider Σ = 1 hereafter for the sake of simplicity, as such a choice does not effect physical content of theory.
As we mentioned before, the usage of a nonlinear symmetry representation determines the set of dynamical variables uniquely.Therefore we establish a correspondence between the wave metric (6.5) and conformal dreibein.ADM foliation and vierbein formalism are related by the following formulae: where ω 0 , ω (i) is the vierbein basis, ω (a)i is correspondent dreibein, and N, N i is the laps function and the shift vector.One can express (6.8) in an equivalent form: As we connected a part of vierbein with the laps function and shift vector, vierbein ω (0)0 ,ω (0)i , and ω (a)0 cannot be taken as dynamical variables in full accordance with the Hamiltonian formalism.Moreover, in accordance with the Section 4, the form ω L is not a dynamical variable (4.18), therefore one needs to consider only connection between the metric and a form ω R .For the nonlinear plane wave metric (6.5) nonzero vierbein are given by the following: ) and the element ω (µ)σ dω σ (ν) is symmetric and given by the following: In such a way one can consider forms ǫ R (k) corresponding to the nonlinear plane wave solution with a (three) wave-vector k and expand an arbitrary form ω R in a series of plane waves.Such an expansion has the following form: where all scalar products are evaluated on dreibein (x • k = k (a) x (a) ) and functions g ± should be interpreted a graviton operators in full analogy with the standard quantum field theory.Similar to a weak wave in general relativity, ǫ (a)(b) (k) is constrained by the following identities A single nonlinear wave is driven by the action (6.7) thereby fix gravitons on zeromass shell in full agreement with the classical theory: One can summarize these results as follows.One should consider a classical single nonlinear plane wave solution (6.5), which is also a general relativity solution, and associate it with a single graviton state.One can treat a single plain wave-like solution as a single graviton state just because one can connect the solution to vierbein which is a dynamical variable carrying model symmetry.A similar thing can hardly be done within general relativity, as it does not provide a unique way to fix dynamical variables.One also can use the expansion (6.13)only because the form ω R is a true dynamical variable that within quantum theory should be expressed as a superposition of gravitons.Within general relativity one is obliged to use the standard vierbein as a dynamical variable and cannot adopt an expansion similar to (6.13).Finally, unlike general relativity our model admits one polarization of a single nonlinear plain wave, while general relativity admits two.Nonetheless the model still has two degrees of freedom, one is encoded in nonlinear plain waves while another one presents in the dilaton.This result appears because of the conformal symmetry of the model.Two (weak) plain wave polarizations within general relativity can be converted in one another because spacetime coordinates admits Lorentz symmetry, but they cannot be identified with one another because gravitational field lacks a proper symmetry.In our model conformal symmetry is admitted both by spacetime coordinates and by gravitational field, therefore these two polarizations can be identified.In such a way one can decouple nonlinear gravitational interaction into a bilinear form (6.7) within free field theory just because of an additional symmetry introduced in the model, in full agreement with our speculation from Sections 2 and 3. However we need to consider multiple gravitons behavior to complete the free theory.
Therefore we need to evaluate the action (6.7) (which is the gravitons action (5.22) presented in terms of ADM foliation) an arbitrary metric expanded in the series (6.13).One can evaluate K ab precisely as the form ω R enters it in a linear way: The factor √ γ remains constant because of the gauge (6.2).To calculate the threecurvature (3) R one should use the definition of the Riemann tensor (4.14) and the form ω R enters it in a bilinear way.Therefore the gravitons action (6.7) evaluated at the expansion (6.13) preserves its bilinear form.As we wrote in the previous paragraph, this feature of the theory appears only because of the additional symmetry that we introduced in the model.

Conformal metric transformations
Conformal metric transformations are defined as the following transformation of the metric g µν : where Ω is a function of coordinates that perform conformal mapping.One can also define conformal metric transformations as the following transformation from the metric g µν to the metric g µν : Following formulae [67] for conformal transformations in D-dimensional spacetime are used: There are many ways to define the connection in vierbein formalism, but they are equivalent up to a complex factor in the connection definition.We use the following connection definition: where φ i is a field of an arbitrary spin, (ω µ ) ab is the connection, and L ab is the Lorentz group generator in a representation suitable for the field φ i .One uses the standard definition of the Riemann tensor in vierbein formalism: (R µν ) j i = ∂ µ (ω ν ) j i − ∂ ν (ω µ ) j i + (ω µ ) k i (ω ν ) j k − (ω ν ) k i (ω µ ) j k . (7.21)

ADM foliation
ADM foliation formalism is stated in papers [28,29], a brief introduction to the formalism may be found in the book [54].
A four-dimensional spacetime can be represented as a foliation of space-like threesurfaces.Zero coordinate x 0 is considered as a time coordinate and for any given moment of time x 1 , x 2 , and x 3 set a coordinate frame of a three-surface.Each surface equipped with a metric γ ab (Latin indices take values 1, 2, 3), a normal vector n µ , and three tangent vectors e a .For all three-dimensional quantities indices must be raised and lowered by the three-metric γ ab .One introduces a scalar function N called the laps and a vector N a called the shift vector.They are defined by the following expression ∂x µ ∂x 0 = Nn µ + N a e µ a , (7.22) where e µ a are coordinates of a tangent vector e a .The four-dimensional metric is given by the following: ) √ −g = N √ γ .(7.25) These three-surfaces are immersed in the four-dimensional spacetime, so one can define two curvatures on the surface.The (internal) curvature is defined by the standard Riemann: (3) R a mn b = ∂ m Γ a nb − ∂ n Γ a mb + Γ a ms Γ s nb − Γ a ns Γ s mb . (7.26) The external curvature K ab is given by the following: The following equation connects the four-dimensional curvature R with the threedimensional curvature (3) R: where G abcd is the DeWitt supermetric: G abcd = √ γ 2 γ ac γ bd + γ ad γ bc − 2γ ab γ cd , (7.29) .31) H in the formula (3.2), then the representation ζ ′ l (ζ, g) acts as an adjoint representation of H on G and u l (ζ, g) becomes a linear representation of H. Therefore one can consider nonlinear representations generated by (3.2) as a spontaneous breaking of the symmetry group G down to H.