On the choice of variable for quantization of conformal GR

The possibility of using spin connection components as basic quantization variables of a conformal version of General Relativity is studied. The considered model contains gravitational degrees of freedom and a scalar dilaton field. The standard tetrad formalism is applied. Properties of spin connections in this model are analyzed. Secondary quantization of the chosen variables is performed. The gravitational part of the model action turns out to be quadratic with respect to the spin connections. So at the quantum level the model looks trivial, i.e., without quantum self-interactions. Meanwhile the correspondence to General Relativity is preserved at the classical level.


Introduction
It is widely known that quantization of General Relativity (GR), we face with the problem of non-renormalizability of the theory if we choose the space-time metric as the basic variables.Non-renormalizability reveals itself starting from the 1-loop order in perturbation theory.Therefore, with this approach, it is possible to construct a quantum theory of gravity only in the one-loop approximation.To solve this problem, various approaches were proposed.The most popular of them are: the effective field theory approach (extending the Lagrangian of general relativity by terms quadratic in curvature and additional scalar fields), string theory, loop quantum gravity, and the method of functional integration.
The approach, which is called effective field theory, is based on the idea that the Lagrangian of general relativity can be considered as a low-energy approximation of a more general theory, which, as a rule, contains terms quadratic in curvature, such as curvature square and the Gauss-Bonnet term, and often some interaction with a scalar field.Such theories can solve the problem of renormalization, but instead of it, ghost and other instabilities can arise and, accordingly, there emerge problems with unitarity.In addition, this approach by default does not address the most important issues of the evolution of topology and the causal structure of space-time.The global hyperbolicity is postulated by default from the very beginning, and the theory of gravity is then treated in the standard way, in analogy with non-gravitational field theories.
In this work, we study the conformal modification of the general theory of relativity (conformal GR) and analyze the possibility of its quantization in variables that were proposed to be used in the works [1,2].The main idea of the conformal modification of GR is that instead of the invariance with respect to the group of general covariant transformations GL(4), the invariance with respect to a broader group of transformations is considered [3].In this case, the action of the original GR undergoes a conformal (Weyl) transformation, as a result of which there appear factors which depend on the dilaton field D. In particular, for the interval one gets where g µν is the conformal metric.It is assumed that the actually observable quantities are not the standard quantities of GR, but quantities that are conformally equivalent to them.
In the standard GR, the Einstein's equations are obtained as the result of variations of the Einstein-Hilbert action When considering the Conformal General Relativity (CGR), the Einstein-Hilbert action is transformed into the action [4,5] 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.Note that action (3) is not equivalent to the Einstein-Hilbert action due to the fact that it is invariant under conformal transformations, while the Einstein-Hilbert action is not.Thus, conformal general relativity is not equivalent to standard GR, i.e., it is actually a modified theory of gravity, as discussed above.Therefore, we will call action (3) the one of conformal GR and denote it S CGR .Varying action (3) leads to equations of three types: Einstein's conformal equations (when varying according to the conformal metric); Klein-Gordon type equation for the dilaton; equations for matter fields when varying according to the corresponding conformally equivalent field variables.Conformal gravitational waves have to be described by Einstein's conformal equations, therefore, just like in the standard GR, they have two physical degrees of freedom.The fact that in action (3) there is a separate term containing the squared derivative of the dilaton field means that in addition to the conformal Einstein's equations and simultaneously with them, the equation describing the dilaton field must be solved.This in turn means that conformal GR has three degrees of freedom: two degrees of freedom describe gravitational waves, similar to the ones in GR, and one describes the propagation of the dilaton field in space-time.Thus, in conformal GR, the dilaton mode is separately identified as an additional physical degree of freedom.
It is important to note that this model is not a special case of the well-known scalartensor Brans-Dicke gravity, since it corresponds to the choice of the Dicke constant equal to −3/2, which leads to a singularity [4].
Phenomenological properties of the conformal modification of GR were considered in works [6,7,8], in which much attention was paid to the study of data from type Ia supernovae.
Based on the results of calculations and subsequent analysis, the authors come to the conclusion that the main contribution to cosmological density comes from matter having a rigid, rather than a vacuum (dark energy), equation of state.This makes it possible to explain the observed accelerated expansion of the universe without using the cosmological constant.
Let us note in advance that below use four types of indices: four-dimensional coordinate indices without brackets will be denoted by Greek letters and run through the values 0 . . .The article is structured as follows.First, the tetrad formalism and spin connections will be considered as applied to conformal general relativity, and it will be shown that the formalism in question is quite general and applicable to many modified theories of gravity.
Then the dilaton degree of freedom will be extracted from the standard GR metric.Next, the metric of a conformal, plane gravitational wave will be considered and the possibility of its quantization in a standard way will be analyzed.

Spin connections
The method that we develop in this work relies heavily on the tetrad formalism and the concept of spin connectivity.In particular, the variables ω R (a)(b),(c) , which we propose below to consider as the basic variables of quantum gravity, are directly related to the spin connection.Therefore, the current section is devoted to a discussion of the concepts of tetrad formalism and spin connectivity as applied to the conformal General Relativity.
In the classical General Relativity, space-time (M, g) is described using a model of a smooth four-dimensional Hausdorff manifold M endowed with a semi-Riemannian metric g µν .Affine connection, which defines the rules for parallel transfer of tensors from one tangent space to another, is considered symmetric and consistent with the metric.In this situation, the affine connection is completely described by a metric and it is represented in components by Christoffel symbols Γ α βγ .Unfortunately, covariant differentiation of spinor fields in the standard coordinate representation turns out to be a poorly defined operation.The solution to this problem is carried out by passing to the orthonormal frame [9].General Relativity can be formulated in terms of tetrads; the resulting method is called the tetrad formalism [10,11] or the moving frame method.Using this method, one can introduce the concept of a spin connections defined on a frame bundle and correctly define the covariant differentiation procedure for spinor fields.A moving frame (tetrad) given on a manifold if a basis of vectors e (a) is chosen in the tangent space to each point of the manifold [12].
Tetrad and co-tetrad can be expanded in a coordinate basis in the tangent and cotangent spaces, respectively.The expansion of tetrad vectors with respect to a basis in the tangent space has the form The decomposition of a co-tetrad in the basis of 1-forms (covectors) from the cotangent space has the form It is assumed that the matrices e α (a) and e α (a) are non-degenerate and are sufficiently smooth functions of points of the manifold.The following conditions on co-tetrads are imposed Let us also determine quantities e ν(a) using the formula where η (a)(d) is the Minkowski metric.Since η (a)(d) are constants and do not depend on the point of the manifold, then η (a)(d) can be freely brought in and out of the derivative.Unlike coordinate basis vector fields, tetrads vector fields, generally speaking, do not commute with each other.This is due, in particular, to the fact that a pair of arbitrary vector fields on a manifold, generally speaking, does not commute with each other.Geometrically, this means that a small parallelogram constructed using two arbitrarily chosen vector fields will, in the general case, be open.The amount of "openness" in the leading order is precisely expressed by the commutator of vector fields e (a) , e (b) .Let us decompose the commutator of a pair of tetrad vector fields into tetrad vectors The coefficients c (a)( are called nonholonomic coefficients.An explicit expression for the nonholomony coefficients in terms of the tetrad components has the following form The connection on the frame bundle is related to the affine connection, which is given on the tangent bundle by the formula Here ω α(a), (b) are so-called spin connection components.We are interested in the case when the connection is metric.That is, when the torsion tensor and the nonmetricity one vanish: Note that the torsion tensor is skew-symmetric in the first two indices, and the nonmetricity tensor is symmetric in the last two indices.In this regard, these indices are separated by commas.In this case, the components of the spin connection can be expressed through the nonholonomic coefficients by the formula [12] where the notation c where ω (a)(b), (c) := e α (a) ω α(b), (c) , ∂ (a) := e α (a) ∂ α , or if the two indices of the two-dimensional direction in which the curvature is defined remain coordinate In paper [1], the introduction of variables ω R (a)(b),(c) was discussed and it was argued that they can only be introduced using a nonlinear representation of the symmetry group, but not within the framework of the classical general relativity.Note that such statements should be treated with caution.Now we will show that in fact it is possible to distinguish ω R (a)(b),(c) and ω L (a)(b),(c) also in the classical GR.And their introduction does not require neither conformal transformations with the release of the dilaton nor the use of a nonlinear symmetry realization.
It is enough to consider the bundle of frames with the structure group SO(1, 3) and use the property that the torsion and nonmetricity tensors are equal to zero.In fact, we omit the index in formula (10) and substitute into Eq.( 12) taking into account the indices, we get and the expression for the spin connection components By rearranging the terms in Eq. ( 18), we can distinguish ω L and ω R in it To derive Eq. ( 19) we also used e In [13] up to the index notation, ω L (a)(c),(α) and ω R (a)(c),(α) are expressed as Note that in deriving these formulas we never used either conformal symmetry or the nonlinear representation of the extended symmetry group [3].However, the introduction of ω R For more information about this relationship, see the review article [14].
Note that further when ω R (b)(a),(c) and ω L (a)(b),(c) are mentioned in the text We omit the indices where this does not lead to loss of meaning and denote them simply as ω R and ω L , respectively.
Let us show that the derivatives of the metric tensor cannot depend on the components of ω L and thus ω L cannot play the role of dynamic variables.The coordinate components of the metric tensor are related to the coordinate components of the tetrads by the formula Using the formulas (23) and (24) we have Substituting ( 27) and ( 28) into (26) we obtain the following expression for the total differential of the metric tensor Although in the modification of GR we are considering, the torsion and non-metricity tensors are equal to zero, congruences of world lines, just like in classical GR, can have nonzero rotation.When considering dynamic variables, we would like to separate the gravitational properties of the theory from properties that are not directly related to gravity and characterize the congruence of world lines (the reference system).Therefore, the disappearance of ω L variables in the equation should be interpreted as the separation of the properties of the reference system (congruence of world lines) associated with rotation from the gravitational properties of the theory.

ADM formalism as applied to conformal general relativity
In this section we will discuss the application of the Arnowitt-Deser-Miesner formalism to conformal general relativity.Here we will limit ourselves to considering only globally hyperbolic space-times (M, g).From the condition of global hyperbolicity of space-time and Geroch's splitting theorem [15,16,17] it follows that (M, g) is representable as a direct product M = R × S, where S is a spacelike Cauchy 3-surface (hypersurface).Consequently, for a given space-time, the Arnowitt-Deser-Misner Formalism is valid, which allows us to represent the conformal metric in the form [18] As already mentioned in the Sect. 1, the standard metric is related to the conformal metric by the formula (1).In tetrad representation, the metric will look like 1-forms e (a) are related to the metric in the Arnowitt-Deser-Miesner representation through the formulas Here e 0 , e (j) are a set of basic co-tetrads (1-forms conjugate to tetrad vectors), e (j)i .The quantities N and N i are called of the lapse function and the shift vector, respectively.The lapse function can be represented as having identified in it a part that depends only on time (global) and a part that depends on coordinates (local part) [19].In (34) N 0 is the global part of the lapse function, and N is the local part of the lapse function.The volume shape then, taking into account (34), can be written as where γ is the determinant of the spatial metric.This allows us to separately highlight integration over space in action.Action (3) will then have the form Thus, (36) is divided into temporal and spatial parts.We also omitted the Λ term, since its presence is not essential for our further analysis.

Analysis of variables proposed for gravity quantization
The purpose of this section is to analyze the possibility of constructing a quantum theory of gravitational waves and their interaction with matter within the framework of conformal General Relativity if we accept the variables ω R as basic variables when quantizing gravity, as was proposed to be done in the works [1,2].
Quantization of gravity in the general case is known to be an extremely difficult problem.In this regard, following [1] in this work, we will consider the simplest special case of a plane wave and some questions that arise when trying to quantize conformal General Relativity.We are interested in the question of quantization of gravitational waves, so we will now focus on the purely gravitational part of the action (3), which we presented in the Sect. 3 as (36), separating explicitly the temporal and spatial parts.
Since in CGR the quantities that are actually observable in experiment are those that are obtained from standard ones using Weyl transformations, we need to consider the conformal metric g µν , which describes the gravitational wave.We will call gravitational waves in conformal General Relativity conformal gravitational waves.The conformal metric is obtained from the standard metric using the appropriate conformal transformation (1).Thus, in conformal GR the main task is to find a conformal transformation that would bring the standard metric to a conformal (actually observable) metric.
In the standard General Relativity there are different metrics that can be interpreted as metrics describing gravitational waves.The simplest of them is the non-linear (strong) plane wave metric, which can be represented in the form [18] Without loss of generality, we can set e Σ = 1 and, accordingly, Σ = 0.It is natural to look for a solution like gravitational waves in CGR, considering metric (37) as an ansatz.Then, taking into account the condition Σ = 0 imposed above, the metric of the conformal g µν gravitational wave (37) can be written in the form Below, the metric and ω R everywhere refers to conformal general relativity, therefore, to simplify the notation, we will further omit the symbol ∼ above all conformal quantities.We also need to impose a gauge condition.We choose the so-called Lichnerovich gauge condition [20], which is that the determinant of the conformal three-dimensional metric is γ = 1.The action for gravity (36) can be written as We can represent (33) in the equivalent form The expression e σ(a) de σ (b) is symmetric in the indices µν, so we can write ω R as: Hence, further, as shown in works [1,2], the following representation is valid for variables where g ± k pretend to be the operators of creation and annihilation of conformal gravitons.
Let us now analyze how gravity interacts with itself and with matter if we accept the variables ω R as the fundamental variables of quantum gravity, as proposed in the works [1,2].Expression (44) was proposed to be considered as a conformal free plane gravitational wave.
First of all, we note that the expression (44) contains momentum k (c) .From this alone we can conclude that this expansion does not describe an ordinary plane wave.Let us first show that in the action (3) there are no terms from which a propagator of a conformal graviton could be obtained.To do this, we will use the expression for the curvature tensor in the nonholonomic basis (13).Make the contraction of in the expression (13) using the indices (b), (d) we obtain the Ricci tensor + ω (b)(c), (e) Raising the index in the expression for the spin connection (19) through ω R and ω L we, up to the designation of the indices, obtain the formula for its components in the form in which we need it in the expression (45).Since the dynamic variables in this case are ω R , and not ω L , we will explicitly write out the formula only for them Then ω (b),(c) (b) is expressed by the formula and ω (a),(c) (b) by the formula Due to the gauge conditions imposed above N = 1, N i = 0 and γ = 1, only the volume shape components √ −g = 1 and ω R can be contained in expression calculated above for scalar curvature.Then, by substituting (48) and ( 49) into (46), taking into account the location of the indices, one can obtain an expression for the derivatives of ω R , from which it is clear that in the Lagrangian there are no terms containing the square of derivatives ω R from which, after integration by parts, a conformal graviton propagator could arise.Thus, if we accept ω R as the fundamental variables of quantum gravity, then we discover the absence of an analogue of the wave equation for the gravitational variables ω R that is familiar to us in quantum field theory.
Let us now find out what the terms of the interaction of gravity with matter look like if we choose ω R as the basic variables when quantizing conformal general relativity.First of all, let us note the following fact.In modified theories of gravity, the interaction with matter fields is either minimal or non-minimal.If the interaction is minimal, then the corresponding Lagrangian terms can contain metric components, but not their derivatives.If the interaction under consideration is non-minimal, then the Lagrangian may contain derivatives of the metric components.
Let's look again at action (50).Due to the gauge conditions already imposed above N = 1, N i = 0 and γ = 1, we have √ −g = 1 and for the action of matter the Lagrangian will have the form Note that if we consider only the interaction terms in (50) then by virtue of (53) we have Accordingly, after passing to tetrad indices using e (c) α , we obtain a system of first order differential equations which must be solved in order to express the components of the metric tensor through the variables ω R , but in the absence of a propagator for ω R it is impossible to construct a standard quantum theory of the interaction of ω R with matter fields.

Conclusion
In this work, some properties of the conformal modification of general relativity related to the spin connections were investigated.In particular, it was shown that the expression for the spin connection components (19) expressed in terms of ω L and ω R can be obtained through formal transformations of the standard formula for spin connection, which is usually written in terms of nonholonomic coefficients in the form (12).It is demonstrated that conformal symmetry and its nonlinear realization if not mandatory for the introduction of the variables ω R,L for construction of the spin connection (19).This provides an extension of the approach presented in [1].Thus, we generalized the expression for the spin connection components (19), written in terms of ω R,L to a much wider range of theories of gravity.Namely, the components of the spin connection can be represented as (19) in any theory of gravity in which the spin connection is a metric one.Note also that our approach is alternative with respect to the attempts of gravity quantization in terms of tetrads as the basic variables.On the other hand, variables ω R,L are related to tetrads through the formulae (24) and ( 23), so the former can be exploited in the general tetrad formalism.
We also performed the analysis of the construction of quantum gravity formulated on the basis of the standard quantum field approach in terms of ω R variables in Sect. 4. It was shown that the gravitational Lagrangian not only lacks interaction terms, as has been noted in [1], but also does not contain terms being quadratic in derivatives of ω R , of which, after integration by parts, a propagator of the conformal graviton would arise in the terms of variables ω R .
Here one can note a similarity to quantization of a free Dirac spinor field.One can see that the constructed (strong) gravitational waves do not interact neither with each other nor with any matter field.So, the obtained quantum theory looks trivial.Nevertheless, there are still minimal couplings (of the classical type) to the matter field and gravity though the metric tensor.Note that the classical GR is reproduced.
As a result, we come to the conclusion that formulation of quantum gravity in terms of ω R as the fundamental quantum variables yields an almost trivial theory which is nevertheless worth for further investigation.In particular, emission, propagation, and detection of the gravitational waves has to be analyzed.Moreover, there is another point for further studies, associated with the fact that quantities ω R do not form a tensor and thus covariant integration by parts in an arbitrary curved space-time turns out to be an ill-defined operation (the results will depend on the parameterization).In any case, we argue that spin connections are definitely useful for studies of its properties in the tetrad formalism, as in General Relativity itself as well as in its modifications including the conformal one.