# Viability of the Loop Quantum Gravity kinematics in Palatini f(R) models

Flavio Bombacigno Physics Department, “Sapienza” University of Rome, P.le Aldo Moro 5, 00185 (Roma), Italy    Simon Boudet Physics Department, “Sapienza” University of Rome, P.le Aldo Moro 5, 00185 (Roma), Italy Department of Physics, University of Trento, Via Sommarive 14, 38123 (Trento), Italy    Giovanni Montani Physics Department, “Sapienza” University of Rome, P.le Aldo Moro 5, 00185 (Roma), Italy ENEA, Fusion and Nuclear Safety Department, C. R. Frascati, Via E. Fermi 45, 00044 Frascati (Roma), Italy
###### Abstract

We consider special classes of Palatini theories, featured by additional Loop Quantum Gravity inspired terms, with the aim of testing the feasibility of a loop-like quantization procedure. In particular, by adopting a scalar-tensor representation, we successfully identify a set of modified Ashtekar canonical variables, preserving the gauge structure of the standard theory by means of a novel Gauss constraint. We discuss in detail the dynamical role of the additional scalar field affecting these models, stressing its effects on the area operator stemming from such a revised theoretical framework. Especially, we show how a canonical rescaling of the scalar field could absorb the Immirzi parameter in the spectrum, eliminating it from the kinematics. We perform then the analysis in the Einstein frame, and we compare our results with earlier studies in literature, outlining possible differences between metric and Palatini approaches.

## I Introduction

The most successful approach to the problem of quantizing gravity is, up to now, the so called Loop Quantum Gravity (LQG) theory bookcqg ; Rovelli:2004tv ; Thiemann . This formulation, of course, still contains a number of non-trivial and unaddressed questions, as for instance the implementation of the quantum dynamics via the scalar constraint, the construction of a classical limit and the intrinsic ambiguity of the appropriate phase space variable to be adopted, i.e. the ambiguity in the meaning and value of the so-called Immirzi parameter Rovelli:1997na ; Gambini:1998ip ; Chou:2005ht ; Perez:2005pm ; Mercuri:2006um ; Date:2008rb ; Mercuri:2009zi ; Nicolai:2005mc . Nonetheless, the great interest for LQG is justified by the possibility to construct a kinematic Hilbert space for the quantum theory, resulting in geometrical operators, like areas and volumes, endowed with discrete spectra Rovelli:1994ge ; Ashtekar:1996eg ; Lewandowski:1996gk . By other words, LQG is able to introduce the concept of space discretization, simply starting from a classical Lagrangian for the gravitational field bookcqg , with the quantum theory just relying on the pre-metric concept of graph. Reasons of this impressive result must be researched in the use of the so-called Ashtekar-Barbero-Immirzi variables Ashtekar:1986yd ; Ashtekar:1987gu ; Barbero:1994ap ; Immirzi:1996di , which allow to pursue the Hamiltonian formulation of the gravitational field by close analogy with non-Abelian gauge theories. In particular, such variables allow to write down the constraint associated to local spatial rotations in the form of a standard Gauss constraint for the group.
In order to achieve this formulation by means of a first order approach, i.e. by considering tetrads and spin connection as independent fields, it is necessary, however, to add to the Einstein-Hilbert Lagrangian new terms, that is Holst or Nieh-Yan contributions Nieh:1981ww ; Holst:1995pc ; Soo:1998ev ; Nieh:2007zz ; Mercuri:2007ki ; Banerjee:2010yn . In particular, they are such that classical dynamics is always recovered, being the former vanishing on of half-shell (where the equation for the connection holds), and the latter a pure topological term. In both cases, therefore, we deal with a restatement of General Relativity, suitable for loop quantization, which admits Einstein equations as classical limit.
In this respect, the recent interest for modifications of General Relativity Sotiriou ; Nojiri:2017ncd , makes very timely questioning about possible LQG extensions of generalized models, especially via their reformulation in term of scalar-tensor theories Bergmann:1968ve ; Capone:2009xk ; Ruf:2017xon . A first attempt in this direction was performed in Cinesi ; CinesiSecondOrder ; Han:2013noa (see also Montani:2009gn ; Bombacigno:2018tbo ; Bombacigno:2018tih ), where the problem was faced according a second order formulation, i.e. by considering the metric as the only independent field, and authors actually followed in defining Ashtekar-like variables an extended phase-space method Thiemann . Conclusions of this study suggest that a suitable set of restated variables can be determine to re-obtain proper Gauss and vector constraints, with the non-minimally coupled scalar dynamics affecting the scalar constraint.
Here, we face the same problem, but on a more general framework and adopting the most natural first order formalism, i.e. we deal with Palatini models Olmo2011 , equipped with Holst and Nieh-Yan terms. In particular, we characterize the classical morphology of the resulting theory and we discuss the reformulation in terms of variables.
In including Holst and Nieh-Yan terms, we actually have two different choices, consisting in inserting these terms either inside or outside the argument of the function , so leading to four distinct cases. We first analyze the classical dynamics of these four models demonstrating they correspond to only two physically distinct scenarios. In fact, both when the Holst term is included in the argument of and when the Nieh-Yan term is added into the Lagrangian at the same level of , we see that the classical dynamics is that one of a standard Palatini model, characterized by a non-dynamical scalar field. Conversely, when the Holst term is added to and the Nieh-Yan one is plugged inside the function, we deal with a classical scalar-tensor theory and the Immirzi parameter enters the definition of the parameter , ruling the kinetic term for the dynamical scalar field. Then, we perform the Hamiltonian formulation for this two distinguished cases, and we discuss the resulting morphology in terms of the constraints emerging after the Legendre transformation. In particular, in the case of a Palatini model and under the assumption of a matter Lagrangian decoupled from connection, we successfully recover the algebraic equation relating the scalar field to the trace of the matter stress energy tensor, leading to a non-dynamical field.
The main merit of this study consists of the determination of suitable generalized Ashtekar-Barbero-Immirzi variables, that for both the cases, i.e. for the all original four choices, allows to deal with a Gauss and vector constraint having exactly the same form as in LQG. The different nature of the two considered scenarios and their departure from LQG are then summarized by the morphology of the scalar field only. By other words, we are always able to construct a kinematic Hilbert space for the canonical quantization a la LQG and we are able to study the spectrum of geometrical operators, with special regard on area operator. Here another crucial result from our analysis comes out, since, differently from what assumed in Fatibene:2010yc , we clarify how the area operator has a clear and unambiguous geometrical nature, being constructed with the real triad of the space. We stress, therefore, how the different link between the real triads and the particular variables considered in every specific model, affects the morphology of the area operator as an gauge-invariant object. Therefore, also the resulting spectrum is influenced by the considered scenario and in all the cases, a scalar field, non-minimally coupled to gravity emerges. In the case of a standard Palatini model, such a field is, as above mentioned, non-dynamical and when it is expressed via the trace of the stress energy tensor, a very intriguing coupling takes place among the size of the area associated to a graph and the nature of the matter filling the space. When we deal with a proper scalar tensor theory, instead, the scalar field is truly dynamical and it must be quantized according to an holonomy-like formalism. This implies the possibility of a canonical rescaling of the scalar field by a constant factor, which allows to set equal to unity the Immirzi parameter in the spectrum and in all the kinematic part of the theory, though it still stands in the Hamiltonian constraint. Thus, we can infer that in extended theories we considered, which are isomorphic to scalar tensor models with depending on the value of the Immirzi parameter, the ambiguity of such a parameter is to some extent weakened. Indeed, since it can be regarded as unity in the kinematic part of the theory, only the quantum dynamics would be sensible of the specific choice of Ashtekar-Barbero-Immirzi variables. Actually, the dependece of the quantum gravitational field on the value of the Immirzi parameter is clearly explained by the fact that we are changing the nature of the scalar tensor theory we are quantizing. This issue of our study seems to suggest that the nature of the ambiguity associated to the Immirzi parameter in LQG could be overcome by more general formulations of the gravitational sector as a gauge theory.
Furthermore, when comparing the metric analysis of CinesiSecondOrder to our Palatini formulation, we observe the emergence of a discrepancy in the value of in the scalar constraints. By other words, starting directly from a metric formulation of gravity with variables provides different dynamical constraints with respect to a first order formulation. In this respect, we outline then the possibility to restore a complete equivalence between these two approaches, by restating our models into the Einstein framework and performing a canonical transformation. We note that similar issues hold also for CinesiFirstOrder , in which the analysis is actually pursued starting from a first order action, which includes however additional contributions which allow to eliminate torsion from the theory at the very beginning.
The paper is structured as follows. In Sec. II the four models are presented and their effective theories are derived, highlighting equivalences and differences with Palatini theory in its scalar-tensor formulation. In Sec. III the new modified variables and the correspondent set of constraints are derived, discussing the different dynamical character of the scalar field in each case analysed. Sec. IV deals with the modifications in the spectrum of the area operator and with the Immirzi field role in the new framework. The analysis performed in the conformal Einstein frame and the comparison with earlier studies in literature are contained in Sec. V. In Sec. VI conclusions are drown, while the detailed computations regarding the results presented in Sec. III can be found in the Appendix A.
Eventually, notation is established as follows. Spacetime indices are denoted by middle alphabet Greek letters , spatial ones by letters from the beginning of the Latin alphabet . Four dimensional internal indices are displayed by capital letters from the middle of the Latin alphabet , while indicate three dimensional internal indices. Spacetime signature is chosen mostly plus, i.e. .

## Ii Extended f(R) actions with Holst and Nieh-Yan terms

The starting point of our analysis is represented by peculiar modifications of Palatini theories, obtained by implementing Loop Quantum Gravity inspired contributions into the action. For the sake of clarity, then it can be useful to briefly recall the scalar-tensor reformulation of ordinary Palatini gravity. Let us consider therefore the action

 S=12χ∫d4x√−gf(R), (1)

where111We set the speed of light . . The Ricci scalar is obtained by the contraction of the Ricci tensor , here considered as a function of the independent connection and related to the Riemann tensor by , with

 R\indicesμνρσ=∂ρ\tensorΓμνσ−∂σ\tensorΓμνρ+\tensorΓμλρ\tensorΓλνσ−\tensorΓμλσ\tensorΓλνρ. (2)

The theory can be thus expressed in the Jordan frame by introducing an auxiliary scalar field , i.e.

 S=12χ∫d4x√−g[f(ξ)+f′(ξ)(R−ξ)], (3)

where the variation with respect to leads to the dynamical constraint , provided , which proves the equivalence of (3) with (1). Moreover, the redefinition allows us to rewrite (1) in the form of a scalar tensor theory, that is

 S=12χ∫d4x√−g[ϕR−V(ϕ)], (4)

where we defined the potential term . Now, such a procedure can be easily implemented for the peculiar actions considered in this work, namely

 Sf(R−H) =12χ∫d4x√−gf(R−H), (5) Sf(R)−H =12χ∫d4x√−g[f(R)−H], (6) Sf(R+NY) =12χ∫d4x√−gf(R+NY), (7) Sf(R)+NY =12χ∫d4x√−g[f(R)+NY], (8)

where we introduced, either in the argument of the function or outside of it, Holst and Nieh-Yan terms, given by, respectively

 H =β2ε\indicesμνρσRμνρσ, (9) NY =β2ε\indicesμνρσ(12T\indicesλμνT\indicesλρσ−Rμνρσ), (10)

with the reciprocal of the Immirzi parameter. Torsion tensor is displayed by and, as outlined in Bombacigno:2018tbo ; Bombacigno:2018tih ; Calcagni , it cannot be a priori neglected in Palatini generalizations of Holst and Nieh-Yan actions, being its form to be determined dynamically, consistently with a first order formulation. This in turn implies that neither (9) nor (10) are vanishing as in the standard approach, since by virtue of torsion the Bianchi identity can be evaded and a possible dynamical coupling with the Nieh-Yan term, stemming from the scalar tensor reformulation, can devoid it of its topological character.
In order to see that, let us extend the technique adopted in (3) to models (5)-(8), i.e.

 Sf(R−H) =12χ∫d4x√−g[ϕ(R−H)−V(ϕ)], (11) Sf(R)−H =12χ∫d4x√−g[ϕR−H−V(ϕ)], (12) Sf(R+NY) =12χ∫d4x√−g[ϕ(R+NY)−V(ϕ)], (13) Sf(R)+NY =12χ∫d4x√−g[ϕR+NY−V(ϕ)], (14)

with now defined in terms of the derivative of the function with respect to its generic argument. Next, if we assume the connection is still metric compatible, then in the presence of torsion it can be written as

 \tensorΓρμν=\tensorCρμν+\tensorKρμν, (15)

 \tensorCνρμ=12gνσ(∂ρgσμ+∂μgσρ−∂σgρμ) (16)

and the independent character of the connection is now encoded in the contortion tensor, given by

 \tensorKρμν=12(\tensorTρμν−\tensorTρμν−\tensorTρνμ). (17)

In standard Palatini gravity Olmo2011 , it is related to the field by

 Kρμν=12(gρν∂μlnϕ−gμν∂ρlnϕ), (18)

which, taking into account (15), allows us to rewrite the Ricci scalar as

 R=R+32ϕ2∂μϕ∂μϕ−3□ϕϕ, (19)

where denotes the Ricci scalar depending on the metric field only and , being the covariant derivative defined from connection (16). Therefore, action (4) can be recast, up to boundary terms, as

 S=12χ∫d4x√−g[ϕR+32ϕ∂μϕ∂μϕ−V(ϕ)]. (20)

The analogous of (20) for actions (11)-(14), can be computed by decomposing the torsion tensor in its independent components according to the Lorentz group. These are the trace vector

 Tμ≡T\indicesνμν, (21)

the pseudotrace axial vector

 Sμ≡ϵμνρσTνρσ (22)

and the antisymmetric tensor satisfying

 ϵμνρσqνρσ=0,q\indicesμνμ=0. (23)

In terms of these quantities the torsion tensor can be written as

 Tμνρ=13(Tνgμρ−Tρgμν)−16ϵμνρσSσ+qμνρ (24)

and, consequently, the actions can be collectively displayed by

 Sb =12χ∫d4x√−g[ϕR+ϕ24SμSμ−23ϕTμTμ+ (25) −2ϕ∇μTμ+b1ϕβ2∇μSμ+b2bϕβ3TμSμ ϕ2qμνρqμνρ+b2bϕβ2ϵμνρσq\indicesτμνqτρσ−V(ϕ)],

where we used relations

 R=R−2∇μTμ−23TμTμ+124SμSμ+12qμνρqμνρ, (26) H=β2∇μSμ+β3TμSμ+β4ϵμνρσqτμν\tensorqτνσ, (27) NY=β2∇μSμ (28)

and we introduced two parameters and which can take values or , together with a field which is either coincident with or identically equal to . The choice of their values allows to select one particular model among the four possible, according table 1.

The equations of motion for stemming from (25) are identically solved by , while varying with respect to and yields, respectively

 ϕTμ−32∂μϕ−b2bϕβ4Sμ=0, (29)
 ϕSμ−6b1β∂μϕ+4b2bϕβTμ=0, (30)

whose solution is given by

 Tμ=[1+b1b2bϕβ2/ϕ1+b2b2ϕβ2/ϕ2]32ϕ∂μϕ, (31) Sμ=[b1−b2bϕ/ϕ1+b2b2ϕβ2/ϕ2]6βϕ∂μϕ. (32)

Substituting back (31) and (32) into (25), actions (11)-(14) can be rearranged into the form

 S=12χ∫d4x√−g[ϕR−Ω(ϕ)ϕ∂μϕ∂μϕ−V(ϕ)], (33)

where depends on the particular model addressed according to table 2.

We see that only and cases take the scalar tensor parameter , corresponding to the effective description (20). By other words, these two alternatives, which are labelled together as case I, are the only models dynamically equivalent to ordinary Palatini theory for all values of the Immirzi parameter. This implies that the field is not an actual degree of freedom, but it is simply determined by the structural equation as in the standard case Sotiriou ; Olmo2011 , i.e.

 2V(ϕ)−ϕV′(ϕ)=χT, (34)

where a prime denotes differentiation with respect to the argument and is trace of the stress energy tensor for some matter contributions, which are assumed do not couple to connection. Case I results also reveal, as one might expect, how the topological character of the Nieh-Yan term be preserved if it is added directly to the Lagrangian. Less trivial, instead, is of course the outcome for the model. In this case, indeed, the vanishing of the Holst term on half shell is to some extent recovered when it is included in the argument of the function .
On the other hand, inserting the Nieh-Yan term in the argument of the function or featuring the Palatini theory with an additional Holst contribution (case II), ensures the equivalence to Palatini gravity only for , as it is expected by the coupling of the Immirzi parameter in (27)-(28). In case II, therefore, the scalar field acquires in general a dynamical character as described by

 (3+2Ω(ϕ))□ϕ+Ω′(ϕ)(∂ϕ)2+2V(ϕ)−ϕV′(ϕ)=χT, (35)

and by virtue of (31) and (32) we actually deal with a theory equipped with propagating torsion degrees, by close analogy with Bombacigno:2018tbo ; Bombacigno:2018tih , where and models were considered in the presence of a dynamical Immirzi field. In this regard, the additional degree of freedom featuring case II does not make the Palatini extensions here analyzed comparable to metric gravity, in that corresponding scalar modes are actually responsible for different gravitational wave polarizations Liang:2017ahj ; Moretti:2019yhs . As a result, even though the model is formally identical to metric gravity for (), they are actually endowed with distinct phenomenology. We also note that the case is singular for . However, this value is not truly pathological, since the initial action simply reduces to the case, provided we fix the Immirzi parameter as .

## Iii Modified Ashtekar variables

The analysis of Hamiltonian systems corresponding to models (5)-(8) can be found in appendix A, which we remind the reader for details. Here, we discuss the main results of the procedure, outlining the emergence of a new set of Ashtekar-like variables still suitable for loop quantization.
The gravitational sector of the phase space turns out to be characterized by the set of canonical variables , where denotes the conjugate momentum to and are defined as

 Kia≡ω\indices0ia, (36) ~Eai≡ϕEai, (37)

with the Riemannian spin connection, and the ordinary densitized triad.
The phase space is subject to a set of first class constraints consisting of the rotational constraint

 Ri≡\tensorεkijKja~Eak≈0, (38)

the vector constraint

 Ha≡2~EbiD(~ω)[aKib]+π∂aϕ≈0, (39)

and the scalar constraint

 H ≡−√ϕ2~Eai~Ebj√~E(3Rijab(~ω)+2Ki[aKjb])+12√ϕ3~EϕΩπ2+ +12√~Eϕ3Ωϕ∂aϕ∂aϕ+12√~Eϕ3V(ϕ)≈0. (40)

where ,

 ~ω\indicesija=~Eb[i(2∂[a~Ej]b]+~Ej]d~Eka∂d~Ebk)+1~E~E[ia~Ej]b∂b~E, (41)

which can be expressed in terms of the Riemannian spin connection and the scalar field as

 ~ω\indicesija=ω\indicesija(E)+1ϕE[iaEj]b∂bϕ. (42)

We point out that models belonging to case I are endowed with an additional second class constraint, which can be recast, provided we properly fix the Lagrangian multipliers, in the form (34), proving the non dynamical character of the field . This result is reinforced by the fact that, in case I, an arbitrary Lagrange multiplier enters the definition of , which can be used for freezing its evolution. In case II, instead, no additional constraints arise and the scalar field and its conjugate momentum are truly dynamical.
Then, performing the canonical transformation

 ~Eai →(β)~Eai=β~Eai, (43) Kia →(β)Aai=1βKia+~Γia, (44)

where , a set of modified Ashtekar variables can be obtained, in terms of which the rotational constraint can be combined with the compatibility condition , satisfied by (41), yielding the Gauss constraint

 Gi=∂a(β)~Eai+εijk(β)Aja(β)~Eak≈0. (45)

This guarantees that Palatini -like models here considered are actually feasible for Loop Quantum Gravity quantization procedure. Especially, by means of the new variables the vector constraint can be rearranged as in standard LQG, along with the additional term associated to the scalar field, i.e.

 Ha=(β)~EbiFiab+π∂aϕ, (46)

where . Conversely, the scalar constraint turns out to be modified with respect to the standard case, namely

 H=−√ϕ2(β)~Eai(β)~Ebj√β(β)~E[β2εijkFkab+(β2+1)3Rijab(~ω)]+ +12√ϕ3(β)~EϕΩπ2+12√(β)~Eϕ3Ωϕ∂aϕ∂aϕ+12√(β)~Eϕ3V(ϕ), (47)

where , reflecting the difference in the dynamics which exists at a classical level between General Relativity and Palatini Gravity.
The preservation of Gauss and vector constraints assure that it is straightforward to extend the usual quantization procedure bookcqg ; Rovelli:2004tv ; Thiemann to the new variables (43)-(44). In particular, we can introduce the fluxes , obtained smearing on a surface , i.e.

 ~Ei(S)=∫Sds(β)~Eaina, (48)

where is the normal vector to the surface, and the holonomies , defined as the path ordered exponential of the smearing of along a path , i.e.

 he[(β)Aia]=Pexp(∫e(β)A), (49)

where , being the tangent vector to the path , and the SU(2) generators. Now, even though they can be promoted to operators acting on spin-network states, just as in the standard case, the quantization of the scalar sector still requires some care, by virtue of the different dynamical character of in cases I and II. Indeed, since for models belonging to case I the scalar field is not an independent degree of freedom, it has not to be actually quantized, but it can be simply considered a function of matter fields by means of (34). In case II on the other hand, it embodies a proper gravitational degree of freedom and it can be quantized using the representation adopted in PolyPhi , based on the definition of point-like holonomies

 Uλ(ϕ(x))=eiλϕ(x), (50)

with and , and smeared momenta

 P(V)=∫Vd3xπ(x), (51)

with a 3-dimensional region in . In particular, their action on scalar network states defined as

 Φ=∏xi∈XUλi(ϕ(xi)), (52)

where is a set of points in , is given by

 ^Uλ(ϕ(x))Φ=eiλϕ(x)Φ, (53)
 ^P(V)Φ=iℏ{P(V),Φ}. (54)

We remark that in this representation only the holonomy (50) is rigorously defined and there exist no operator associated to the scalar field. However, as outlined in Cinesi , it can be defined by the limit process

 ^ϕ(x)Φ=limϵ→012iε[^Uε(ϕ(x))−^U−ε(ϕ(x))]Φ=ϕ(x)Φ, (55)

where (53) is taken into account.

## Iv Area spectrum and Immirzi ambiguity

In standard LQG the classical expression for the area of a surface is written in terms of densitized triads as

 A(S)=∫Sds√EaiEbinanb (56)

and then quantized computing the action of fluxes on spin-network basis states. The area operator turns out to be diagonal in this basis, with the spectrum given by222We consider here only the simple case in which there are no nodes of the graph belonging to the surface nor edges laying on it.

 a=8πℓ2Pβ∑p√jp(jp+1), (57)

where is the Planck length and the sum runs over punctures of the surface due to edges of the spin-network, colored by spin quantum numbers .
However, in the theories we analyzed, the phase space variable to be quantized is , whereas the physical metric is still associated to the ordinary densitized triad . Thus, equation (56) still holds and, in view of its quantization, it has to be rewritten in terms of , namely

 A(S)=1β∫Sds√(β)~Eaina(β)~Ebinbϕ. (58)

Now, the square root can be quantized via a regularization procedure as in the standard case, and as long as case I is considered, the scalar field appearing in the last expression should be interpreted as a function of the trace of the stress energy tensor of matter fields . Then, in this case the spectrum of the area operator is simply given by

 a(I)ϕ=8πℓ2Pβ∑p1ϕ(T)√jp(jp+1). (59)

When case II is taken into account, instead, the presence in (58) of the dynamical quantity in the denominator raises the issues about the proper quantization procedure for the scalar field. This problem can be overcome by performing the canonical transformation

 ϕ′=1ϕ,π′=−ϕ2π (60)

which leaves the Gauss and vector constraints invariant333Transformation (60) affects, of course, the scalar constraint (47), but in this work we do not address its quantization. but causes the area expression (58) to depend on the scalar field rather than its reciprocal. Then, one can perform the standard regularization procedure for quantizing fluxes and adopt the definition (55) for , computing the action of both and on a state obtained by the direct product of spin-network and scalar-network states. That results in the modified area spectrum

 a(II)ϕ=8πℓ2Pβ∑pϕ′(p)√jp(jp+1), (61)

where the only non vanishing contributions are those in which the scalar field is computed in the punctures.
This implies that the scalar field contribution to the area operator does not spoil the discrete character of its spectrum, since only a discrete set of its eigeinvalues, i.e. , gives non vanishing contribution, contrary to what argued in ConformalImmirzi1 ; ConformalImmirzi2 .
Thus, in both cases a modification in the spectrum of the area operator arises. This is in contrast with the results of Fatibene:2010yc , where it is argued that the preservation of Gauss and vector constraints is sufficient to ensure the equivalence of the spectrum with the standard one.
Moreover, the dependence of (58) on (or on ), can be exploited in order to absorb the Immirzi parameter, eliminating it from the spectrum. Indeed, we can redefine the scalar field and its conjugate momentum as

 (β)ϕ=βϕ, (62) (β)π=πβ, (63)

yielding the new classical expression for the area

 A(S)=∫Sds√(β)~Eaina(β)~Ebinb(β)ϕ, (64)

whose quantization can be addressed as in (59) and (61), according the case. Therefore, by virtue of (64), the area spectrum does not contain any dependence on the Immirzi parameter, while the scalar field retains the same meaning as in the cases previously discussed. In this way, though the Immirzi parameter still enters the theory as an arbitrary real parameter ruling (43)-(44), it does not characterize the definition of kinematic quantities like the area444Analogous considerations are expected to hold also for the volume operator.. Such an outcome seems to suggest that in Palatini extensions of LQG, the Immirzi parameter can be conveniently set to unity in definitions of geometrical objects as in (56), and its effects on dynamics absorbed in the value of . We note that in ConformalImmirzi1 ; ConformalImmirzi2 analogous results are achieved in the context of Conformal-LQG, where an additional conformal transformation is included into the symmetries of the theory. In that case, indeed, it is possible to build an area operator invariant under conformal transformation and independent on the Immirzi parameter, which acquires the role of gauge parameter for the new conformal symmetry. For such a purpose, however, we are forced to consider the conformal rescaled metric as the physical one, in contrast to our assumptions and by analogy with Fatibene:2010yc .

## V Formulation in the Einstein frame

Here we compare the results of the Hamiltonian approach discussed in Sec. III to analogous studies present in literature CinesiSecondOrder ; CinesiFirstOrder , where a different set of canonical variables was obtained, denoted by hatted characters and related to ours by the canonical transformation

 ^Eai=1ϕ~Eai,^Kia=ϕKia, (65) ^π=π−1ϕ~EaiKia,^ϕ=ϕ. (66)

These results are to some extent controversial, since performing on (38)-(III) such a transformation reproduces the same set of constraints of CinesiSecondOrder ; CinesiFirstOrder only if we replace by hand with . Furthermore, in CinesiSecondOrder starting from a second order analysis of (33), a LQG formulation of scalar tensor theories was achieved via a symplectic reduction technique Thiemann . Then, this seems to point out that the implementation of Ashtekar variables could be affected by the peculiar choice of the formalism adopted, when extensions to General Relativity are taken into account, as it occurs for the Jordan frame formulation of Palatini -like models we considered. In this sense, the fact that in CinesiFirstOrder were actually derived results equivalent to CinesiSecondOrder according a first order approach, can be traced back to the choice of including additional contributions in the action, featuring a Holst term, with the aim of eliminating torsion from the theory.
Now, we want to show how the phase space structure obtained in CinesiFirstOrder ; CinesiSecondOrder could be reproduced by means of the canonical transformation (65)-(66), provided the analysis of Sec. II be pursued in the so called Einstein frame, endowed with the conformally rescaled metric . Of course, we note that such an equivalence does not exclude a priori the existence, even at the level of the Jordan frame, of a different, possibly gauged canonical transformation (see Lulli ), able to tackle with this problem.
Let us therefore rewrite in the Einstein frame actions (11)-(14), that is

 Sf(R−H) (67) Sf(R)−H (68) Sf(R+NY) (69) Sf(R)+NY (70)

where the potential is redefined as and the Ricci scalar is obtained from

 ~R=~gμν~Rμν, (71)

with denoting the conformally rescaled Ricci tensor, which we assume to depend on the connection . Analogously, we introduced the rescaled Holst and Nieh-Yan terms

 ~H =β2~εμνρσ~Rμνρσ, (72) ˜NY =β2~εμνρσ(12~gλτ\tensor~Tλμν~Tτρσ−~Rμνρσ), (73)

with . Now, assuming the compatibility condition for the conformal metric with respect to , namely

 ~∇μ~gνρ=0, (74)

the connection can be again written as

 ~Γ\indicesρμν=~C\indicesρμν+~K\indicesρμν, (75)

where is the Christoffel symbol of the conformal metric and is defined as in (17) but with indices now raised, lowered and contracted using the conformal metric. Torsion tensor can be therefore displayed as

 ~Tμνρ=13(~Tν~gμρ−~Tρ~gμν)−16~ϵμνρσ~Sσ+~qμνρ. (76)

and the effective action for the four models can be derived, i.e.

 S=12χ∫d4x√−~g{˜R−~Ω(ϕ)ϕ~gμν∂μϕ∂νϕ−U(ϕ)}, (77)

where is the Ricci scalar depending only on the conformal metric and the function takes the values in table 3.

In particular, by comparing (77) with (33), we see that and are related by

 ~Ω=Ω+3/2ϕ, (78)

which shows how the value , associated to non dynamical configurations for the field in the Jordan frame, correspond to , in agreement with the representation in the Einstein frame of scalar tensor theories. Therefore, models belonging to case I are dynamically equivalent to Palatini theory for every value of the Immirzi parameter, while in case II the equivalence is correctly recovered only for .
Then, following the line of appendix A, we can perform the Hamiltonian analysis, which reveals a phase space coordinatized by the set of variables , where the densitized triad is now defined in terms of the spatial part of the conformal tetrad , and a rotational constraint coincident for all models with the expression derived in the Jordan frame. Regarding the vector and scalar constraints instead, for case I they read, respectively

 Ha=2~EbiD(~ω)[aKib]≈0 (79) H=−~Eai~Ebj2√~E(3Rijab(~ω)+2Ki[aKjb])+√~EU(ϕ)≈0, (80)

where is the spin connection compatible with . In this case, the conjugated momentum to the scalar field is weakly vanishing, leading to the primary constraint , whose conservation along the dynamics simply amounts to impose the variation of the potential term with respect to to weakly vanish. If matter is included as in Appendix A, this reproduces the structural equation as a secondary constraint, as can be shown using (131).
In models belonging to case II, instead, the scalar field is dynamical, so that the vector constraints takes the form (39) and the scalar constraint reads

 H =−~Eai~Ebj2√~E(3Rijab(~ω)+2Ki[aKjb])+12√~Eϕ~Ωπ2+ +12~Ωϕ~Eai~Ebi√~E∂aϕ∂bϕ+√~EU(ϕ)≈0. (81)

If we then perform the canonical transformation (65)-(66), the difference consisting in the shift by of is now compensated taking into account relation (78), and the final expression for the constraints is in agreement with CinesiFirstOrder , both for and for ( and ). Specifically, in the latter case the primary constraint becomes

 ^π+1^ϕ^Eai^Kia≈0, (82)

which reproduces the so called conformal constraint present in CinesiFirstOrder , and proves the equivalence of first and second order approaches in the Einstein frame.

## Vi Concluding remarks

In this work, we adopted the point of view that the most natural way to introduce Ashtekar-Barbero-Immirzi variables into modified contexts consists of dealing with a first order (Palatini) formulation of the action. This perspective implies the need to add to the Lagrangian the typical Holst and Nieh-Yan contributions, as discussed for the standard Einstein-Hilbert action in Holst:1995pc . As shown in Sec. II, we have four possible combinations for the Lagrangian, corresponding to include the aforementioned terms either inside the argument of the function or simply outside. When the equations of motion are calculated, and the torsion field is expressed via the metric and the scalar field variables, we see that two different physical formulations comes out. In fact, plugging the Holst term in the function argument or adding the Nieh-Yan contribute directly into the Lagrangian, simply corresponds to a standard Palatini theory. Conversely, in the opposite case a truly scalar-tensor model is obtained, whose parameter turns out to depend on the Immirzi parameter. In these two cases we are able to define variables of the Ashtekar-Barbero-Immirzi type, in terms of which Gauss and vector constraints of LQG are recovered. The crucial difference between the two cases consists of the non-dynamical role of the scalar mode in the standard Palatini model, with respect to its non-minimal dynamical coupling with gravity in scalar-tensor models with . Particularly, in the Palatini model the scalar field can be expressed via the trace of the stress energy tensor, where the matter Lagrangian is necessarily assumed to be connection independent. This difference is also reflected into the morphology of the area operator, obtained starting from its expression in natural geometrical variables, i.e. the natural tetrad fields, and then expressed via the proper variables, suitable for loop quantization. As a result, the area spectrum, even if discrete, depends now on the scalar field properties as well. In particular, in the case of a Palatini theory, we have to deal with the intriguing feature that the geometrical structure of the space depends on the nature of the matter by which it is filled. The most interesting issue, however, comes out for scalar-tensor models, when the scalar field must also be quantized via an holonomy-like representation and the area operators eigenvalues must contains feature of the scalar mode spectrum. This feature is a consequence of the non-minimal coupling of the scalar field to gravity and suggests that geometrical properties of the quantum space are influenced by the particular considered form of the function , i.e. of the potential term . Thus, the form of the Lagrangian one adopted to describe gravity seems to directly influence the space quantum kinematics, differently from the classical scenario, where only the space metric fixes the geometry kinematics, disregarding the Lagrangian form.
We also showed that by a canonical rescaling of the scalar field, it is possible setting the Immirzi parameter equal to one in all the kinematic constraints, while it still affects the scalar constraint morphology. This is not surprising since we are able to directly link the Immirzi parameter to the scalar-tensor parameter . However, the morphology of the theory and its kinematic properties must not be affected by the parameter, since we can re-absorbe its value into the Ashtekar-Barbero-Immirzi variable definition. In turn, the theory dynamics can be instead affected by the value of , i.e. of , since the theories are not dynamical equivalent and these parameters can be constrained by experimental observations Zakharov:2006uq ; Chiba:2006jp ; Schmidt:2008qi ; Berry:2011pb .
In this sense, we can claim that the Immirzi parameter ambiguity is here completely solved, by its link to the physical scalar-tensor parameter and by the independence of the theory kinematics on its specific value. This is, in our opinion, a very relevant result, opening a new perspective for the solution of some of the LQG shortcomings into a revised and extended formulation of the gravitational interaction.
We conclude by stressing a technical issue concerning the possibility to obtain equivalent formulations, when starting from a second order approach as in CinesiSecondOrder and according the present Palatini approach. In fact, the scalar constraint appears different in the two analysis and the possibility to restore a complete equivalence implies the choice of a Einstein framework for our formulation, i.e. a conformal rescaling of the tetrad field. This technical evidence suggests a possible physical interpretation for the dynamics in the Einstein framework of a theory (on the present level it must be considered simply a mathematical tool to make the scalar dynamics minimally coupled), which deserves further investigation.

## Appendix A Derivation of the constrained Hamiltonian system

In this appendix we provide an in-depth analysis of the Hamiltonian formulation of (25), which, by means of tetrad fields and spin connections , can be rewritten, modulo surface terms, as

 S=12χ∫d4xe[ϕeμIeνJRIJμν+ϕ24SμSμ−23ϕTμTμ +2Tμ∂μϕ−b1β2Sμ∂μϕ+b2bϕβ3TμSμ−V(ϕ)], (83)

where and is the strength tensor of the spin connection. Terms in have been neglected since they would eventually turn out to yield vanishing contributions as argued further on.
The spacetime splitting is achieved via a foliation of the manifold into a family of 3-dimensional hypersurfaces defined by the parametric equations , where . The submanifold is globally defined by a time-like vector normal to the hypersurfaces, such that , and an adapted base on is then given by , satisfying the conditions . Defining the deformation vector as , it can be decomposed on the basis vectors as

 tμ=Nnμ+Nμ, (84)

where , is the lapse function and the shift vector. The following completeness relation holds

 hμν=gμν+nμnν, (85)

where is the projector on the spatial hypersurfaces and the 3-metric, related to the triads by .
Hence, assuming the time gauge conditions ,
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