Immirzi Parameter Without Immirzi Ambiguity: Conformal Loop Quantization of Scalar-Tensor Gravity

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provided by Aberdeen University Research Archive PHYSICAL REVIEW D 96, 084011 (2017) Immirzi parameter without Immirzi ambiguity: Conformal loop quantization of scalar-tensor gravity

† Olivier J. Veraguth* and Charles H.-T. Wang Department of Physics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, United Kingdom (Received 25 May 2017; published 5 October 2017) Conformal loop quantum gravity provides an approach to loop quantization through an underlying conformal structure i.e. conformally equivalent class of metrics. The property that general relativity itself has no conformal invariance is reinstated with a constrained scalar field setting the physical scale. Conformally equivalent metrics have recently been shown to be amenable to loop quantization including matter coupling. It has been suggested that conformal geometry may provide an extended symmetry to allow a reformulated Immirzi parameter necessary for loop quantization to behave like an arbitrary group parameter that requires no further fixing as its present standard form does. Here, we find that this can be naturally realized via conformal frame transformations in scalar-tensor gravity. Such a theory generally incorporates a dynamical scalar gravitational field and reduces to general relativity when the scalar field becomes a pure gauge. In particular, we introduce a conformal Einstein frame in which loop quantization is implemented. We then discuss how different Immirzi parameters under this description may be related by conformal frame transformations and yet share the same quantization having, for example, the same area gaps, modulated by the scalar gravitational field.

DOI: 10.1103/PhysRevD.96.084011

I. INTRODUCTION fundamental issue of LQG—the Immirzi ambiguity—using a constrained or dynamical gravitational scalar field. This Loop quantum gravity (LQG) offers a major nonpertur- ambiguity arises when Ashtekar’s original complex “new bative approach, through mathematically tractable and ” conceptually appealing constructions, to quantizing general variables [14,15] for an SU(2) spin-gauge connection formalism of GR were revised by Barbero to be real relativity (GR) that circumvents its nonrenormalizability “ ” using conventional quantum field theory [1–3]. Although Ashtekar-Barbero connection variables [16] as a basis LQG is not a unified theory per se, recent demonstrations for the subsequent LQG [3]. Immirzi was quick to point out “ ” of its coupling to other fields are important necessary that these variables involve a free (Barbero-)Immirzi β features as a viable theory of quantum gravity. The for- parameter and went on to show that a different choice of mulated interactions with Yang-Mills fields do not follow this parameter leads to inequivalent quantum theories of – straightforwardly despite the gauge structure inherent in the gravity having different eigenspectra of operators [17 19]. symmetries of LQG, since LQG has developed its own Indeed, the discrete volumes and areas formulated by β extensive unique representations in terms of spin networks Rovelli and Smolin [20] depend on the values of entering and spin foams. into LQG, resulting in unitarily unrelated quantizations. Physically indispensable couplings have also been While this somewhat unexpected theoretical ambiguity has established recently with fermions and most recently with persisted, to date the mainstream view seems to be taking a scalar bosons [4,5]. The coupling to scalar fields in “pragmatic” approach by fixing β [21] to phenomenologi- particular has wide implications since they go far beyond cally match the black hole entropies predicted by the merely a form of matter. Scalar fields are responsible for resulting LQG with the Bekenstein-Hawking values. generating mass through the Higgs mechanism, and induc- However, some felt such a practice unnatural as exemplified ing cosmic inflation as inflatons may serve to resolve the by Rovelli’s opinion that “the result is not entirely satisfac- problem of time [6,7] and provide models for a variety of tory,” remarking “the sense that there is something important problems in physics and cosmology [8–10]. which is not yet understood is unavoidable” [2], while some Leaving matter couplings aside, the incorporation of others have developed models with the Immirzi parameter scalars has made possible the extensions of LQG to turned into a scalar field [22–24]. Given gradually impend- beyond-Einstein fðRÞ and scalar-tensor (ST) theories of ing contact of LQG with the real world [1,2] and increased gravity [11–13]. The purpose of this paper is to address a understandingof quantumgravity effects such asdecoherence through the emission and absorption of gravitons [25–30],the *[email protected] physical ramifications of the Immirzi parameter on spacetime † Corresponding author. fluctuations towards the deep Planckian domain have been [email protected] receiving ever more interest and attention [31].

2470-0010=2017=96(8)=084011(11) 084011-1 © 2017 American Physical Society OLIVIER J. VERAGUTH and CHARLES H.-T. WANG PHYSICAL REVIEW D 96, 084011 (2017) At the outset, Immirzi suggested resolving this ambi- more gravity oriented with the Jordan frame more matter guity may require “a group larger than SU(2)” [18]. Some oriented. time ago, motivated by York’s conformal analysis of To relate more directly with the standard loop quantiza- dynamical freedoms of gravity [32] and the scaling tion of gravity, we find it useful to start from the Einstein properties of loop-quantized geometry [33], one of us frame as adopted in Ref. [8] rather than the Jordan frame as considered extending the kinematics of LQG to accom- adopted in Refs. [12,13]. Furthermore, a global conformal modate conformal symmetry [34,35] leading to the invariance of the gravitational action to be made clear development of “conformal loop quantum gravity” [36] below allows us to formulate a new loop quantization using conformally transformed variables. As GR is not scheme where different Immirzi-type parameters can be conformally invariant, its conformally extended phase conformally related. space is subject to a new conformal constraint [34,35]. In the Einstein frame, using the Einstein metric g¯αβ, with Further recent theoretical motivations and justifications scalar curvature R½g¯, and g¯ ¼jdet g¯αβj, and the scalar field for the conformally invariant LQG variables can be found ϕ¯ , the total Lagrangian (density) for a general ST gravity is in Refs. [37–39]. given by Ref. [8] to be The purpose of this paper is to report on a new L L L L conformal quantization scheme for general ST gravity ¼ ES þ SP þ M; ð1Þ coupled to matter, containing general relativity as a special case, that is amenable to LQG implementations. where By taking advantage of the freedom of changing con- 1 pffiffiffi pffiffiffi formal frames and the conformal invariance of the L ¯ ¯ − 2 ¯ ¯αβϕ¯ ϕ¯ Einstein gravitational action in the sense of ST theory, ES ¼ 2κ ½ gR½g gg ;α ;βð2Þ we show that when LQG is implemented with a conformally transformed Einstein metric, different values of is here referred to as the Einstein-scalar Lagrangian, the corresponding Immirzi parameter are related by a 2 pffiffiffi global change of conformal frame. This novel feature L ¼ − g¯Vðϕ¯ Þð3Þ suggests the resulting conformal loop quantum ST SP κ gravity will be free from the Immirzi ambiguity associated with standard LQG and many of its variants in the is a scalar potential Lagrangian for some potential function ϕ¯ literature. In particular, we discuss the prospect of Vð Þ, and quantized areas with different choices of β to have the L L Ω2 ϕ¯ ¯ ψ same discrete spectrum albeit modulated by a power of M ¼ M½ ð Þgαβ; ð4Þ the scalar field that could arise from the microscopic gravitational constant and new quantum behavior of is a matter Lagrangian for some metric coupling function ¯ geometry at the Planck scale. ΩðϕÞ and matter fields ψ. In the above, κ is a coupling 4 We use metric signature ð−; þ; þ; þÞ with a; b; … ¼ 1, constant which may be identified as κ ¼ 8πG=c if the ST α β … 0 ¯ 2, 3 and ; ; ¼ , 1, 2, 3 as spatial and spacetime theory reduces to GR with pϕ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficonst. For the Brans-Dicke coordinate indices, respectively, and i; j; … ¼ 1, 2, 3 and theory, Ω2ðϕ¯ Þ ∝ exp½−ϕ¯ = ω þ 3=2 using the Brans- … 0 I;J; ¼ , 1, 2, 3 as triad and tetrad indices, respectively. Dicke coupling constant ω. GR is also obtained in the limits ω → −3=2 and Ω → const. The combination II. CONFORMAL EINSTEIN FRAME IN ¯ ΩðϕÞg¯αβ is called the “physical metric” [42] as it defines SCALAR-TENSOR GRAVITY the spacetime geometry that matter “feels”. Among various potential physical effects of the Immirzi Below we will focus on the Einstein-scalar Lagrangian L parameter is that it may effectively shift the gravitational ES and use it to derive a set of conformal spin-gauge constant [40,41]. Combined with ongoing interest in the variables for ST gravity. Starting from the scalar-tensor ¯ possible role of conformal properties in LQG including its variables ðg¯αβ; ϕÞ in the Einstein frame, one can always implications for the Immirzi ambiguity, this consideration change to an alternative conformal frame with variables ¯ ¯ leads to the following framework for the loop quantization ðgαβ; ϕÞ by reparametrizing the scalar field ϕ ¼ ϕðϕÞ of ST gravity that is invariant under changes of conformal and conformally transforming the metric tensor g¯αβ ¼ frames. 2 F ðϕÞgαβ for some function FðϕÞ. Analogous to previous Two types of conformal frame for ST gravity have works on conformal loop quantum gravity [34–37] with a received particular attention in the literature [42]: (a) the ϕ ϕ Einstein frame in which the gravitational action has a constrained scalar field , we choose Fð Þ so that gαβ is leading term identical to the Hilbert action and (b) the given by Jordan frame in which the matter action is unaffected by the ¯ ϕ2 scalar field. In other words, the Einstein frame is somewhat gαβ ¼ gαβ: ð5Þ

084011-2 IMMIRZI PARAMETER WITHOUT IMMIRZI AMBIGUITY: … PHYSICAL REVIEW D 96, 084011 (2017) In addition, we adopt the reparametrization ϕ2 pffiffiffi 2 pffiffiffi L ab − 2 ϕϕ ES ¼ 2κ hNðKabK K þ R½hÞ þ κ h ;0K ¯ ϕ ¼ ln ϕ ð6Þ 2Na pffiffiffi 2N pffiffiffi 2 pffiffiffi − hϕϕ K − hðϕϕ Þ;a − h κ ;a κ ;a κN of the scalar field, which is dynamical, except for GR 2 a 2 ab a b × ϕ − 2N ϕ 0ϕ − N h − N N ϕ ϕ : reduction, so that the Einstein-scalar Lagrangian (2) ½ ;0 ; ;a ð Þ ;a ;b becomes ð13Þ 1 ffiffiffi ffiffiffi 4p 2 p αβ This gives rise to the canonical momenta for the metric L ¼ ½ϕ gR½ϕ gαβ − 2 gg ϕ αϕ β: ð7Þ R R ES 2κ ; ; ab δ L 3 δ π δ L 3 δϕ p ¼ ESd x= hab;0 and scalar ϕ ¼ ESd x= ;0 as follows: We will refer to the conformal frame with variables ¯ ðgαβ; ϕÞ obtained from the Einstein frame variables ðg¯αβ; ϕÞ 1 pffiffiffi ab ab c p ¼ − hh ϕðϕ 0 − ϕ N Þ through Eqs. (5) and (6) as the conformal Einstein frame. κN ; ;c Its usefulness in addressing the Immirzi ambiguity follows ϕ2 pffiffiffi − ab − ab from the form of Lagrangian (7) being invariant under the 2κ hðK h KÞ; ð14Þ following global conformal transformations: where h ¼ det h , and ϕ → Λ−1ϕ → Λ2 ab ;gαβ gαβ ð8Þ pffiffiffi 2 4 c Λ πϕ ¼ h ϕK − ðϕ 0 − ϕ N Þ ; ð15Þ for any positive constant . κ κN ; ;c As will become clear later, unlike standard approaches, – our idea is to introduce the Ashtekar-Barbero type varia- yielding bles in the conformal Einstein frame and show that any choice of the corresponding Immirzi parameter can be κN N c ffiffiffi ϕ;0 ¼ ϕ;cN − p πϕ þ ϕK: ð16Þ mapped to a different value, such as unity, in an alternative 4 h 2 conformal Einstein frame transformed using Eq. (8). It also follows from Eq. (12) that III. CANONICAL ANALYSIS IN THE 4κ 1 CONFORMAL EINSTEIN FRAME Nffiffiffi hab;0 ¼ p pab − habϕπϕ þ Na;b þ Nb;a: ð17Þ ϕ2 h 4 Prior to canonical quantization of ST gravity in a conformal Einstein frame, the preceding Lagrangian for- By using Eqs. (12), (13), (16), and (17), up to a total malism will in this section be transcribed to the corre- divergence we can derive sponding Arnowitt-Deser-Misner canonical formalism. ¯ First, using Eq. (5), we can map the spatial metric hab, H ab π ϕ − L ES ¼ p hab;0 þ ϕ ;0 ES lapse function N¯ , and shift vector N¯ a associated with the a a ¼ NC⊥ þ N C ð18Þ Einstein metric g¯αβ to their counterparts hab, N, and N a according to with the Hamiltonian and diffeomorphism constraints as ¯ 2 follows: hab ¼ ϕ hab; ð9Þ 2κ ϕ2 pffiffiffi 2 pffiffiffi ¯ ϕ ffiffiffi −2 ab ab N ¼ N; ð10Þ C⊥ ¼ p ϕ pabp − hR½hþ hh ϕϕ;a;b h 2κ κ ¯ a a κ 2 κ −1 N ¼ N : ð11Þ þ pffiffiffi πϕ − pffiffiffi ϕ πϕp; ð19Þ 4 h h We then evaluate the extrinsic curvature in the conformal C −2 bc ϕ π Einstein frame using the above to get a ¼ habp ;c þ ;a ϕ ð20Þ

ab 1 in terms of p ¼ habp . K ¼ ð−h 0 þ N ; þ N ; Þ: ð12Þ ab 2N ab; a b b a From the totally constrained nature of the Hamiltonian (18), C⊥ and Ca are required to vanish weakly: After some calculations involving conformal change rela- C⊥ ≈ 0; Ca ≈ 0. The consistency of this is ensured by the tions [43], the above then yields the Einstein-scalar Dirac algebra of C⊥ and Ca under their Poisson brackets as Lagrangian up to a total divergence of the form established in Appendix A, which is satisfied also by the

084011-3 OLIVIER J. VERAGUTH and CHARLES H.-T. WANG PHYSICAL REVIEW D 96, 084011 (2017) pffiffiffi κ Hamiltonian and diffeomorphism constraints for canoni- −2 i i ffiffiffi −2 Kab ¼ −κ hϕ K E þ p ϕ Chab ð27Þ cal GR. ða bÞ 2 h The canonical generator of the local conformal trans- i a formation that preserves the Einstein metric g¯αβ is given by in terms of arbitrary Ka and Ei . Then by using Eqs. (23) and (27) we can calculate that C ¼ ϕπϕ − 2p: ð21Þ h pab ¼ ðhadhbc þ hachbd − 2habhcdÞKi Ei : ð28Þ For the GR case, we require in addition the weakly 4 c d vanishing C ≈ 0 as well, with the consistent closure condition under Poisson brackets previously established in By contracting the above with hab,wehave Refs. [34,35]. In general, the uniformed smeared C gen- L p ¼ −hhabKi Ei ¼ −Ki Ea: ð29Þ erates the discussed global conformal invariance of ES a b a i using Eq. (8) as will be discussed further below. Then using (28) and (24) we can evaluate that

IV. CONFORMAL ASHTEKAR-BARBERO ab j b i i a b p h 0 ¼ −K E − K E E E : ð30Þ VARIABLES AND CONFORMAL ab; b j;0 ½a b j j;0 IMMIRZI PARAMETER i i This implies an additional constraint K½aEb or equivalently Having obtained the Hamiltonian formalism of ST gravity in the conformal Einstein frame with variables C ϵ a −ϵ l l a b k ¼ kijKa½iEj ¼ kijK½aEbEi Ej ð31Þ ðgαβ; ϕÞ, we can now proceed to finding their counterpart for the standard Ashtekar-Barbero variables and explore the called the spin constraint [34]. It is equivalent to the property of the resulting Immirzi parameter in what rotation constraint defined in Ref. [3] and generates local follows. pffiffiffi a a a SU(2) transformations [44]. From Refs. [13,34], this In terms of the triad ei and its densitization Ei ¼ hei constraint is first class, forming a closed Poisson bracket associated with hab, we have the standard relation algebra with the Hamiltonian, diffeomorphism, and conformal constraints. δ i j δ i j hab ¼ ijeaeb ¼ h ijEaEb: ð22Þ The following variables then form canonical pairs:

Substituting (16) into (14) we obtain κ i κ−1 a ϕ π ð Ka; Ei Þ and ð ; ϕÞ: ð32Þ 1 ϕ2 pffiffiffi ab − C ab − ab − ab p ¼ 2 h 2κ hðK h KÞ: ð23Þ Using these canonical variables and (29), we see that the conformal constraint for GR (21) becomes It then follows from Eq. (22) that i a C ¼ ϕπϕ þ 2KaE : ð33Þ 1 i c d h 0 ¼ ðh h − h h − h h ÞE E ð24Þ ab; h ab cd ac bd ad bc i i;0 For any positive constant β, a trivial canonical transformation from (32) yields and hence βκ i β−1κ−1 a ϕ π ab a i − a i ð Ka; Ei Þ and ð ; ϕÞ: ð34Þ p hab;0 ¼ Ei Ka;0 ðEi KaÞ;0; Since Γi commute with Ea under Poisson brackets, we can where a i further perform a canonical transformation from (32) to 1 1 yield Ki ¼ − pffiffiffi ϕ2K Eb þ Ch Eb: ð25Þ a κ ab i 2h ab i h i Γi κ i κ−1 a ϕ π ðAa ¼ a þ Ka; Ei Þ and ð ; ϕÞð35Þ i Contracting (25) with Ec we get and alternatively from (34) to yield pffiffiffi κ K −κ hϕ−2Ki Ei pffiffiffi ϕ−2Ch : 26 ab ¼ a b þ ab ð Þ 0i Γi βκ i β−1κ−1 a ϕ π 2 h ðA a ¼ a þ Ka; Ei Þ and ð ; ϕÞ: ð36Þ

i i i i Using Kab ¼ Kba we see from Eq. (25) that KaEb ¼ KbEa. We refer to canonical variables (35) and more generally i a β However, we will from now on treat Ka and Ei as (36) as conformal Ashtekar-Barbero variables and independent variables without imposing this condition. involved as the conformal Immirzi parameter. For any β, i Instead we define the variable Aa has the same construction as the SU(2) spin

084011-4 IMMIRZI PARAMETER WITHOUT IMMIRZI AMBIGUITY: … PHYSICAL REVIEW D 96, 084011 (2017) a connection with densitized triad Ei as the conjugate Therefore the two sets of canonical variables Eqs. (35) momentum. As they have the same structure as the standard and (36) are equivalent. Ashtekar-Barbero variables of LQG, they are amenable to At the quantum level the global conformal transforma- loop quantization based on a spin-network representation tion of a quantum state Ψ½ϕ;A is generated by the H – with a Hilbert space here denoted by SN [1 3]. uniformly smeared conformal constraint To quantize the scalar-spin variables (36) as a whole, we Z follow the kinematic quantization recently developed in C ¼ Cd3x ð43Þ Ref. [5], where a diffeomorphism invariant representation H using a Hilbert space SF for the scalar field in which the field operator ϕ is diagonal. This leads to the total Hilbert as follows: space iϵC 1 − Ψ½ϕ;Aa¼Ψ½ϕ þ ϵϕ;Aa þ 2ϵκKa; ð44Þ H H ⊗ H ℏ i i i ¼ SF SN ð37Þ ϵ C as with the treatment of LQG coupled to a scalar field in for an infinitesimal , where given by Eq. (33) can be ’ i a Ref. [5]. The quantum states are therefore expressed as implemented through Thiemann s quantization of KaEi ’ superpositions of terms [3,45] and Lewandowski and Sahlmann s quantization of πϕ → −iℏδ=δϕ terms [5]. Ψ½ϕ;A¼Ψ½ϕ ⊗ Ψ½Að38Þ The quantum implementation of the invariance under global conformal transformation (41) causes no ordering β in terms of the cylindrical functions of A: issues, as it contributes only to commuting powers of as a c number in e.g. the Hamiltonian constraint C⊥ leaving an Ψ ψ … overall transformation according to Eq. (B15) as ½A¼ ðhe1 ½A; ;hen ½AÞ ð39Þ C0 β1=2C … ⊥ ¼ ⊥: ð45Þ involving holonomies he1 ½A; ;hen ½A over edges … e1; ;en. C C0 i a Therefore Dirac quantization using ⊥ or ⊥ yields the In terms of the variables ðAa;Ei Þ, the spin constraint (31) same physics irrespective of the choice of β. Specifically, can be rewritten as the familiar Gauss constraint we have invariant discrete areas and volumes using the Einstein frame densitized triad under the global conformal C a a ϵ i a k ¼ DaEk ¼ Ek;a þ kijAaEj ð40Þ transformation that generates rotations. Since the scalar states Ψ½ϕ are ¯ a ϕ2 a ϕ02 0a Ei ¼ Ei ¼ E i ð46Þ SU(2) invariant and by construction Ψ½A satisfy the quantum Gauss law, the spectra of invariant operators such by using Eq. (41), which is clearly independent of the value as areas on Ψ½ϕ;A are unchanged under actions from of conformal Immirzi parameter β. Eq. (40). As with Eq. (31), from Refs. [13,34], the above Gauss constraint is also first class, being closed under V. CONCLUSION AND OUTLOOK Poisson bracket algebra with the Hamiltonian, diffeomorphism, and conformal constraints [13,34]. In this paper we address loop quantization in a wider The new conformal properties of the conformal context of ST gravity. Our main motivation has not been Ashtekar-Barbero variables mean that the standard just to extend LQG beyond GR, but to seize on the freedom Immirzi ambiguity no longer has a direct analogy as will of conformal frame transformations available in ST gravity be explained below. By using that may help rendering the ambiguous Immirzi parameter in current LQG into a more natural conformal gauge 0 1=2 0 −1=2 parameter having no preferred values. For this purpose ϕ ¼ β ϕ; πϕ ¼ β πϕ ð41Þ we have found it useful to start from the Einstein frame of 0a β−1 a 0i β i E i ¼ Ei ;Ka ¼ Ka ST gravity followed by certain conformal frame transformation into the conformal Einstein frame in Sec. II, C C C and the structures of ⊥ and a, and k (see Appendix B for since the resulting scalar-gravitational action specified by details), the canonical variables (36) are equivalent to the Einstein-scalar Lagrangian (2) used to define canonical variables of gravity has a global conformal symmetry (8). 0i Γi κ 0i κ−1 0a ϕ0 π0 ðA a ¼ a þ K a; E i Þ and ð ; ϕÞ: ð42Þ This indicates that loop gravitational variables built in such a conformal frame may inherit a global conformal sym- With a global conformal transformation, the canonical metry under which the corresponding Immirzi parameter is variables (42) are in turn equivalent to Eq. (35). the gauge parameter.

084011-5 OLIVIER J. VERAGUTH and CHARLES H.-T. WANG PHYSICAL REVIEW D 96, 084011 (2017) We have therefore been led by the above observation Finally, although a fully quantum description of the to the construction of the Hamiltonian formalism of ST conformal LQG formalism without a free Immirzi param- gravity in the conformal Einstein frame in Sec. III as a eter proposed in this work is yet to be completed, one may prerequisite for canonical quantization. Like canonical GR, already wonder how the well-accepted Bekenstein- the resulting Hamiltonian ST system is also totally con- Hawking entropy of black holes could be recovered. strained, having a more involved set of the Hamiltonian Admittedly, here we see little immediate phenomenological constraint (19) and diffeomorphism constraints (20).Furthe- analogy of matching the Immirzi parameter as done in rmore, these constraints satisfy the same Dirac algebra standard LQG. Nonetheless, given our revised spacetime under their Poisson brackets with respect to the conformal dynamics with an extended conformal gauge structure and Einstein frame canonical variables as explicitly established a coupled scalar field, it seems reasonable to expect a in Appendix A. We then build on this canonical structure a different kinematical and perhaps more dynamical new set of conformal Ashtekar-Barbero variables with a approach to the quantum black hole entropy problem. corresponding conformal version of the Immirzi parameter β This would involve reformulating the ensembles of micro- in Sec. IV. Remarkably, this conformal Immirzi parameter states of quantum geometry that incorporate the effects of does indeed represent the global conformal gauge parameter the additional scalar field and redefine the corresponding whose value should not appear in physical observables such state counting. It might also be useful to identify a quantum as area operators after quantization. These main findings also dissipator to allow relaxation to thermal equilibrium, apply to GR as a special case of ST gravity where the ideally to achieve the corresponding Hawking temperature conformal constraint equation C ≈ 0 must be satisfied for black holes or Unruh temperature for accelerating making the scalar gravitational field a pure gauge. frames. It would then be interesting to calculate the Additionally, we remark that the above canonical treat- resulting entropies. Last but not least, given the math- ment may also be approached using a conformal version of ematical similarity between the conformal constraint (33) the Holst action [46], relevant for the spin foam extension and Thiemann’s complexifier [3], our suggested reformu- to this work. The new starting point would be to consider lation may allow to implement recently proposed new the Hilbert part of Lagrangian (2) in its Palatini form mechanism of obtaining the Bekenstein-Hawking entropy 1 formula in LQG from an analytic continuation to β ¼ i L ¯ ¯α ¯β ¯ IJ – P ¼ 2κ ee Ie JFαβ ð47Þ [47 50] in a more natural way [51]. The progress of the above continued work is deferred for future publications. α in terms of the tetrad e¯ I with determinant e¯ and curvature ¯ IJ 2 Fαβ associated with g¯αβ ¼ ϕ gαβ as in Eq. (5). A plausible ACKNOWLEDGMENTS candidate for the conformally modified Palatini-Holst Lagrangian of Eq. (47) would then be C. W. wishes to thank G. Immirzi and C. Rovelli for 1 early discussions and brief correspondence, respectively, L ¯ ¯α ¯β ¯ IJ and appreciates the EPSRC GG-Top Project and the PH ¼ 2κ ee Ie JFαβ Cruickshank Trust for financial support. O. V. is grateful 1 to the Aberdeen University College of Physical Sciences − α β ϵIJ KL 2 ee Ie J KLFαβ ð48Þ for a research studentship.

α in terms of the tetrad e I with determinant e and curvature IJ θ2 Fαβ associated with gαβ ¼ gαβ using another (multi- APPENDIX A: DIRAC ALGEBRA OF plier) scalar field θ. CONSTRAINTS IN THE CONFORMAL Note that in Eq. (48), the extra Holst term is associated EINSTEIN FRAME g with a conformally transformed metric ab, nonidentical to In this Appendix, we show that the Dirac algebra for the g¯ as used in Eq. (47) and so this action is different from ab Hamiltonian constraint C⊥ and diffeomorphism constraints that in Ref. [37], which is recovered from Eq. (48) by Ca is satisfied using the Poisson bracket f·; ·g with respect equating ϕ and θ. On the other hand, the standard Palatini- ab to the conformal Einstein frame variables ðh ;p ; ϕ; πϕÞ, Holst action is recovered from Eq. (48) through conformal ab −1=2 where both metric and scalar fields are dynamical. gauge fixing with constants ϕ ¼ 1 and θ ¼ β yielding δ β We use the Dirac function given by Refs. [52,53] as a the usual Immirzi parameter . Furthermore, Lagrangian bidensity of weight zero in the first and weight one in the (48) can be shown to reduce to the Einstein-Hilbert action second argument, having the properties [54,55] for GR after varying the scalars ϕ and θ and the connection α ϕ θ 1-forms of the tetrad e I. However, leaving and free 0 0 δ 0 ðx; x Þ¼−δ ðx; x Þ; ðA1Þ retains the freedom of conformal frame transformation in ;a ;a the quantum dynamics to be explored in the context of the 0 δ 0 δ 0 δ 0 Immirzi ambiguity. fðx Þ ;aðx; x Þ¼fðxÞ ;aðx; x Þþf;aðxÞ ðx; x Þ: ðA2Þ

084011-6 IMMIRZI PARAMETER WITHOUT IMMIRZI AMBIGUITY: … PHYSICAL REVIEW D 96, 084011 (2017) 0 1. Poisson bracket fC⊥ðxÞ;C⊥ðx Þg where ξðxÞ is an arbitrary smearing function. For the Poisson bracket between two Hamiltonian Applying the functional derivatives to the Hamiltonian constraints given by Eq. (19), it is useful to consider the constraints, by the antisymmetrical property of the Poisson smeared version of the Hamiltonian constraint bracket, all terms not containing any derivative of the ξ μ Z smearing function and cancel out, and therefore what remains, after integration by parts and inserting the diffeo- Σ ξ ξ C 3 ½ ¼ ðxÞ ⊥ðxÞd x; ðA3Þ morphism constraint, is

Z 00 00 ab 00 00 00 00 ab 00 00 fΣ½ξ; Σ½μg ¼ f−ξ;bðx Þμðx Þh ðx ÞCaðx Þþξðx Þμ;bðx Þh ðx ÞCaðx Þ

− ξ 00 μ 00 ϕ−1 00 ϕ 00 8 ab 00 − 4 ab 00 00 ab 00 ϕ 00 π 00 ;ðaðx Þ ðx Þ ðx Þ ;bÞðx Þ½ p ðx Þ h ðx Þpðx Þþh ðx Þ ðx Þ ϕðx Þ ξ 00 μ 00 ϕ−1 00 ϕ 00 8 ab 00 − 4 ab 00 00 ab 00 ϕ 00 π 00 3 00 þ ðx Þ ;ðaðx Þ ðx Þ ;bÞðx Þ½ p ðx Þ h ðx Þpðx Þþh ðx Þ ðx Þ ϕðx Þgd x : ðA4Þ

The Poisson bracket between the nonsmeared constraints is δF recovered by take the double functional derivative with δμ x0 δξ x ðZ Þ ð Þ respect to both smearing functions: ϕ−1 ϕ ¼ f ðxÞ ;ðaðxÞ

δ δ ab ab ab 0 × 8p x − 2h x p x h x ϕ x πϕ x fC⊥ðxÞ; C⊥ðx Þg ¼ fΣ½ξ; Σ½μg: ðA5Þ ½ ð Þ ð Þ ð Þþ ð Þð ð Þ ð Þ δξ δμ 0 ðxÞ ðx Þ − 2 δ3 0 3 pðxÞÞ ðx; x Þg;bÞd x: ðA6Þ

Let us consider the second term of Eq. (A4): Following the same procedure on the last term of (A4) but taking the functional derivatives in the opposite order yields Z Z ξ μ − ξ 00 μ 00 ϕ−1 00 ϕ 00 δ 0 F½ ; ¼ ;ðaðx Þ ðx Þ ðx Þ ;bÞðx Þ F −1 0 0 ¼ fϕ ðx Þϕ; ðx Þ δξðxÞδμðx0Þ ða × ½8pabðx00Þ − 4habðx00Þpðx00Þ × ½8pabðx0Þ − 2habðx0Þpðx0Þ ab 00 ϕ 00 π 00 3 þ h ðx Þ ðx Þ ϕðx Þd x: ab 0 0 0 þ h ðx Þðϕðx Þπϕðx Þ 0 3 0 3 − 2pðx ÞÞδ ðx ;xÞg; 0 d x: ðA7Þ Integrating by parts with respect to the first smearing b Þ function and removing the surface term, we get By Schwarz’s theorem, both derivatives can be inter- Z changed and therefore Eqs. (A6) and (A7) cancel each other ξ μ ξ μ ϕ−1 ϕ out such that the Poisson bracket (A4) reads F½ ; ¼ ðxÞf ðxÞ ðxÞ ;ðaðxÞ δ δ ab ab 0 × ½8p ðxÞ − 4h ðxÞpðxÞ fC⊥ðxÞ; C⊥ðx Þg ¼ δξðxÞ δμðx0Þ ab ϕ π 3 Z þ h ðxÞ ðxÞ ϕðxÞg;bÞd x: 00 00 ab 00 00 × f−ξ;bðx Þμðx Þh ðx ÞCaðx Þ

Taking the variational derivative with respect to ξ, the 00 00 ab 00 00 3 00 þ ξðx Þμ;bðx Þh ðx ÞCaðx Þgd x : expression becomes ðA8Þ Z δ F −1 ab ab Finally, the functional derivatives are applied on the last ¼ fμðxÞϕ ðxÞϕ; aðxÞ½8p ðxÞ − 2h ðxÞpðxÞ δξðxÞ ð two terms to obtain the looked for Poisson bracket between ab ϕ π − 2 3 two Hamiltonian constraints: þ h ðxÞð ðxÞ ϕðxÞ pðxÞÞg;bÞd x: C C 0 ab C δ3 0 f ⊥ðxÞ; ⊥ðx Þg ¼ h ðxÞ aðxÞ ;bðx ;xÞ The functional derivative with respect to the second ab 0 0 3 0 − h x C x δ 0 x; x : A9 smearing function at a point x0 leads to ð Þ að Þ ;b ð Þ ð Þ

084011-7 OLIVIER J. VERAGUTH and CHARLES H.-T. WANG PHYSICAL REVIEW D 96, 084011 (2017)

0 2. Poisson bracket fCaðxÞ;Cbðx Þg Between two diffeomorphism constraints given by Eq. (20), the Poisson bracket is also solved using smeared constraints of the form Z a c 3 Σ½ξ ¼ ξ ðxÞCcðxÞd x: ðA10Þ

Here ξaðxÞ is a function for the smeared diffeomorphism constraint. Calculating the functional derivatives of the above smeared constraints leads to the expression Z Σ ξa Σ μb −4ξða 00 μb 00 00 cÞd 00 − 2ξða 00 μb 00 00 cÞd 00 f ½ ; ½ g ¼ f ðx Þ ;aðx Þhbcðx Þp;d ðx Þ ðx Þ ðx Þhac;bðx Þp;d ðx Þ

4ξa 00 μðb 00 00 cÞd 00 2ξa 00 μðb 00 00 cÞd 00 − ξa 00 μb 00 ϕ 00 π 00 þ ;bðx Þ ðx Þhacðx Þp;d ðx Þþ ðx Þ ðx Þhbc;aðx Þp;d ðx Þ ;aðx Þ ðx Þ ;bðx Þ ϕðx Þ − ξa 00 μb 00 ϕ 00 π 00 ξa 00 μb 00 ϕ 00 π 00 ξa 00 μb 00 ϕ 00 π 00 3 00 ðx Þ ðx Þ ;bðx Þ ϕ;aðx Þþ ðx Þ ;bðx Þ ;aðx Þ ϕðx Þþ ðx Þ ðx Þ ;aðx Þ ϕ;bðx Þgd x to be used below. To recover the sought after Poisson We see that the Poisson bracket is symmetric under the brackets, the functional derivatives with respect to both exchange of the two indices; therefore, all the antisym- smearing functions ξa and μb has to be taken: metric terms are equal to zero and so we have

0 3 0 0 3 0 δ δ fC ðxÞ; C ðx Þg ¼ C ðxÞδ ðx ;xÞ − C ðx Þδ 0 ðx; x Þ: fC ðxÞ; C ðx0Þg ¼ fΣ½ξc; Σ½μcg: a b b ;a a ;b a b δξaðxÞ δμbðx0Þ Finally, the Poisson bracket between two diffeomor- Using the definition of the diffeomorphism constraint in phism constraints reads Eq. (20), the calculation reduces to C C 0 C 0 δ3 0 C δ3 0 f aðxÞ; bðx Þg ¼ aðx Þ ;b0 ðx; x Þþ bðxÞ ;aðx ;xÞ: C C 0 C δ3 0 − C 0 δ3 0 f aðxÞ; bðx Þg ¼ bðxÞ ;aðx ;xÞ aðx Þ ;b0 ðx; x Þ ðA14Þ − 4 dc δ3 0 hd½a;bðxÞp;c ðxÞ ðx ;xÞ 0 2ϕ 0 π 0 δ3 0 3. Poisson bracket fCaðxÞ;C⊥ðx Þg þ ;½aðx Þ ϕðx Þ ;b0ðx; x Þ 3 0 To derive the Hamiltonian-diffeomorphism Poisson þ 2ϕ; ðxÞπϕðxÞδ ðx ;xÞ ½a ;b bracket, we use the fact that the Poisson bracket 2ϕ π δ3 0 þ ;½aðxÞ ϕ;bðxÞ ðx; x Þ: ðA11Þ between the diffeomorphism constraint and any weight- one element f, as e.g. the Hamiltonian constraint, gives the same result: Following the same procedure but exchanging the indices a and b yields to the Poisson bracket 0 3 0 fCcðxÞ;fðx Þg ¼ fðxÞδcðx; x Þ: ðA15Þ

0 3 0 0 3 0 fC ðxÞ; C ðx Þg ¼ C ðxÞδ ðx ;xÞ − C ðx Þδ 0 ðx; x Þ b a a ;b b ;a Using its linearity, we can obtain the Poisson bracket by 4 dc δ3 0 þ hd½a;bðxÞp;c ðxÞ ðx ;xÞ calculating it for every term of the Hamiltonian constraint − 2ϕ 0 π 0 δ3 0 separately. Considering the first term of Eq. (19), ;½aðx Þ ϕðx Þ ;b0ðx; x Þ 3 0 1 − 2ϕ; aðxÞπϕðxÞδ; ðx ;xÞ ffiffiffi −2 ab ½ b S ¼ p ϕ pabp ; ðA16Þ h − 2ϕ π δ3 0 ;½aðxÞ ϕ;bðxÞ ðx; x Þ: ðA12Þ and using the properties of the Dirac δ function (A1) and Taking half of the sum of Eqs. (A11) and (A12), we obtain (A2), the first Poisson bracket can be calculated. It is derived by separating it into simpler terms using the C C 0 C δ3 0 − C 0 δ3 0 property f ðaðxÞ; bÞðx Þg ¼ ðaðxÞ ;bÞðx ;xÞ ðaðx Þ ;b0Þðx; x Þ: ðA13Þ ff; ghg¼ff; ggh þ gff; hg: ðA17Þ

084011-8 IMMIRZI PARAMETER WITHOUT IMMIRZI AMBIGUITY: … PHYSICAL REVIEW D 96, 084011 (2017) We therefore obtain using the variation of h−1=2 that 2κ C 0 C pffiffiffiffiffiffiffiffiffiffi ϕ−2 0 0 0 ab 0 cd 0 f eðxÞ;Sðx Þg ¼ eðxÞ; ðx Þhaðcðx ÞhdÞbðx Þp ðx Þp ðx Þ hðx0Þ 2κ pffiffiffiffiffiffiffiffiffiffi C ϕ−2 0 0 0 ab 0 cd 0 þ f eðxÞ; ðx Þghaðcðx ÞhdÞbðx Þp ðx Þp ðx Þ hðx0Þ 2κ pffiffiffiffiffiffiffiffiffiffi ϕ−2 0 C 0 0 ab 0 cd 0 þ ðx Þf eðxÞ;haðcðx ÞghdÞbðx Þp ðx Þp ðx Þ hðx0Þ 2κ 4κ pffiffiffiffiffiffiffiffiffiffi ϕ−2 0 0 C 0 ab 0 cd 0 pffiffiffiffiffiffiffiffiffiffi ϕ−2 0 0 C ab 0 þ ðx Þhaðcðx Þf eðxÞ;hdÞbðx Þgp ðx Þp ðx Þþ ðx Þpabðx Þf eðxÞ;p ðx Þg: hðx0Þ hðx0Þ

This expression after some calculations reduces to the transformation given by Eq. (8), we have the following Poisson bracket relations:

0 0 → Λ2 fCaðxÞ;Sðx Þg ¼ SðxÞδ;aðx; x Þ: ðA18Þ hab hab; ðB1Þ

−2 This shows that the first term follows the weight-one hab → Λ hab; ðB2Þ element rule (A15). By extending linearly to the other ffiffiffi ffiffiffi p 3p terms, we extrapolate to the global Hamiltonian constraint h → Λ h; ðB3Þ and therefore have N → ΛN; ðB4Þ 0 0 fCaðxÞ; C⊥ðx Þg ¼ C⊥ðxÞδ;aðx; x Þ: ðA19Þ Na → Na; ðB5Þ Therefore, combining all results in this Appendix, we have the following Poisson brackets Eqs. (A9), (A14), N → Λ2N : ðB6Þ and (A19): a a Using the above and Eq. (12) we have C C 0 ab C δ3 0 f ⊥ðxÞ; ⊥ðx Þg ¼ h ðxÞ aðxÞ ;bðx ;xÞ → Λ − ab 0 C 0 δ3 0 Kab Kab; ðB7Þ h ðx Þ aðx Þ ;b0 ðx; x Þ; ðA20Þ ab → Λ−3 ab C C 0 C 0 δ3 0 K K ; ðB8Þ f aðxÞ; bðx Þg ¼ aðx Þ ;b0 ðx; x Þ −1 3 0 → Λ þ CbðxÞδ;aðx ;xÞ; ðA21Þ K K: ðB9Þ

0 3 0 Furthermore, from Eqs. (14) and (15) we have fCeðxÞ; C⊥ðx Þg ¼ C⊥ðxÞδ;eðx; x Þ; ðA22Þ ab → Λ−2 ab which all consistently vanish weakly if the constraints p p ; ðB10Þ C C ⊥ðxÞ and aðxÞ vanish weakly. The above relations form → Λ2 the same Dirac algebra of constraints as with the metric pab pab; ðB11Þ tensor-only theory of GR [56]. p → p; ðB12Þ Furthermore, since the conformal Ashtekar-Barbero variables are constructed from the conformal Einstein πϕ → Λπϕ; ðB13Þ frame ST variables using a set of canonical transformations, the argument about the previous Poisson brackets is also R½h → Λ−2R½h: ðB14Þ valid [57] in these new variables in Eq. (35) or (36). From Eqs. (19)–(21) we see that APPENDIX B: GLOBAL CONFORMAL −1 TRANSFORMATION RELATIONS C⊥ → Λ C⊥; ðB15Þ Here we summarize how various physical quantities used C → C in this work undergo changes with a global conformal a a; ðB16Þ transformation used in the main text of this work. It follows C → C directly from Eq. (5) that, under a global conformal ; ðB17Þ

084011-9 OLIVIER J. VERAGUTH and CHARLES H.-T. WANG PHYSICAL REVIEW D 96, 084011 (2017) and hence i −2 i Ea → Λ Ea; ðB22Þ L → L ES ES; ðB18Þ Ea → Λ2Ea; ðB23Þ H → H i i ES ES: ðB19Þ i → Λ−2 i In terms of the triad variables, we find that Ka Ka: ðB24Þ

i i ea → Λea; ðB20Þ Finally, from Eqs. (31), (B23), and (B24) we have

a → Λ−1 a C → C ei ei ; ðB21Þ k k: ðB25Þ

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