Extended loop quantum gravity L Fatibene, M Ferraris, M Francaviglia To cite this version: L Fatibene, M Ferraris, M Francaviglia. Extended loop quantum gravity. Classical and Quan- tum Gravity, IOP Publishing, 2010, 27 (18), pp.185016. 10.1088/0264-9381/27/18/185016. hal- 00625165 HAL Id: hal-00625165 https://hal.archives-ouvertes.fr/hal-00625165 Submitted on 21 Sep 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Confidential: not for distribution. Submitted to IOP Publishing for peer review 1 July 2010 Extended Loop Quantum Gravity∗ by L.Fatibenea,b,M.Ferrarisa,M.Francavigliaa,b,c a Department of Mathematics, University of Torino (Italy) b INFN - Iniziativa Specifica Na12 c LCS, University of Calabria (Italy) Abstract: We discuss constraint structure of extended theories of gravitation (also known as f(R)theories) in the vacuum selfdual formulation introduced in [1]. 1. Introduction We have recently investigated a formulation of f(R)theories(inametric-affineframework) based on non-linear actions similar to the Holst Lagrangian; see [1].Theseactionsareinfact written in terms of the scalar curvature βR of the Barbero-Immirzi connection with parameter β (see [2], [3])andaredynamicallyequivalenttothecorresponding“classical”f(R)theory.For the linear case f(βR)=βR one obtains the standard Holst action. Hence these new actions are to be understood as Barbero-Immirzi formulations of the corresponding classical f(R)theory. This could be interesting for at least two reasons: from the point of view of LQG this new formulation provides a family of models which are classically well–understood and investigated in detail (see [4], [5]). There are many classical effects known in f(R)theoriesthatshouldbe traced in their quantum genesis. The minisuperspace of these models is quite well–understood and should be studied in loop quantum cosmology (LQC) formulation (see [6]), to contribute to a better understanding of the classical limit of LQG models. Moreover, as in all metric- affine models, matter has a non-trivial feedback on the gravitational field which would be also interesting to trace in its quantum origin. It is often said that matter in LQG simply adds new labels to spin networks, while in these models one could expect a more complicated mechanism that would be certainly interesting to be discussed in detail. Finally, there are a number of equivalences, e.g. with scalar tensor models (see [7]), that again would be interesting to be discussed in detail at quantum level. Let us stress that these equivalences are known to hold at the classical level and, as usual, one should investigate whether they still hold at the full quantum level or just emerge classically. From the classical viewpoint we shall here provide a route to define a quantization `alaloop of f(R)theories.Ofcourseclassicaleffectsoftheseextendedtheoriesofgravitationhavebeen extensively investigated. It is therefore interesting to investigate also their quantum effects. For example it would be interesting to see whether the removal of singularities that has been shown to hold in standard loop quantization of GR is preserved generically in these extended gravitational models. For the sake of simplicity we shall here restrict our attention to the Euclidean signature and to the selfdual formulation (which in the Euclidean sector is in fact a special case of the ∗ This paper is published despite the effects of the Italian law 133/08 (http://groups.google.it/group/scienceaction). This law drastically reduces public funds to public Italian universities, which is particularly dangerous for free scientific research, and it will prevent young researchers from getting a position, either temporary or tenured, in Italy. The authors are protesting against this law to obtain its cancellation. 1 Barbero-Immirzi formulation) and show that one can apply LQG methods (see [8])alsotothe quantization of these theories. In vacuum we shall obtain something similar to Einstein gravity with a cosmological constant. This is very well expected on the basis of a classical equivalence (see [9]); however, let us stress that our result seems to establish a stronger equivalence at the quantum level and not only at the classical level. Moreover, let us stress that the classical equivalence holds only in vacuum, while the equiv- alence is broken when generic matter is considered and the extended models are equivalent to scalar tensor theories; see [7].Tracingthemechanismwhichleadstothisshiftofequivalenceat the quantum level would be therefore rather interesting and will be investigated in forthcoming papers. We shall follow the notation introduced in [1] and [8]. The aim of this paper is to go towards a quantum description of f(R)theories;wehavehowever to mention the reverse problem of giving a (semi)-classical account of the quantum effect of ordinary standard LQG models; see [10].Thetwoapproachesaresomehowcomplementary and based on similar techniques. 2. Selfdual Formulation for Extended Theories In [1] we introduced β ab µ ν ab cµ dν R := R µν ea eb + βR µν e e cdab (2.1) µ ab ab where ea is a spin frame (see [11]), R µν is the curvature of a spin connection ωµ on a 4 dimensional (spin) manifold M and β =0isarealparameter.Indices6 a,c,... =0..3and µ, ν, . =0..3whilei,j,...=1..3. 1 In the Euclidean sector one obtains for β = 2 the standard selfdual curvature + ab µ ν 1 ab cµ dν R := R µν ea eb + 2 R µνe e cdab (2.2) i i ab which can be written in terms of the curvature Fµν := pabR µν of the usual selfdual connection i i ab 0i 1 i jk Aµ := pabωµ = ωµ + 2 jkωµ as follows 1 + 1 cd a b 1 ab µ ν cd i ab µ ν ab i µ ν 2 R = 2 R µν δ[cδd] + 2 cd ea eb = R µν pcdpi ea eb = pi Fµν ea eb =: F (2.3) ab Here pi denotes the algebra projector p : spin(4) → su(2) on selfdual forms. It is given by 0j 1 j j0 − 0j jk 1 jk pi = 2 δi pi = pi pi = 2 i (2.4) i and the inverse projector pab is defined by i 1 i i i i 1 i p0j = 2 δj pj0 = −p0j pjk = 2 jk (2.5) One can easily prove that i ab i i cd 1 a b 1 ab pabpj = δj pabpi = 2 δ[cδd] + 2 cd (2.6) 2 One is then led to consider the following family of Lagrangians + 1 L = 2κ ef(F )+Lm (2.7) where κ =8πG, e is the determinant of the frame matrix, f is a generic analytic function and Lm encodes the matter contribution. Usually matter is assumed to couple only with g (and possibly to its derivatives up to some finite order; usually, in view of minimal coupling principle, ab at most 1) and not to the connection ωµ .Hereafterweshalljustconsiderthevacuumsector, i.e. we set Lm =0. In the special case f(F )=F one obtains an equivalent formulation of the usual selfdual action + 1 + ab ∧ c ∧ d 1 ab 1 ab ef c d µνρσ L = 8κ R e e abcd = 16κ R µν + 2 ef R µν eρeσ abcdds = 1 ef a b 1 ab c d µνρσ 1 ef ab i c d µνρσ = 14κ R µν δ[eδf] + 2 ef eρeσ abcdds = 8κ R µν pi pef eρeσ abcdds = (2.8) 1 ab i c d µνρσ e ab i µ ν e = 8κ pi F µν eρeσ abcdds = 2κ pi F µν ea eb ds = 2κ Fds where ds is the standard local basis of 4-forms on M induced by coordinates. Field equations of the Lagrangian L+ are ab i c µνρσ ( p F e abcd =0 i µν ρ (2.9) i a b µνρσ pab∇µ eνeρ =0 Let us now consider a Cauchy (boundary) surface i : S → M : kA 7→ xµ(k), A, B, . =1..3; µ A A A µ µ in coordinates x =(t,k )adaptedtothesubmanifoldS one has i : k 7→ k and ∂Ax = δA. The unit covector normal to S is given by n = dx0.Onecanuseantiselfdualtransformationsto µ a a µ define a canonical adapted frame ea = e a ∂µ and coframe e = e µdx (see [12])givenby 0 −1 0 0 0 e 0 = N e i =0 e 0 = N e i =0 (2.10) e j = N−1Nj e j = αi e j = −Nlαj e j = αi 0 i j 0 l i j A Tetrads (or better spin frames; [11])adaptedtoS define triads i = ei = αi ∂A on S.Alsothe i i µ selfdual connection can be projected onto S to define a connection AA = Aµ∂Ax on S.Letus i i µ ν denote by FAB = Fµν ∂Ax ∂Bx the projected curvature (which is the same as the curvature of the projected connection); for later convenience let us also define the tangent-normal projection of i i µ ν the curvature FA = Fµν ∂Ax n (of course the normal-normal projection vanishes due to the skew symmetry of F ). A A i Let us also set Ei = i for the momentum conjugated to the connection AA written in terms of A i the triad i tangent to S,with the determinant of the (co)triad A. Field equations (2.9) can be projected onto S to obtain a number of evolution equations and the following constraints on S: A A ∇AEi =0 i A (2.11) FABEi =0 jk i A B i FABEj Ek =0 These constitute the starting point of LQG quantization scheme; the first equation is related to gauge covariance, the second to Diff(S)–covariance; while the third equation is called the 3 Hamiltonian constraint,whenquantizeditbecomestheso-calledWheeler-deWitt equation and it encodes the (quantum) dynamics.