Conformal Hamiltonian Dynamics of General Relativity ∗ A.B

Physics Letters B 691 (2010) 230–233

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Physics Letters B

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Conformal Hamiltonian dynamics of general relativity ∗ A.B. Arbuzov a,B.M.Barbashova, R.G. Nazmitdinov a,b, V.N. Pervushin a, , A. Borowiec c, K.N. Pichugin d,A.F.Zakharove a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia b Department de Física, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain c Institute of Theoretical Physics, University of Wrocław, Pl. Maxa Borna 9, 50-204 Wrocław, Poland d Kirensky Institute of Physics, 660036 Krasnoyarsk, Russia e Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya str. 25, 117259 Moscow, Russia article info abstract

Article history: The General Relativity formulated with the aid of the spin connection coeﬃcients is considered in the Received 17 March 2010 ﬁnite space geometry of similarity with the Dirac scalar dilaton. We show that the redshift evolution Accepted 23 June 2010 of the General Relativity describes the vacuum creation of the matter in the empty Universe at the Available online 3 July 2010 electroweak epoch and the dilaton vacuum energy plays a role of the dark energy. Editor: S. Dodelson © 2010 Elsevier B.V. All rights reserved. Keywords: General relativity Cosmology Dilaton gravity

The distance-redshift dependence in the data of the type Ia su- Within the Dirac approach the Einstein–Hilbert action takes the pernovae [1] is a topical problem in the standard cosmology (SC). form As it is known, the SNeIa distances are greater than the ones pre- 4 1 dicted by the SC based on the matter dominance idea [2].There W Hilbert =− d x |−e| R(e) are numerous attempts to resolve this problem with a various de- 6 e=e−D e˜ − gree of success (see for review [3]). One of the popular approaches |−e˜|e 2D =− 4 ˜ − −D |−˜|˜ μν −D is the -Cold-Dark-Matter model [4]. It provides, however, the d x R(e) e ∂μ e g ∂νe . − 6 present-day slow inflation density that is less by factor of 10 57 than the fast primordial inflation density proposed to include the (1) Planck epoch. 2 3 = Hereafter, we use the units MPlanck 8π 1. The interval is defined μ Approaches to the General Relativity (GR) with conformal sym- via diffeo-invariant linear forms ω(α) = e(λ)μ dx with the tetrad metry provide a natural relation to the SC [5]. The Dirac version coefficients [6] of the geometry of similarity [7] is an efficient way to include 2 μ ν the conformal symmetry into the GR. In fact, the latter approach ds = gμν dx dx = ω(α)(d) ⊗ ω(β)(d)η(α)(β); allows to explain the SNeIa data without the inflation [8].Inthe η = Diag(1, −1, −1, −1). (2) present Letter, the Dirac formulation of the GR in the geometry of (α)(β) similarity is adapted to the diffeo-invariant Hamiltonian approach The geometry of similarity [6,7] means the identification of mea- by means of the spin connection coefficients in a finite space–time, sured physical quantities F (n), where (n) is the conformal weight, − developed in [9]. In this way we study a possibility to choose with their ratios in dilaton units e nD variables and their initial data that are compatible with the ob- ˜ (n) nD (n) ˜ 2 2D 2 servational data associated with the dark energy content. We find F = e F , ds = e ds . (3) integrals of motion of the metric and matter fields in terms of the We define the measurable space–time coordinates in the GR as variables distinguished by the conformal initial data. the scale-invariant quantities in the framework of the Dirac–ADM 4 = 1 + 3 foliation [10,11]

˜ 2 ˜ ˜ ˜ ˜ ds = ω(0) ⊗ ω(0) − ω(b) ⊗ ω(b), (4) Corresponding author. * μ E-mail address: [email protected] (V.N. Pervushin). where the linear forms ω˜ (α) = e˜(α)μ dx are

˜ = −2D 0 R = [ − ] ω(0) e Ndx , (5) where ε(ab)(k) diag 1, 1, 0 in the orthogonal basis of spatial ± ˜ j 0 vectors [ε (1)(k), ε (2)(k), k]. Here, g are the holomorphic variables ω(b) = ω(b) + e(b) j N dx , (6) √ ofthesingledegreeoffreedom,ω = k2 is the graviton energy = i k ω(b) e(b)i dx . (7) normalized (like a photon in QED) on the units of a volume and j time Here N is the Dirac lapse function, N are the shift vector com- √ ponents, and e are the triads corresponding to the unit spatial (b)i ± 8π ± (3) g = g . (16) metric determinant |g˜ |≡|e e |=1. k 1/2 k ij (b) j (b)i M V (3) Planck 0 The Dirac dilaton D =−(1/6) log |g |=D+D,istaken ij The triad velocities in the Lichnerowicz gauge [12]. The Dirac lapse function N = 0 N −1 = −1 1 N0(x ) (τ , x) is split on the global factor N0 N which + l ⊥ ⊥ v(ab) = ω ∂0 − N ∂l + ∂(a)N + ∂(a)N (17) determines all time intervals used in the observational cosmol- N (a)(b) (b) (b) ogy: the redshift interval dτ = N dx0 [13], the conformal one + 0 depend on the symmetric forms , and the shift vector compo- = −2D = −D = −3D ω(ab) dη dτ e , and the world interval dt e dη dτ e . ⊥ −1 3 = In this case the dilaton zeroth mode D=V d xD (defined nents ∂(b)N(b) 0 are treated as the non-dynamical potentials. This 0 V 0 − in the finite diffeo-invariant volume) coincides with the logarithm means that the anti-symmetric forms ω(ab) are not dynamically in- of the redshift of spectral line energy Em dependent variables but are determined by a matter distribution. Following Dirac [10,16] one can define such a coordinate sys- D=log(1 + z) = log Em(η0 − η)/Em(η0) , (8) tem, where the covariant velocity v D of the local volume element and the momentum where η0 is the present-day conformal time interval, and η0 − η = r/c is the SNeIa distance. In accord with the new Poincaré group 2 l l P = 2v = ∂0 − N ∂l D + ∂l N /3 = 0 (18) classification, the “redshift” (8) is treated as one of the matter D D N components, on the equal footing with the matter. The key point of our approach is to express the GR action di- are zero. As a result, the dilaton deviation D can be treated as a rectly in terms of the redshift factor. The action can be represented static potential. The dilaton contribution to the curvature (11) with as a sum of the dilaton and the graviton terms: matter sources yield the Schwarzschild solution of classical equations [exp{−7D/2}N ]=0 and exp{−D/2}=0. The solutions 2 {− }N = + {− }= − 0 (∂0 D ) −2D are exp 7D/2 1 rg /(4r) and exp D/2 1 rg /(4r) in W Hilbert = dx − + N0e Lg , (9) N0 the isotropic coordinates of the Einstein interval ds, where rg is the gravitation radius of a matter source. These solutions double v2 (3) 2D 3 2 (ab) −4D R the angle of the photon beam deflection by the Sun field, exactly Lg = e d x N −(v ) + − e . (10) D 6 6 as the Einstein’s metric determinant. Note that the GR theory pro- Here, vides also the Newtonian limit in our variables (see details in [9]). Furthemore, in empty space without a matter source (rg = 0),the 4 − N = = l = R(3) = R(3)(e) − e7D/2 e D/2, (11) mean field approximation ( 1, D 0, N 0) becomes exact. 3 If there are no matter sources one can impose the condition − + (3) = is the curvature, where R (e) is expressed via the spin-connection ω(a)(b) 0, since the kinetic term (17) depends only on ω(ab) com- coefficients ponents. In this case the curvature (11) takes the bilinear form + + ± 1 j j i i (3) = ω (∂(c)) = e ∂(c)e ± e ∂(c)e , (12) R (e) ω(ab)(∂(c))ω(ab)(∂(c)). (19) (ab) 2 (a) (b) (b) (a) j The variation of the Hilbert action with respect to the lapse and = ∂ [ei e ∂ ] is the Laplace operator. i (a) (a) j function leads to the energy constraint [17] The dependence of the linear forms 2 − ∂ D = ρ Ω + e 2 D H /V , (20) = i = − i τ cr D g 0 ω(b)(d) e(b)i dx dX(b) X(c)e(c) de(b)i (13) where the dilaton integral of motion ρcrΩD is added, ρcr = i i 2 2 on the tangent space coordinates X(b) ≡ dx e(b)i = x e(b)i by H0 MPl3/(8π) is the critical density, and means of the spin connection coefficients can be obtained by − = − e 4D R(3) virtue of the Leibniz rule AdB d(AB) (AB) d log(A) (in par- = 2D 3 N 2 + [ i ] T = [ i T ]− i [ T ] Hg e d x 3p(ab) (21) ticular d x ebi d x ebi x d ebi ). The difference between this 6 approach to gravitation waves and the accepted one [14,15] is that is the graviton Hamiltonian, p = v /3 is a canonical momen- the symmetry with respect to diffeomorphisms is imposed on spin (ab) (ab) tum (see Eq. (17)). connection coefficients. Straightforward calculations define a set of evolution equations The linear graviton form (12) can be expressed via two photon- (α) for the Lagrangian Lg (10) and the Hamiltonian Hg (21) like polarization vectors ε(a) (k). By virtue of the condition ∂DHg = 2Lg , (22) (α) (α) = − k(a)k(b) ε (k)ε (k) δ(a)(b) , (14) = −2D (a) (b) k(2) ∂DTg 2e Lg , (23) α=1,2 −2D ∂DLg = 2Hg − 2e Tg , (24) one obtains ikX = 2 − 2 + e + − where Tg Hg Lg . ω (∂ ) = √ k εR (k)g (η) + εR (−k)g (η) , (ab) (c) (c) (ab) k (ab) k Note, the GR equations in terms of the spin-connection coef- 2ωk k2=0 ficients (22)–(24) coincide with the evolution equations for the (15) parameters of squeezing rb and rotation θb [18] 232 A.B. Arbuzov et al. / Physics Letters B 691 (2010) 230–233

∂Drb = cos 2θb, (25)

ωso − ∂Dθb = coth 2rb sin 2θb (26) + + − of the Bogoliubov transformations A = B cosh reiθ + B sinh reiθ ± ± ∓ for a squeezed oscillator (SO) ∂D A =±iωso A + A .Indeed, Eqs. (25), (26) establish similar relations for the expectation val- ± ues of various combinations of the operators A with respect to − the Bogoliubov vacuum B |=0(seedetailsin[17]) − + − cosh 2rb 1 −1 Nb ≡ A A = ≡ ω :Hb:, (27) 2 so i − − + + sinh 2rb sin 2θb −1 A A − A A = ≡ ω Tb, (28) 4 2 so 1 + + − − sinh 2rb cos 2θb −1 A A + A A = ≡ ω Lb. (29) 4 2 so On the other hand, Eqs. (10), (15), (19), and (21) show up that the graviton action (9) has a bilinear oscillator-like form [ = ] ωk + − − + Fig. 1. The creation of the Universe distribution Nk Nb (27) versus dimensionless = H H = + = = Hg k, k gk g−k gk g−k , time η and energies 0.5 ωk at the initial data Nk(η 0) 0 and the Hubble 2 parameter H(η) = 1/(1 + 2η) = (1 + z)2. k ω + + − − k 2 Lg = L , L = g g + g g , k k k −k k −k H0 8 2 ΩgI = ωc · · (1 + zI) (33) k M2 Pl. iωk + + − − Tg = T , T = g g− − g g− , (30) k k 2 k k k k defined by the Casimir energy. The question which remains to an- k swer is how to define zI? + where In order to estimate the instance of creation (1 zI),one → = √ can add the Hamiltonian of the Standard Model (SM): Hg H ± √ √ =[ ∓ ] Hg + HSM—when in the limit (1 + zI) →∞ and a → 0 all particles gk gk ωk ipk/ ωk / 2 (31) become nearly massless k2 + a2 M2 → ω . In this case, the same are the classical variables in the holomorphic representation [15]. 0 k ± mechanism of intensive particle creation works also for any scalar The form (31) suggests itself to replace the variables gk by creation and annihilation graviton operators. Evidently, in this case fields including four Higgs bosons [21] we have to postulate the existence of a stable vacuum |0.Asa = consequence, it is reasonable to suppose that the classical graviton ΩI Higgs 4ΩgI. (34) Hamiltonian (see Eqs. (30)) is the quantum Hamiltonian averaged The decays of the Higgs sector including longitudinal vector W and over coherent states [19]. One may speculate that such proce- Z bosons approximately preserve this partial energy density for dure reflects a transformation of a genuine quantum Hamiltonian the decay products. These products are Cosmic Microwave Back- (describing the initial dynamics of the Universe) to the classical ground (CMB) photons and nν neutrino. Therefore, one obtains Hamiltonian, associated with present-day dynamics. Having the correspondence between two sets of equations (22)– (1 + n )Ω ≈ 4Ω . (35) (24) for the GR and (27)–(29) for the SO, we are led to the ansatz ν CMB gI that the SO is the quantum version of our graviton Hamiltonian In our model there is the coincidence of two epochs: (see also [14]). This is a central point of our construction. As a result, the normal ordering of the graviton Hamiltonian yields • the creation of SM bosons in the Universe in electroweak epoch ωc Hg = Hb =:Hb:+ , Lg = Lb, Tg = Tb, (32) 2 1/3 15 1 + zW =[MW /H0] = 0.37 · 10 , (36) where ω = ω e2 D [17]. The normal ordering creates the Casimir- c so = + 2 = + 2 · type vacuum energy ω = 0.09235/(2r ) [20], where r is the when the horizon H(zW ) (1 zW ) H0 (1 zW ) 1.5 c h h −42 radius of the sphere defined by the Hubble parameter. 10 GeV contains only a single W boson; • The solution of Eqs. (22)–(24) is shown at Fig. 1. In accordance and the CMB origin time − with this solution, at the tremendous redshift 1 + z = e D = a 1, + =[ ]−1/2 = −29 · 1/2 z →∞, a = 0, Eq. (20) is reduced to the zeroth mode dilaton inte- 1 zCMB λCMB H0 10 2.35/1.5 gral of motion ΩD which corresponds to the z-dependence of the = 0 39 · 1015 (37) 2 . , Hubble parameter H(z) = H0(1+ z) . At this moment, the Universe was empty, and all particle densities had the zero initial data. The when the horizon contains only a single CMB photon with 2 same dilaton vacuum regime H(z) = H0(1 + z) is compatible with mean wave length λCMB that is approximately equal to the in- −1 = = · −13 the SNeIa data [1] in the geometry of similarity (3) [8]. verse temperature λCMB TCMB 2.35 10 GeV. The next step is the creation of gravitons induced by the di- rect dilaton interaction. A hypothetic observer being at the first In the same epoch zI ≈ zW ≈ zCMB, if the primordial graviton den- = = 3 instance at rI 1/HI in the primordial volume V I 4πrI /3ob- sity (33) coincides with the CMB density normalized to a single serves the vacuum creation of these particles with the primordial degree of freedom (as it was supposed in [14]). The coincidence density of the Planck epoch zI with the first two ones solves cosmological A.B. Arbuzov et al. / Physics Letters B 691 (2010) 230–233 233 problems with the aid of the geometry of similarity (3),without means of the Dirac constraint (18).Wehaveprovidedafewar- the inflation (see also [8]). guments in favour that the exact evolution of the GR as a theory While adding the SM sector to the theory in order to preserve of spontaneous conformal symmetries breaking is related to the the conformal symmetry, we should exclude the unique dimen- equations for the quantum squeezed oscillator. We found that the sional parameter from the SM Lagrangian, i.e. the Higgs term with dilaton evolution yields the vacuum creation of matter. a negative squared mass. However, following Kirzhnits [22],wecan include the vacuum expectation of the Higgs field (its zeroth har- Acknowledgements monic) φ. The latter appears as a certain external initial data or a condensate. In our construction we can choose it in the most The authors thank D. Blaschke, K. Bronnikov, D.V. Gal’tsov, simple form: φ=Const =φI = 246 GeV which could be con- A.V. Efremov, N.K. Plakida, and V.B. Priezzhev for useful discus- sider as the initial condition at the beginning of the Universe. The sions. V.N.P. thanks Yu.G. Ignatev and N.I. Kolosnitsyn for the dis- fact, that the Higgs vacuum expectation is equal to its present day cussion of experimental consequences of the General Relativity. value, allows us to preserve the status of the SM as the proper quantum field theory during the whole Universe evolution. 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