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Marija Dimitrijević Ćirić "Matter Fields in SO(2,3)⋆ Model of Noncommutative Gravity"

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SEENET-MTP Balkan Workshop BW2018
10 - 14 June 2018, Niš, Serbia

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Marija Dimitrijević Ćirić "Matter Fields in SO(2,3)⋆ Model of Noncommutative Gravity"

  1. 1. BW2018 workshop ”Field Theory and the Early Universe” June 10-14 , 2018, Niˇs, Serbia Matter fields in SO(2, 3) model of noncommutative gravity Marija Dimitrijevi´c ´Ciri´c University of Belgrade, Faculty of Physics, Belgrade, Serbia with: D. Goˇcanin, N. Konjik, V. Radovanovi´c; arXiv:1612.00768, 1708.07437, 1804.00608
  2. 2. NC geometry and gravity Early Universe =⇒ Quantum gravity =⇒ Quantum space-time Noncommutative (NC) space-time =⇒ Gravity on NC spaces. General Relativity is based on the diffeomorphism symmetry. This concept (space-time symmetry) is difficult to generalize to NC spaces. Different approaches: NC spectral geometry [Chamseddine, Connes, Marcolli ’07; Chamseddine, Connes, Mukhanov ’14]. Emergent gravity [Steinacker ’10, ’16]. Frame formalism, operator description [Buri´c, Madore ’14; Fritz, Majid ’16]. Twist approach [Wess et al. ’05, ’06; Ohl, Schenckel ’09; Castellani, Aschieri ’09; Aschieri, Schenkel ’14]. NC gravity as a gauge theory of Lorentz/Poincar´e group [Chamseddine ’01,’04, Cardela, Zanon ’03, Aschieri, Castellani ’09,’12; Dobrski ’16].
  3. 3. Overview Review of SO(2, 3) NC gravity General Action NC Minkowski space-time Adding matter fields Spinors U(1) gauge field NC Landau problem Discussion
  4. 4. SO(2, 3) NC gravity: General SO(2, 3) NC gravity is based on: -NC space-time → Moyal-Weyl deformation with small, constant NC parameter θαβ = −θβα; [ˆxµ, ˆxν] = iθµν. -gravity → SO(2, 3) gauge theory with symmetry broken to SO(1, 3), [Stelle, West ’80; Wilczek ’98]. - -product formalism: Moyal-Weyl -product. -Seiberg-Witten (SW) map → relates NC fields to the corresponding commutative fields. Our goals: -consistently construct NC gravity action, add matter fields, calculate equations of motion and find NC gravity solutions; investigate phenomenological consequences of the constructed model. -diffeomorphism symmetry broken by fixing constant θαβ, give physical meaning to θαβ.
  5. 5. SO(2, 3) NC gravity: Action SO(2, 3) gauge theory: gauge field ωµ and field strength tensor Fµν of the SO(2, 3) gauge group: ωµ = 1 2 ωAB µ MAB = 1 4 ωab σab + 1 2 ea µγa , (1) Fµν = 1 2 FAB µν MAB = ∂µων − ∂νωµ − i[ωµ, ων] = 1 4 Rab µν − 1 l2 (ea µeb ν − eb µea ν ) σab + 1 2 Fa5 µνγa , (2) with [MAB , MCD] = i(ηADMBC + ηBC MAD − ηAC MBD − ηBDMAC ) , (3) ηAB = (+, −, −, −, +), A, B = 0, . . . , 3, 5 MAB → (Mab, Ma5) = ( i 4 [γa, γb], 1 2 γa), a, b = 0, . . . 3 Rab µν = ∂µωab ν − ∂νωab µ + ωa µc ωcb ν − ωb µc ωca ν , Fa5 µν = 1 l µe a ν − νe a µ , µe a ν = ∂µe a ν + ωa µbe b ν .
  6. 6. The decomposition in (1, 2) very much resembles the definitions of curvature and torsion in Einstein-Cartan gravity leading to General Relativity. Indeed, after introducing proper action(s) and the breaking of SO(2, 3) symmetry (gauge fixing) down to the SO(1, 3) symmetry one obtains this result [Stelle, West ’80]. To fix the gauge: the field φ transforming in the adjoint representation: φ = φA ΓA, δ φ = i[ , φ] , with ΓA = (iγaγ5, γ5) and γa and γ5 are the usual Dirac gamma matrices in four dimensions.
  7. 7. Inspired by [Stelle, West ’80; Wilczek ’98] we define the NC gravity action as SNC = c1S1NC + c2S2NC + c3S3NC , (4) with S1NC = il 64πGN Tr d4 x µνρσ ˆFµν ˆFρσ ˆφ , S2NC = 1 64πGN l Tr d4 x µνρσ ˆφ ˆFµν ˆDρ ˆφ ˆDσ ˆφ + c.c. , S3NC = − i 128πGN l Tr d4 x εµνρσ Dµ ˆφ Dν ˆφ ˆDρ ˆφ ˆDσ ˆφ ˆφ . The action is written in the 4-dimensional Minkowski space-time, as an ordianry NC gauge theory. It is invariant under the NC SO(2, 3) gauge symmetry and the SW map guarantees that after the expansion it will be invariant under the commutative SO(2, 3) gauge symmetry.
  8. 8. Using the SW map solutions for the fields ˆFµν and ˆφ and the Moyal-Weyl -product, we expand the action (4) in the orders of NC parameter θαβ. In the commutative limit θαβ → 0 and after the gauge fixing: φa = 0, φ5 = l, these actions reduce to S (0) 1NC → − 1 16πGN d4 x l2 16 µνρσ abcd R ab µν R cd ρσ + e(R − 6 l2 ) , S (0) 2NC → − 1 16πGN d4 xe R − 12 l2 , S (0) 3NC → − 1 16πGN d4 xe − 12 l2 , with e a µ = 1 l ωa5 µ , e = det ea µ, R = R ab µν e µ a e ν b . The constants c1, c2 and c3 are arbitrary and can be determined from some consistency conditions.
  9. 9. Comments: -main adventage of SO(2, 3) approach: basic fields are not metric and/or vielbeins but gauge fields of (A)dS group; consequences also in the NC setting. -after the symmetry breaking: spin connection ωµ and vielbeins eµ. They are independent, 1st order formalsim. -varying (4) with respect to ωµ and vielbeins eµ gives equations of motion for these fields. The spin connection in not dynamical (the equation of motion is algebraic, the zero-torsion condition) and can be expressed in terms of vielbeins, 2nd order formalism. -(4) written in the 2nd order formalism has three terms: Gauss-Bonnet topological term, Einstein-Hilbert term and the cosmological constant term. -arbitrary constants c1, c2 and c3: EH term requires c1 + c2 = 1, while the absence of the cosmological constant is provided with c1 + 2c2 + 2c3 = 0. Applying both constraints leaves one free parameter (can be used later in the NC generalization).
  10. 10. Calculations show that the first order correction S (1) NC = 0. Already known result [Chamseddine ’01,’04, Cardela, Zanon ’03, Aschieri, Castellani ’09]. The first non-vanishing correction is of the second order in the NC parameter; it is long and difficult to calculate. However, the second order corrections can be anayzed sector by sector: high/low energy, high/low/zero cosmological constant, zero/non-zero torsion. In the low energy sector, i.e., keeping only terms of the zeroth, the first and the second order in the derivatives of vierbeins (linear in Rαβγδ, quadratic in Ta αβ), we calculate NC induced corrections to Minkowski space-time. This calculation cen be generalized to other solutions of vacuum Einstein equations.
  11. 11. SO(2, 3) NC gravity: NC Minkowski space-time A solution of the form: g00 = 1 − R0m0nxm xn , g0i = − 2 3 R0minxm xn , gij = −δij − 1 3 Rimjnxm xn , (5) where Rµνρσ ∼ θαβθγδ: the Reimann tensor for this solution. The coordinates xµ we started with, are Fermi normal coordinates: inertial coordinates of a local observer moving along a geodesic; can be constructed in a small neighborhood along the geodesic (cylinder), [Manasse, Misner’63; Chicone, Mashoon’06; Klein, Randles ’11]. The measurements performed by the local observer moving along the geodesic are described in the Fermi normal coordinates. He/she is the one that measures θαβ to be constant! In any other reference frame, observers will measure θαβ different from constant. Fixed NC background: gauge fixed diffeomorphism symmetry, prefered reference frame given by Fermi normal coordinates.
  12. 12. Adding matter fields: spinors Spinors are naturally coupled to gravity in the first order formalism. NC generalization of an action for Dirac spinor coupled to gravity in SO(2, 3) model: Sψ = i 12 d4 x µνρσ ¯ψ (Dµφ) (Dν φ) (Dρφ) (Dσψ) − (Dσ ¯ψ) (Dµφ) (Dν φ) (Dρφ) ψ (6) + i 144 m l − 2 l2 d4 x µνρσ ¯ψ Dµφ Dν φ Dρφ Dσφ φ − Dµφ Dν φ Dρφ φ Dσφ + Dµφ Dν φ φ Dρφ Dσφ ψ + h.c. , where Dσψ = ∂σψ − iω ψ is the SO(2, 3) covariant derivative in the defining representation. Expanging the action (6) ( -product, SW-map) gives a nontivial first order correction for a Dirac fermion coupled to gravity.
  13. 13. Phenomenological consequences: in the flat space-time limit we find a deformed propagator: iSF (p) = i /p − m + i + i /p − m + i (iθαβ Dαβ) i /p − m + i + . . . , (7) Dαβ = 1 2l σ σ α pβpσ + 7 24l2 ε ρσ αβ γργ5pσ − m 4l2 + 1 6l3 σαβ . (8) A spinor moving along the z-axis and with only θ12 = θ = 0 has a deformed dispersion relation (analogue to the birefringence effect): v1,2 = ∂E ∂p = p Ep 1 ± m2 12l2 − m 3l3 θ E2 p + O(θ2 ) , (9) with Ep = m2 + p2 z . These results are different from the usual NC free fermion action/propagator in flat space-time: Sψ = d4 x ¯ψ i /∂ψ − mψ = d4 x ¯ψ i /∂ψ − mψ .
  14. 14. Adding matter fields: U(1) gauge field Metric tensor in SO(2, 3) gravity is an emergent quantity. Therefeore, it is not possible to define the Hodge dual ∗H. Yang-Mills action S ∼ F ∧ (∗HF) cannot be defined. A method of auxiliary field f [Aschieri, Castellani ’12], with δε f = i[Λε , f ]. An action for NC U(1) gauge field coupled to gravity in SO(2, 3) model: SA = − 1 16l Tr d4 x µνρσ ˆf ˆFµν Dρ ˆφ Dσ ˆφ ˆφ + i 3! ˆf ˆf Dµ ˆφ Dν ˆφ Dρ ˆφ Dσ ˆφ ˆφ + h.c. . (10) After the expansion ( -product, SW-map) and on the equations of motion fa5 = 0, fab = −eµ a eν b Fµν the zeroth order of the action reduces to SA = − 1 4 d4 x e gµρ gνσ FµνFρσ . (11) describing U(1) gauge field minimally coupled to gravity.
  15. 15. NC Landau problem Phenomenological consequences of our model and NC in general: the NC Landau problem: an electron moving in the x-y plane in the constant magnetic field B = Bez. Our model in the flat space-time limit gives i /∂ − m + /A + θαβ Mαβ ψ = 0 , (12) where θαβMαβ is given by θαβ Mαβ = θαβ − 1 2l σ σ α DβDσ + 7i 24l2 ρσ αβ γργ5Dσ − m 4l2 + 1 6l3 σαβ + 3i 4 Fαβ /D − i 2 Fαµγµ Dβ − 3m 4 − 1 4l Fαβ . (13) For simplicity: θ12 = θ = 0 and Aµ = (0, By, 0, 0). Assume ψ = ϕ(y) χ(y) e−iEt+ipx x+ipz z . (14) with ϕ, χ and E represented as powers series in θ.
  16. 16. Deformed energy levels, i.e., NC Landau levels are given by En,s =E (0) n,s + E (1) n,s , (15) E (0) n,s = p2 z + m2 + (2n + s + 1)B , E (1) n,s = − θs E (0) n,s m2 12l2 − m 3l3 1 + B (E (0) n,s + m) (2n + s + 1) + θB2 2E (0) n,s (2n + s + 1) . Here s = ±1 is the projection of electron spin. In the nonrelativistic limit and with pz = 0, (15) reduces to En,s = m − sθ m 12l2 − 1 3l3 + 2n + s + 1 2m Beff − (2n + s + 1)2 8m3 B2 eff + O(θ2 ) , (16) Beff = (B + θB2 ) . Consistent with string theory interpretation of noncommutativity as a Neveu-Schwarz B-field.
  17. 17. In addition, the induced magnetic dipole moment of an electon is given by µn,s = − ∂En,s ∂B = −µB (2n + s + 1)(1 + θB) , (17) where µB = e 2mc is the Bohr magneton. Some numbers: -θ = 2c2 Λ2 NC and ΛNC ∼ 10TeV , -accuracy of magnetic moment measurements δµn,s ∼ 10−13, -for observable effects in µn,s, B ∼ 1011T needed. This is the magnetic field of some neutron stars (magnetars), in laboratory B ∼ 103T.
  18. 18. Discussion A consistend model of NC gravity coupled to matter fields. Pure gravity: -NC as a source of curvature and torsion. -The breaking of diffeomorphism invariance is understood as gauge fixing: a prefered refernce system is defined by the Fermi normal coordinates and the NC parameter θµν is constant in that particular reference system. Coupling of matter fields: spinors and U(1) gauge field. -deformed propagator and dispersion relations for free fermions in the flat space-time limit. -nonstandard NC Electrodynamics (QED), new terms compared with the standard QED on the Moyal-Weyl NC space-time: phenomenological consequences, renormalizability. Study: corrections to GR solutions (cosmological, Reissner-Nordstr¨om black hole,. . . ); fermions in curved space-time (cosmological neutrinos); . . .

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