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Maxwell symmetries and some applications 2 1 0 2 José A. de Azcárraga1, Kiyoshi Kamimura2 and Jerzy Lukierski3 n a J 3 1 1 Department of Theoretical Physics, University of Valencia and IFIC (CSIC-UVEG), ] 46100-Burjassot (Valencia), Spain h 2 Department of Physics, Toho University Funabashi, 274-8510, Japan t - 3 Institute for Theoretical Physics, University of Wroc law, pl. Maxa Borna 9, 50–205 Wroc law, p e Poland h [ 1 v 0 5 8 2 Abstract . 1 The Maxwell algebra is the result of enlarging the Poincaré algebra by six addi- 0 tional tensorial Abelian generators that make the fourmomenta non-commutative. 2 1 We present a local gauge theory based on the Maxwell algebra with vierbein, spin : v connection and six additional geometric Abelian gauge fields. We apply this ge- i ometric framework to the construction of Maxwell gravity, which is described by X the Einstein action plus a generalized cosmological term. We mention a Friedman- r a Robertson-Walker cosmological approximation to the Maxwell gravity field equations, with two scalar fields obtained from the additional gauge fields. Finally, we outline further developments of the Maxwell symmetries framework. 1 1 Introduction Maxwell symmetry was introduced around 40 years ago [1,2], but it is only recently that has attracted more attention. The D = 4 Maxwell algebra, with sixteen generators (P ,M ,Z ), isobtainedfromPoincaréalgebraifwereplaceitscommuting fourmomenta a ab ab by noncommuting ones [P ,P ] = ΛZ , [P ,Z ] = 0 , a,b = 0,1,2,3 , (1) a b ab a bc where thesixadditionalgeneratorsZ = −Z areAbeliananddefineaLorentz-covariant ab ba tensor, [M ,Z ] = −(η Z −η Z ) . (2) ab cd c[a b]d d[a b]c Further, we assume that the Z are dimensionless, which implies that the parameter ab Λ has mass dimensions [Λ] = M2. The Maxwell algebra M is the semidirect sum M = so(3,1) A MI, where MI is the Maxwell ideal generated by P ,Z (eq. (1)), in a ab which Λ is the central extension parameter. M can be obtained from the so(3,2) algebra (MAB;A,B = 0,1,2,3,4)aftertherescalingMab = β2Zab; a,b = 0,1,2,3,Ma4 = βΛ−21Pa in the contraction limit β → ∞ (β is a dimensionless parameter). The global Maxwell symmetries have been introduced in order to describe Minkowski space with constant e.m. background [1–4] in models of relativistic particles interacting with a constant e.m. field1. In this paper, following [6], we present the construction of a local D = 4 gauge theory based on the Maxwell algebra (eqs. (1,2)) and apply it to generalize Einstein gravity. Such a theory will accordingly contain six additional geometric Abelian gauge fields, playing the role of vectorial inflatons2 and which in Maxwell gravity contribute to a generalization of the cosmological term. Further, we shall mention a one-dimensional FRW cosmological approximation describing the cosmic scale factor a(t) in Maxwell gravity and comment briefly on other uses of Maxwell symmetries. 2 Gauging the Maxwell algebra to generalize Ein- stein gravity In order to introduce geometrically the Maxwell gauge vector fields we consider the Maxwell algebra-valued one-form h = h dxµ, where µ 1 1 h = ea P + ωab M + Aab Z . (3) µ µ a 2 µ ab 2 µ ab TheMaxwell multiplet (ea (x),ωab (x),Aab (x))includes thevierbein, thespin connection µ µ µ and the new Abelian gauge fields Aab, which we interpret as geometrical inflaton vector µ 1For the non-relativistic case see also [5], Sec. 8.3. 2For vector inflatons described by SU(2) gauge fields and non-abelian ‘gauge-flation’ models see e.g. refs. [7–10]. 2 fields. Their associated curvatures are the components of the M-valued curvature two- form R = R dxµ ∧dxν, µν 1 1 R = Ta P + Rab M + Fab Z , (4) µν µν a 2 µν ab 2 µν ab which defines the two-forms corresponding to the torsion Ta, the Lorentz curvature Rab and the field strength Fab for Aab, with dimensions [T] = M−1, [R] = M0 = [F] (the one- forms ea = ea dx, ωab = ωab dxµ, Aab = Aab dxµ have dimensions [ea] = M−1, [ωab] = µ µ µ M0 = [Aab]). Explicitly, it follows from (3) and R = dh+ 1[h,h] that the Ta, Rab, Fab 2 spacetime components are given by Ta = ∂ ea +ωa eb ≡ Da eb , (5) µν [µ ν] b[µ ν] b[µ ν] Rab = ∂ ωab +ωa ωcb = (Dωab) = −Rba , (6) µν [µ ν] c[µ ν] µν µν Fab = Da Acb +Λ ea eb , (7) µν c[µ ν] [µ ν] where Da = δa ∂ +ωa . We observe that the torsion and the Lorentz curvature are the bµ b µ bµ same as in standard Einstein gravity, which is the particular choice of the Einstein-Cartan gravity described by the Poincaré gauge theory; the Λ-dependent additional term, which recalls the contribution to Rab in (A)dS gravity, enters through the new gauge curvature Fab. The following two geometric (metric independent) Lagrangian densities were considered in detail in ref. [6], namely 1) The Lagrangian density that leads to Einstein gravity 1 L = − ε Rab∧Fcd. (8) 2 2κΛ abcd Indeed, using ε Rab∧(DA)cd = d(ε Rab∧Acd) , (9) abcd abcd it follows that, modulo the above boundary term, the L in eq. (8) gives the Einstein 2 gravity Lagrangian L , E 1 L ≃ L ≡ − ε Rab∧ec∧ed . (10) 2 E abcd 2κ 2) The generalized cosmological term. Let us recall that the Lagrangian density for the standard geometric Einstein cosmological (EC) term, proportional to the cosmological constant λ ([λ] = M2), is λ L = ε ea∧eb∧ec∧ed . (11) EC abcd 4κ 3 The generalized cosmological term depends both on the standard cosmological constant λ and on the parameter Λ in eq. (1). It is defined by λ L = ε Fab∧Fcd = L +∆L , (12) C abcd EC C 4κΛ2 where the additional piece ∆L with respect to eq. (11) is given by C λ ∆L = ε [(DA)ab∧(DA)cd +2Λ(DA)ab∧ec∧ed] . (13) C abcd 4κΛ2 Our basic Maxwell gravity action L is then given [6] by the Lagrangian density M L = L +L = L +L +∆L , (14) M E C E EC C which includes a new contribution, ∆L , to the standard cosmological term (11). By C varying with respect to the independent field variables the following three equations of motion are obtained λ δωab : T[a∧eb] + F[a ∧Ac|b] = 0 , (15) c Λ2 λ δea : ε eb ∧ Rcd − Fcd = 0 , (16) abcd (cid:18) Λ (cid:19) 1 δAab : (De)[a∧eb] + R[a ∧Ae|b] = 0 . (17) e Λ It is useful to introduce a shifted Lorentz curvature by λ Jcd = Rcd − Fcd. (18) Λ If we assume Jab = 0 the equations (15) and (17) become identical, but this assumption describes only special solutions of these equations. In the general Jab 6= 0 case, the field equations (15-17) can be rewritten in simpler form. Using Jab, one obtains δωab : (DJ)ab ≡ dJab +ω[a|cJ b] = 0, (19) c where ωab = ωab − λAab ies a shiftedespin connection,eand Λ e δea : ε eb ∧Jcd = 0, (20) abcd δAde : ε JabAc = 0. (21) abc[d e] Equation (20) is the generalization of the Einstein equation. Passing from tangent to world spacetime indices (Jµν=e µe νJab = 1Jµν dxρ ∧ dxσ, a b 2 ρσ Fµν = e µe νFab, etc.), eq. (20) can be written as follows a b 1 λ 1 λ Jµ − δµ J ≡ Rµ − Fµ − δµ (R− F) = 0, (22) ν ν ν ν ν 2 Λ 2 Λ 4 where Jµ ≡ Jµν , J ≡ Jµ and, similarly, Rµ ≡ Rµν , R ≡ Rµ , Fµ ≡ Fµν , F ≡ ρ ρν µ ρ ρν µ ρ ρν Fµ . More explicitly, eq. (22) can be written in a more familiar form as µ 1 Rµ − Rδµ −3λδµ = ν ν ν 2 λ = e µe σ(D A )ab −δµ e ρe σ(D A )ab . (23) a b [ν σ] ν a b ρ σ Λ (cid:0) (cid:1) Weseethatthesourceaddedtothestandardgravityequationswithcosmologicalconstant λ contains linear contributions from the new gauge fields. The second term in the rhs of (23) provides a field-dependent modification of the cosmological constant at the lhs of the equation. We wish to add that: 1) Using equations (15) and (16), the spin connection may be expressed as a function ωab(e,A) of the vierbein and the new gauge fields (for perturbative solutions see [6], Appendix). In such a way we obtain the second order formulation of Maxwell gravity, with independent fields ea and Aab. µ µ 2) Using eqs. (15), (17) and the Bianchi identities for the Fab curvature, we obtain the free field equation for new gauge fields Aab, µ (DF)ab = 0 . (24) Eq.(24)canbemodifiedbyasourcetermifweaddtotheLagrangian(14)non-geometrical contributions containing the fields Aab as e.g., a kinematical term proportional to the µ density Fab∧∗F , similar to the free Lagrangian −1F ∧∗F of Maxwell electrodynamics. ab 2 3)Inordertoestimatetheeffectofthenewgaugefieldsonthedynamicsoftheuniverse we have considered recently [11] a cosmological FRW approximation to the field equations of Maxwell gravity. After introducing a function a(t) describing the time dependence of the cosmic scale factor ea(x) = (ea(x),ea(x)) F−R→W (δaa(t),0) , (25) µ i 0 i where a = 0,1,2,3 and i,j = 1,2,3, the six Abelian gauge fields Aab are approximated in µ terms of the one-dimensional inflaton fields φ , φ , as follows 1 2 Ars(x) = (Ars(x),Ars(x)) F−R→W (ǫ rsφ (t),0), µ i 0 i 1 (26) A0r(x) = (A0r(x),A0r(x)) F−R→W (δ aφ (t),0) . µ i 0 i 2 The usual way of introducing the vector inflaton fields is based on Yang-Mills gauge fields [9,10] with internal symmetry indices. In our case these internal indices are replaced bytangentspacetimeindices, andthethree-dimensionaltensorsappearinginformula(26), ǫrs and δa (for a = 1,2,3), are genuine three-dimensional so(3) tensors. i i 5 3 Outlook To conclude, we make the following comments: a) The action defining Maxwell gravity was chosen to obtain a generalization of the cosmological term. Nevertheless, as for the standard Einstein-Hilbert Lagrangian (10), the action that follows from (14) is only invariant under local Lorentz transformations and spacetime diffeomorphisms, not under the full local Maxwell algebra. We would like to mention at this point that other Maxwell generalizations of Ein- stein gravity, invariant under the local Abelian gauge symmetries associated with the Z ab generators, have been proposed recently. The locally Maxwell-invariant gravity model in ref. [12] contains a rather controversial torsion squared term, and the Maxwell-invariant extensions proposed in ref. [13] differ from the Einstein Lagrangian (10) only by a topo- logical term i.e., the Einstein field equations remain unaltered. We also note that another modification of Einstein gravity (see ref. [14]), obtained by gauging a deformation of the Maxwell algebra3, so(3,1) ⊕ so(3,2), has been proposed recently4. The action of the deformed Maxwell gravity in [14] is invariant under local deformed Maxwell transformations, but in the contraction limit that leads to the Maxwell Lie algebra the Abelian local Maxwell symmetries are also broken, as in our case. b) The Maxwell symmetries have been generalized to Maxwell supersymmetries [16]; in particular, the N-extended Maxwell superalgebras were recently described in detail in [17]. When N=1 one obtains three different models of lowest dimensional Maxwell superalgebras, containing a pair of two-component Weyl charges. c)Itisknownthatthehigherspin(HS)freefieldscanbedescribed asafreefieldtheory on enlarged, tensorial spaces which contains the D-dimensional ‘physical’ spacetime as a submanifold [18–22]. It turns outthat forD=4the freeHSfields canbeobtained fromthe first quantization of spinorial particle model on a ten-dimensional tensorial space [19–21]. However, the ten-dimensional group manifold generated by the Maxwell ideal MI (eq. (1)) of the Maxwell Lie algebra also defines a ten-dimensional extended D = 4 spacetime that we call Maxwell D = 4 tensorial space. One can consider as well a spinorial particle model on this new Maxwell tensorial space which, after first quantization, should also provide an infinite-dimensional multiplet ofD = 4HSfields; such amodel isunder consideration [23]. Acknowledgements One of the authors (J.L.) would like to thank the organizers of the III Galilei-Guang-Xi Meeting (Beijing, October 2011) for their warm hospitality. This paper is supported by a research grant from the Spanish Ministry of Science and Innovation (FIS2008-01980) and by the Polish NCN grant 2011/B/S12/03354. 3For the classification of the Maxwell algebra deformations see ref. [15]. 4Such a model can be called AdS-Maxwell gravity. 6 References [1] H. Bacry, P. Combe, J.L. Richard, Nuovo Cim. A67, 267 (1970). [2] R. Schrader, Fortschr. Phys. 20, 701 (1972). [3] S. Bonanos, J. Gomis, J. Phys.A43, 015201 (2010) [arXiv:0812.4140v3 [hep-th]]. [4] S. Bonanos, J. Gomis, K. Kamimura, J. Lukierski, J. Math. Phys. 51, 102301 (2010) [arXiv:1005.3714v2 [hep-th]]. [5] J.A. de Azcárraga and J. M. Izquierdo, Lie groups, Lie algebras, cohomolgy and some applications in Physics, Camb. Univ. Press, 1985 [6] J.A. de Azcárraga, K. Kamimura, J. Lukierski, Phys.Rev. D83, 12403 (2011) [arXiv:1012.4402 [hep-th]]. [7] L.H. Ford, Phys. Rev. D40, 967 (1989). [8] A. Golovnev, V. Mukhanov, V. Vanchurin, J. Cosmol. Astropart. Phys. 08060, 009 (2008) [arXiv:0802.2068 [astro-ph]]. [9] A. Maleknejad, M.M. Sheikh-Jabbari, arXiv:1102.1513 [hep-ph]. [10] D.V. Gal’tsov, E.A. Davydov, Proc. Steklov Inst. Math. 272, 119 (2011) [arXiv:1012.2861 [gr-qc]]. [11] A. Borowiec, J. Lukierski, M. Woronowicz, in preparation. [12] D.V. Soroka, V.A. Soroka, Phys. Lett. B707, 160 (2012) [arXiv:1101.1591 [hep-th]]. [13] R. Durka, J. Kowalski-Glikman, M. Szczachor, Mod. Phys. Lett. A26, 2689 (2011) [arXiv:1107.4728 [hep-th]]. [14] R. Durka, J. Kowalski-Glikman, arXiv:1110.6812 [hep-th]. [15] J. Gomis, K. Kamimura, J. Lukierski, JHEP 0908, 039 (2009) [arXiv:0906.4464 [hep-th]]. [16] S. Bonanos, J. Gomis, K. Kamimura, J. Lukierski, Phys. Rev. Lett. 104, 090401 (2010). [17] K. Kamimura, J. Lukierski, Phys. Lett. B707, 292 (2012) [arXiv:1111.3598 [math- ph]]. 7 [18] C. Fronsdal, Massless particles, ortosymplectic symmetry and another type of Kaluza- Klein theory, PreprintUCLA/85/TEP/10, inEssays on Supersymmetry, Reidel, 1986 (Mathematical Physics Studies, v. 8), p. 163. [19] I.Bandos, J.Lukierski, D.Sorokin, Phys. Rev.D61, 045002(2000)[hep-th/9907113]. [20] M. Vasiliev, Phys. Rev. D66, 06606 (2002), hep-th/0106149; Fortsch. Phys. 52, 702 (2004) [hep-th/0401177]. [21] M. Plyushchay, D. Sorokin, M. Tsulaia, JHEP 0304, 013 (2003) [hep-th/0301067]. [22] I. A. Bandos, J. A. de Azcárraga, M. Picoń and O. Varela, Phys. Rev. D69, 085007 (2004) [hep-th/0307106]. [23] S. Fedoruk, J. Lukierski and D. Sorokin, in preparation. 8