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Dynamical Spacetime Symmetry Utah State University DigitalCommons@USU Physics Student Research Physics Student Research 4-15-2016 Dynamical spacetime symmetry Benjamin Lovelady Utah State University James Thomas Wheeler Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/phys_stures Part of the Physics Commons Recommended Citation Lovelady, B. C., & Wheeler, J. T. (2016). Dynamical spacetime symmetry. Phys. Rev. D, 93(8), 085002. http://doi.org/10.1103/PhysRevD.93.085002 This Article is brought to you for free and open access by the Physics Student Research at DigitalCommons@USU. It has been accepted for inclusion in Physics Student Research by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. PHYSICAL REVIEW D 93, 085002 (2016) Dynamical spacetime symmetry † Benjamin C. Lovelady,* and James T. Wheeler, Department of Physics, Utah State University, 4415 Old Main Hill, Logan, Utah, 84322-4415, USA (Received 9 December 2015; published 1 April 2016) According to the Coleman-Mandula theorem, any gauge theory of gravity combined with an internal symmetry based on a Lie group must take the form of a direct product in order to be consistent with basic assumptions of quantum field theory. However, we show that an alternative gauging of a simple group can lead dynamically to a spacetime with compact internal symmetry. The biconformal gauging of the conformal symmetry of n-dimensional Euclidean space doubles the dimension to give a symplectic manifold. Examining one of the Lagrangian submanifolds in the flat case, we find that in addition to the expected SOðnÞ connection and curvature, the solder form necessarily becomes Lorentzian. General coordinate invariance gives rise to an SOðn − 1; 1Þ connection on the spacetime. The principal fiber bundle character of the original SOðnÞ guarantees that the two symmetries enter as a direct product, in agreement with the Coleman-Mandula theorem. DOI: 10.1103/PhysRevD.93.085002 I. INTRODUCTION by Ne’eman and Regge [6,7] who used it as a systematic way to study general relativity and supergravity. Later, The Coleman-Mandula theorem [1] and generalizations Ivanov and Niederle used the same method to develop a [2] show that, given certain assumptions likely true of a variety of gauge theories, including the biconformal gaug- satisfactory quantum field theory, any unification of general ing we use here [8,9], though the appropriate gravity field relativity with internal symmetries based on Lie groups equations appear to arise from the curvature-linear action must take the form of a direct product of the Poincaré or found by Wehner and Wheeler [10]. conformal group with a compact internal symmetry group. Starting from the generators of the conformal group of By extending to a graded Lie algebra, supersymmetric Euclidean n-space, we present the biconformal gauging theories escape this conclusion and give a nontrivial (developed in [9–11] and presented concisely in [12]). unification of gravity with the standard model interactions. Rather than continuing by introducing Cartan curvatures, In the present examination, we find an alternative to such we work directly with the homogeneous quotient manifold, unification, using a quotient of the conformal group of using a known solution to the flat Maurer-Cartan equations. Euclidean space which doubles the original dimension to a This solution is naturally expressed in terms of a pair of symplectic manifold. We show that the solution of the field involute 1-forms which together span a 2n-dimensional equations dynamically produces a Lorentzian metric on a symplectic manifold. The Killing metric induces a Lagrangian submanifold. The class of orthonormal frame Lorentzian metric on one of the resulting Lagrangian fields of this Lorentzian metric is invariant under submanifolds ([13], also see [12,14]), and we interpret − 1 1 SOðn ; Þ and enters as a direct product with the this submanifold as spacetime. SOðnÞ symmetry of the original principal fiber bundle, In previous work studying these solutions [12,14], the satisfying the Coleman-Mandula theorem. Of course, while SOð4Þ connection was separated into an SOð3; 1Þ piece and this approach satisfies the Coleman-Mandula theorem additional fields. The additional fields, constructed from without supersymmetry, it does not preclude supersym- certain invariant scalar fields, were interpreted as contri- metric extension. butions to dark matter and dark energy. The SOð3; 1Þ piece Our method uses the standard construction of a Cartan of their decomposition becomes compatible with the usual geometry [3], in which a principal fiber bundle is produced soldering of the base manifold to the fiber. from the quotient of a Lie group by a Lie subgroup. The Lie The novelty of our approach is to leave the SOð4Þ subgroup provides the local symmetry over the quotient symmetry intact and distinct from the Lorentzian symmetry manifold. Group quotients allow group-theoretic insights that necessarily [13] develops for the metric. Because the into gravity models, and thus improve on the early work of gauging includes the SOðnÞ symmetry of the original Utiyama [4] and Kibble [5], who extended global Lorentz Euclidean space, the fiber symmetry includes its own and Poincaré symmetries, respectively, to local symmetries. SOðnÞ connection. However, the solder form now fails The group quotient method was first employed for gravity to “solder”, displaying instead a Lorentzian inner product and inducing the spacetime metric. General coordinate *[email protected] covariance induces local Lorentz symmetry on orthonormal † [email protected] frame fields, so the final model has both SOðnÞ and 2470-0010=2016=93(8)=085002(11) 085002-1 © 2016 American Physical Society BENJAMIN C. LOVELADY and JAMES T. WHEELER PHYSICAL REVIEW D 93, 085002 (2016) SOðn − 1; 1Þ symmetries and corresponding connections, The quotient of G by a Lie subgroup divides the in the direct product form required by the Coleman- Maurer-Cartan into horizontal and vertical parts. Thus, if A a m Mandula theorem. We introduce adapted coordinates that we write the connection forms as ω~ ¼ðω~ H; ω~ MÞ with a m enable us to clearly display the simultaneous presence of ω~ H;a;b;c¼ 1; …K spanning H and ω~ M;m;n;p¼ the two symmetries. 1; …N − K spanning the homogeneous quotient manifold, In a further new contribution to understanding these MN−K ¼ G=H, then we have spaces, we include a discussion of the meaning of the full biconformal manifold. We show that the homogeneous p CA ¼ cA ω þ cA ωb solution permits identification of the full 2n-dimensional B pB M bB H space as the cotangent bundle of spacetime with orthogonal m 0 Lagrangian submanifolds, one of which may be made flat with the condition c ab ¼ guaranteeing the subgroup ω~ a ω~ a by conformal transformation. condition. The structure equations for H and M are While we deal with only the flat case here, we expect further examination to permit curvatures of both connec- 1 dω~ a ¼ − Ca ∧ ωB tions. For the n ¼ 4 case, the SOð4Þ symmetry is naturally H 2 B replaced by SUð2Þ × SUð2Þ and interpreted as a left-right 1 dω~ m − Cm ∧ ωB symmetric electroweak model. Symmetry breaking of one M ¼ 2 B : of the SUð2Þ groups should then give a grav-electroweak unification. Larger n could incorporate SOðnÞ or SpinðnÞ m 0 Because c ab ¼ , the second of these takes the special GUT models. form II. BICONFORMAL GAUGING 1 dω~ m − cm ω~ n ∧ ω~ p − cm ω~ n ∧ ω~ b M ¼ 2 np M M nb M H A. Quotient manifold method 1 To develop the biconformal space, we use the quotient m ~ p m ~ b ~ n ¼ − c pnωM − c ωH ∧ ωM manifold method [3,6,7]. Starting with a Lie group G, (in 2 bn our case the conformal group, C), we construct a principal m fiber bundle by taking the quotient by a Lie subgroup, H [in thereby placing ω~ M in involution. This involution means W m our case the Euclidean Weyl group, , comprised of there exist submanifolds found by setting ω~ M ¼ 0, and we SOðnÞ transformations and dilatations]. This subgroup recover the subgroup Lie structure equations of the fibers. becomes the local symmetry of the 2n-dimensional quo- The final step in the construction of a Cartan geometry is tient manifold. to allow horizontal curvature 2-forms, ΣA ¼ðΣa; ΩmÞ. This ωA a m Lie groups have natural connection 1-forms dual to changes the connection 1-forms, ðω~ H; ω~ MÞ to a new the group generators G , defined by the linear mapping a m A connection, ðωH; ωMÞ, and we have the Cartan equations, ωA δA ðGBÞ¼ B, with the coordinate bases being dual, dωA ¼ − 1 cA ωB ∧ ωC þ ΣA,or ∂ d ν δν 2 BC h∂xμ ; x i¼ μ. Rewriting the commutation relations of the generators 1 dωa − Ca ∧ ωB Σa H ¼ B þ C 2 ½GA;GB¼cAB GC; ð1Þ 1 m m B m dωM ¼ − C ∧ ω þ Ω : A 2 B where c BC are the group structure constants, in terms of these dual 1-forms, we arrive at the Maurer-Cartan structure equations, Separating out the subgroup components, 1 1 1 dωA − A ωB ∧ ωC dωa ¼ − ca ωb ∧ ωc − ca ωm ∧ ωp ¼ 2 c BC : ð2Þ H 2 bc H H 2 mp M M − ca ωm ∧ ωb þ Σa The integrability conditions of Eq. (2) follow from the mb M H 2 1 Poincare lemma, d ¼ 0, and exactly reproduce the Jacobi m m n p m n b m dωM ¼ − c npωM ∧ ωM − c ωM ∧ ωH þ Ω : ð3Þ identity, 2 nb H ½GA; ½GB;GC þ ½GB; ½GC;GA þ ½GC; ½GA;GB ¼ 0 The curvature forms are tensorial under , and because they are horizontal, describe curvature of M only.
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