Dynamical Spacetime Symmetry

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4-15-2016

Dynamical spacetime symmetry

Benjamin Lovelady Utah State University

James Thomas Wheeler Utah State University

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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. so that Eq. (2) is equivalent to the Lie algebra. Consistency requires the integrability of these equations, CA ≡ A ωC It is convenient to define B c CB so the structure which is no longer guaranteed by the Jacobi identity. dωA 1 ωC ∧ CA dωA − 1 A ωB ∧ ωC ΣA equation becomes ¼ 2 C. Integrability of ¼ 2 c BC þ requires

085002-2 DYNAMICAL SPACETIME SYMMETRY PHYSICAL REVIEW D 93, 085002 (2016) 0 ≡ d2ωA 1 1 − A dωB ∧ ωC A ωB ∧ dωC dΣA ¼ 2 c BC þ 2 c BC þ 1 1 1 1 − cA − cB ωD ∧ ωE ΣB ∧ ωC cA ωB ∧ − cC ωD ∧ ωE ΣC dΣA ¼ 2 BC 2 DE þ þ 2 BC 2 DE þ þ 1 1 1 cA cB ωC ∧ ωD ∧ ωE dΣA − cA ΣB ∧ ωC cA ωB ∧ ΣC ¼ 2 B½C DE þ 2 BC þ 2 BC dΣA A ωB ∧ ΣC ¼ þ c BC ≡ DΣA

A B ≡ 0 1 where we use the Jacobi identity, c B½Cc DE , and define a c δcδaδf η δcηfa ηacη δf ½M b;M d¼ ½ b e d þ bd e þ ed b the covariant exterior derivative. The result holds for both 2 Σa Ωa δaη ηfc e and , leaving us with þ d be M f Ma ;P ΔadP DΣa ¼ 0 ½ b c¼ cb d Ma ;Kc −ΔacKd DΩa ¼ 0: ½ b ¼ db b 2Δbd c − δb ½Pa;K ¼ caM d aD These are the Bianchi identities in gravitational models. −δc A locally H-invariant physical theory is now found ½D; Pa¼ aPc a a c by writing any scalar Lagrange density built from the ½D; K ¼δcK : tensors available from the construction of this principal bundle, Ωa, Σm, along with tensors from the original linear m representation. The nonvertical basis forms ωM also Then we introduce basis 1-forms dual to the generators become tensors because the curvature breaks the corresponding symmetries. a ω~ c 2Δac hM b; di¼ db ~ b b B. Conformal structure equations hPa; ω i¼δa and the Cartan geometry a ω~ δa hK ; bi¼ b Applying the quotient method to the conformal group of hD; ω~ i¼1: a compactified space with flat metric, ηab, of signature ðp; qÞ, we express the generators as 1 Notice that index position corresponds to conformal Ma xa∂ − ηacη xd∂ ω~ a ω~ b ¼ 2 ð b bd cÞ weight, so and a are distinct fields. These lead directly to the Maurer-Cartan structure equations ≡ Δac d∂ dbx c P ¼ ∂ a a dω~ a ω~ c ∧ ω~ a 2Δadω~ ∧ ω~ c 1 b ¼ b c þ cb d a 2ηab∂ − 2 a c∂ K ¼ ðx b x x cÞ ~ a ~ c ~ a ~ ~ a 2 dω ¼ ω ∧ ω c þ ω ∧ ω c∂ ~ ~ c ~ ~ ~ D ¼ x c dωa ¼ ω a ∧ ωc þ ωa ∧ ω ~ ~ c ~ dω ¼ ω ∧ ωc: ð4Þ where the A index on the generators GA includes all possible antisymmetric pairs for the group, i.e., A ∈ a · a · Δac ≡ 1 δaδc − ηacη fðbÞ; ðaÞ; ð · Þ; ð·Þg and db 2 ð d b bdÞ is the anti- With the quotient C=W, these equations describe a 2n- 1 c symmetric projection operator on ð1Þ tensors. The operators dimensional homogeneous manifold spanned by ω~ and a a ω~ dω~ ω~ c ∧ ω~ M b, Pa, K , D generate SOðp; qÞ transformations, trans- c. Notice that ¼ c is a symplectic form since it lations, special conformal transformations, and dilatations, is manifestly both closed and nondegenerate. respectively. To complete the construction of a Cartan geometry, these ω~ a ω~ a ω~ ω~ → The commutators of the generators then give the Lie connection forms are now generalized, ð b; ; a; Þ ωa ωa ω ω algebra, ð b; ; a; Þ, to permit horizontal curvature 2-forms,

085002-3 BENJAMIN C. LOVELADY and JAMES T. WHEELER PHYSICAL REVIEW D 93, 085002 (2016) pffiffiffiffiffiffiffi dωa ωc ∧ ωa 2Δadω ∧ ωc Ωa a 2 0 0 b ¼ b c þ cb d þ b to x ¼þ jx jð1; 0; …; 0Þ gives signature ðp ;qÞ¼ 2 a c a a a ðp þ 1;q− 1Þ for b , while if xa is spacelike x > 0, dω ¼ ω ∧ ω c þ ω ∧ ω þ T ab pffiffiffiffiffiffiffi a 2 0 1 0 0 dω ωc ∧ ω ω ∧ ω S boosting and rotating to x ¼þ jx jð ; ; ; Þ at any a ¼ a c þ a þ a 00 00 given point gives bab signature ðp ;q Þ¼ðq þ 1;p− 1Þ. c 0 0 00 00 dω ¼ ω ∧ ωc þ Ω: ð5Þ Equating ðp ;qÞ¼ðp ;q Þ requires p ¼ q. Therefore, a bab only has a consistent signature for all x if ηab is The final step in producing a gravity model is to write the Euclidean or if p ¼ q and in the Euclidean case bab is most general action functional linear in the curvatures [10], necessarily Lorentzian. This is in agreement with the Z signature theorem of [13]. For the remainder of this αΩa βδaΩ γωa ∧ ω ε be…fωd ∧ … investigation, we will take η ¼ δ ¼ diagð1; …; 1Þ so S ¼ ð b þ b þ bÞ ac…d ab ab that bab ¼ 2xaxb − δabx2 is Lorentzian for all xa. c ∧ ω ∧ ωe ∧ … ∧ ωf: We show in the next section that bab is the restriction of the Killing metric to certain Lagrangian submanifolds. The We will be concerned only with the solution to the argument above then shows how the uniqueness of the homogeneous geometry, Eqs. (4), but the complete signature theorem occurs. If the initial signature differs Ta ¼ 0 class of solutions has been shown to reduce to from Euclidean or p ¼ q, the restriction of the Killing locally scale-invariant general relativity on a Lagrangian metric to the Lagrangian submanifolds is degenerate. submanifold [15]. For the rest of our discussion, we return to the homo- C. The Killing form geneous manifold described by Eqs. (4). An extended derivation, similar to that in [10,15] but straightforward to The Killing metric of the biconformal manifold is the verify by direct substitution, gives the general solution to restriction of the conformal Killing form KAB ¼ trðGAGBÞ C D these structure equations up to gauge and coordinate (equal to c ADc BC in the adjoint representation) to the choices. The solution takes the form first given in [11], quotient manifold. The nondegeneracy of KAB allows us to but a coordinate choice removes the function αaðxÞ present define the inner product ~ a ~ there. Here we use the symmetry between ω and ωa to recast that general solution as hωA; ωBi ≡ KAB ð11Þ

ω~ a −2Δac dd b ¼ dbx yc ð6Þ in either the curved ðωAÞ or the homogeneous ðω~ AÞ case. AB 1 Restricting K to its form on the quotient manifold, we ω~ a d a − a b − ηab 2 d have ¼ x x x 2 x yb ð7Þ

a b ~ hω ; ω i¼0 ωa ¼ dya ð8Þ ωa ω ω ωa δa h ; bi¼h b; i¼ b ~ c ω ¼ x dyc ð9Þ hωa; ωbi¼0 ð12Þ where x2 ≡ η xaxb. For convenience, we define ab which is readily seen to be nondegenerate. This means that, bab ≡ 2xaxb − ηabx2 ð10Þ unlike the Poincaré case, the group Killing form provides a metric on the quotient manifold. 1 xa which simplifies Eq. (8) to ω~ a ¼ dxa − babdy . It is useful to express this inner product in terms of the 2 b and y coordinates as well. Substituting the solution of Our Euclidean starting point gives us a specific model a Eqs. (6)–(9) into the Killing metric, Eq. (12), we find the within which we can explicitly verify and more clearly inner product of the coordinate basis forms to be understand the signature theorem of [13]. Note that when ab ηab ¼ δab is Euclidean, b has Lorentzian signature hdxa; dxbi¼bab 0 −1 1 hdxa; dy i¼hdy ; dxai¼δa B 1 C b b b bab ¼ −jx2j@ A 1 hdya; dybi¼0: 1 In the next section, we show that there is a Lagrangian but other initial signatures ðp; qÞ for ηab do not give submanifold of flat SOðnÞ biconformal space with natural consistent signature for bab. Specifically, for non- Lorentzian signature and both SOðnÞ and spacetime a 2 Euclidean ηab,ifx is timelike ðx < 0Þ, a transformation connections.

085002-4 DYNAMICAL SPACETIME SYMMETRY PHYSICAL REVIEW D 93, 085002 (2016) SO n 4 III. AN ð Þ FIELD ON SPACETIME Δacω~ d − ΔacΔdeδ gd f db c ¼ 2 db fc egx x We now come to our principal results: the simultaneous x 4 presence on a Lagrangian submanifold of an SOðnÞ − Δae δ gd f ¼ 2 fb egx x connection and its Yang-Mills field strength, together with x ω~ a Lorentzian metric, connection and curvature. The result is ¼ b achieved by using a different involution from that of [12,14]. the infinitesimal change in the Euclidean metric δab From the Maurer-Cartan equations, Eqs. (4), we see that produced by ω~ a vanishes: ~ a ~ b both basis forms, ω and separately, ωa, are in involution. This implies the existence of complementary submanifolds, δ ω~ c þ δ ω~ c ¼ 0: and the vanishing of the symplectic form when one or the ac b bc a other basis form vanishes shows that the submanifolds are ω~ a SO n Lagrangian. In this section, we use a general solution to the Therefore, b is a generator of ð Þ. flat structure equations to develop the Killing metric on Evidently, though the biconformal bundle as a whole has ~ ~ a SOðnÞ fiber symmetry, and this symmetry together with both the ωa ¼ 0 submanifold and the ω ¼ 0 submanifold and explore properties of the solution, showing the pres- dilatations preserve the structure equations, the symmetry ence of an SOðnÞ Yang-Mills field on spacetime from the of the metric restricted to this submanifold is Lorentzian. elements of the solution. We develop this further as follows. Let the SOðnÞ gauge be fixed, and consider coordinate ω~ A. The structure equations on the spacetime transformations. The vielbein, a, gives rise to a submanifold Lorentzian metric: Consider the involution of the basis forms, ω~ a. This hω~ ; ω~ i¼4b ð13Þ means that there exist coordinates, found in the next a b ab ~ a a b a section, such that ω ¼ α bdu , and holding u constant sets ω~ a ¼ 0 and selects a submanifold spanned by the as do the coordinates remaining basis forms. Setting ω~ a 0, Eq. (7) shows that ¼ α β αβ b hdx ; dx i¼b : dya ¼ 2babdx and the structure equations reduce to

4 Given a metric manifold we may construct the frame ω~ a ¼ − Δacx dxd b x2 db c bundle, Bðπ;G;MÞ. There then exists the subbundle of ~ b orthonormal frames [16], Oðπ;SOðn − 1; 1Þ; MÞ, a prin- ωa ¼ 2babdx 2 cipal fiber bundle with Lorentz symmetry group. Therefore, ω~ ¼ dðln x Þ: implicit in the set of general coordinate transformations, we have all Lorentz transformations of the corresponding These describe what we will call the spacetime submani- orthonormal frame fields. In this way, we generate a fold; it is a Lagrangian submanifold since the symplectic representation of SOðn − 1; 1Þ which is clearly indepen- dω~ d2 2 ≡ 0 form ¼ ðln x Þ vanishes. dent of the SOðnÞ fiber symmetry. The bundle structure guarantees that these symmetries are independent, satisfy- B. Symmetries ing the Coleman-Mandula theorem. We regard the com- ~ − 1 1 The inner products of the restricted basis forms ωa are bined bundle as having symmetry SOðnÞ × SOðn ; Þ × now SOð1; 1Þ. α We define the Christoffel connection, Γ μν, and Riemann ~ ~ c d α hωa; ωbi¼h2bacdx ; 2bbddx i curvature, R βμν, in the usual way, from the metric bαβ and 4 d c d d without reference to the SOðnÞ bundle symmetry. The ¼ bacbbdh x ; x i ω~ a orthogonal bundle symmetry has its own connection, b, cd ¼ 4bacbbdb and corresponding Yang-Mills field, ¼ 4b ab Fa dω~ a − ω~ c ∧ ω~ a b ¼ b b c: so the basis and coordinate differentials specify a Lorentzian inner product. This development within the solution of a Lorentzian Unlike previous treatments, this metric symmetry is metric is in accordance with the signature theorem proved different from the symmetry of the connection, which is in [13] and developed further in [12,14]. constructed from SOðnÞ invariants including the antisym- We now compute the spacetime curvature and the Yang- Δac metric projection db. Since Mills field strength of this model.

085002-5 BENJAMIN C. LOVELADY and JAMES T. WHEELER PHYSICAL REVIEW D 93, 085002 (2016) Fa 0 C. Spacetime curvature of the Lagrangian submanifold b ¼ (see Appendix A). Naturally, this will change The inverse spacetime metric is when curved biconformal spaces are considered. We end the section by finding a set of coordinates ω~ a hdxα; dxβi¼bαβ adapted to the involution of . ¼ 2xαxβ − δαβx2: E. Adapted coordinates for the involution of the αβ translational gauge fields In these coordinates, δ ¼ diagð1; …; 1Þ. Inverting, The involution of ωa means that we may write ωa ¼ 1 α β α α β 2 α βdu for n coordinates u . To find α β and u , we need to gαβ ¼ ð2xαxβ − δαβx Þ: ðx2Þ2 solve 1 The Christoffel connection is readily found to be, αα d β d α − αβd β u ¼ x 2 b yβ: ð14Þ 1 Γα αδ − δα − δα βμ ¼ 2 ðx βμ βxμ μxβÞ x With the metric bαβ given by and the curvature is 1 2 bαβ ¼ ð2xαxβ − δαβx Þ ðx2Þ2 α Γα − Γα − Γα Γσ Γα Γσ R βμν ¼ βν;μ βμ;ν σν βμ þ σμ βν β 1 1 1 where we define xα ≡ δαβx , we rewrite Eq. (14) as δα − α δ − ¼ 2 μ 2 x xμ βν 2 xβxν αμ d β d μ − 1 d d μ x x x bαμ β u ¼ bαμ x 2 yα and note that bαμ x ¼ −δ d xμ 1 1 αμ ð 2Þ. Therefore, − δα − α δ − x ν 2 x xν βμ 2 xβxμ : x x μ μ β 2x μβ 2bαμα βdu ¼ −δαμd þ δ yβ ; α ≡ δα − 1 α x2 Defining the projection operator P μ ð μ x2 x xμÞ, orthogonal to xα, we may write this as and we may identify 1 α α − α 2 α R βμν ¼ 2 ðP μPβν P νPβμÞ: α x αβ x u ≡ þ δ yβ x2

D. The Yang-Mills field strength and require Finally, we compute the Yang-Mills field strength, μ 2bαμα β ¼ δαβ: Fa dω~ a − ω~ c ∧ ω~ a b ¼ b b c Inverting the metric on this expression now shows that where the SOðnÞ connection is given by 1 αμ α − δα 2 4 β ¼ x xβ 2 βx : ω~ a ¼ − Δacx dxd: b x2 db c In terms of the adapted coordinates uα, we may solve for α We may find this directly from the SOðnÞ structure and replace x , equation, 2 α − δαβ α α ðu yβÞ dω~ a ω~ c ∧ ω~ a 2Δadω~ ∧ ω~ c x ðu ;yβÞ¼ 2 α 2 b ¼ b c þ cb d u − 2u yα þ y which shows immediately that so that the full structure equations take the form

a ad c a ac d ν μ α F −2Δ ω~ ∧ ω~ c 0: ω~ −2Δ δ δ d b ¼ cb d jω~ ¼0 ¼ b ¼ db μ cx ðu ;yβÞ yν ω~ a δad β The Yang-Mills field strength therefore vanishes on the ¼ β u spacetime Lagrangian submanifold. ~ β ωa ¼ δadyβ We easily check this directly. Substituting the submani- μ α − 4 Δac d d ω~ ¼ x ðu ;yβÞdyμ; ð15Þ fold form, x2 dbxc x leads after some algebra to

085002-6 DYNAMICAL SPACETIME SYMMETRY PHYSICAL REVIEW D 93, 085002 (2016) ~ and we note that the second involution, for ωa, is simulta- integrable. In general, this is not the case. If we substitute a a a neously expressed in adapted coordinates yα. It is straight- dx → κ and dza → λ in the structure equations and α a a a forward to check both directly and in terms of the ðx ;yβÞ solve for dκ and dλ , they are not involute. For dκ , for inner products that hduα; duβi¼0. example, we find

2dκa κc ∧ ω~ a κa ∧ ω~ − acd κb ω~ ∧ κa IV. PHASE SPACE ¼ c þ b bcb þ aeω~ c ∧ κb − cbλ ∧ ω~ a − ω~ ∧ abλ The full biconformal space is a symplectic manifold, but þ b e bcb b b c b b d abλ abω~ c ∧ λ abλ ∧ ω~ unlike our usual basis for phase spaces, the inner product of þ b b þ b b c þ b b : ~ a ~ the ðω ; ωaÞ basis is off diagonal, Eq. (12). To make a full a connection between the background biconformal manifold Therefore, in order for dx and dza to span a pair of and a one-particle phase space, we introduce a further Lagrangian submanifolds, we require the λa ∧ λb terms to change of basis. vanish. Equivalently, we must have

Λa ≡ − cbλ ∧ ω~ a − ω~ ∧ abλ d abλ abω~ c ∧ λ A. An orthogonal basis of Lagrangian submanifolds b b c b b þ b b þ b b c ab ~ Having both metric and symplectic form, we may find a þ b λb ∧ ω subspace orthogonal to the spacetime manifold. To find it, b a a a we introduce new coordinates za such that hdza; dx i¼0. linear in κ , with a similar condition Σ ∼ λ arising for the d ξ bd Ξ d b Making the ansatz za ¼ a yb þ ab x , we solve for involution of dλb. b ξa and Ξab As we show in detail in Appendix B, these conditions do hold for the homogeneous solution above—substituting c Λa Σa 0 ¼hdza; dx i the form of the connection into and so in the α model considered ðx ;zβÞ do characterize Lagrangian ¼ ξ bδc þ Ξ bbc: a b ab submanifolds. c c Restricting to the constant zα submanifold, the solution is This is satisfied if ξa ¼ δa and Ξab ¼ −bab, giving

b ω~ a ¼ −2Δacxdb dxe dza ¼ dya − babdx b db ce 1 ω~ a d a for the differential of the coordinates. ¼ 2 x These subspaces are better aligned with our usual notion 1 1 2 of a single particle phase space. In addition to the inner ω~ ¼ x dxa ¼ d ln x x2 a 2 product of the spacetime coordinate differentials dxa; dxb bab, and orthogonality of the spacetime ~ ~ c h i¼ with metric 4bab and SOð4Þ connection and ωa ¼ 2bacω , and momentum directions, we have a metric on the as before. If instead, we hold xα constant, then the momentum space as well, solution is

c d hdza; dzbi¼hdya − bacdx ; dyb − bbddx i ω~ a −2Δac dd b ¼ dbx0 zc ¼ −b : ~ ab ωa ¼ dza α ω~ ad In terms of the ðx ;zβÞ coordinates, the solution for the ¼ x0 za connection is ~ a 1 ab ab c with ω ¼ − 2 b dzb, constant metric −b ðx0Þ and con- ω~ a −2Δac d d d e SO 4 b ¼ dbx ð zc þ bce x Þ stant ð Þ connection. 1 While the constancy of the SOð4Þ connection does not ω~ a d a − abd ¼ 2 ð x b zbÞ imply vanishing of the Yang-Mills field, it may be gauged to zero by a conformal transformation. The constancy of the ω~ d d c a ¼ za þ bac x metric, SOð4Þ connection, and Weyl vector show the 1 momentum space to be a Lagrangian submanifold with ω~ ¼ xadz þ x dxa: a x2 a vanishing curvature and vanishing Yang-Mills field. It is therefore consistent to identify the entire biconformal α However, in order for ðx ;zβÞ coordinates to fully match manifold with the cotangent bundle. our expectations for canonically conjugate variables, space- It is unclear which of these properties survive in curved time and momentum space must be Lagrangian submani- biconformal spaces. Torsion-free biconformal spaces are α folds. Therefore the differentials dx and dzα need to be known to be fully determined by the spacetime solder form

085002-7 BENJAMIN C. LOVELADY and JAMES T. WHEELER PHYSICAL REVIEW D 93, 085002 (2016) and Weyl vector with ωa spanning the cotangent spaces, We outlined the quotient manifold method and applied it but the solutions apply to different submanifolds than we to the biconformal gauging of the conformal group. By consider in this section. Still, it is conceivable that the starting from the generators, we constructed the Maurer- cotangent interpretation is always possible. Cartan structure equations for the conformal group. The It is amusing to speculate on the meaning of the known solution to these equations was introduced, though momentum space if it is found to be curved in some we choose to use different coordinates better adapted to the solutions. If so, it might provide a novel approach to Lagrangian submanifolds. canonical field theory by allowing canonically conjugate Our principal contribution was to identify a comple- fields to coexist on what is essentially a particle phase mentary pair of Lagrangian submanifolds on which the space. Expressing a relativistic field theory on a particle connection (of the principal fiber bundle) is orthogonal phase space has had only measured success. Born [17] [SOðnÞ] while the restriction of the Killing metric is suggested introducing a curved momentum space to Lorentzian. The Lorentzian metric allows calculation of complement gravitating spacetime, but with no clear its Christoffel connection and curvature, while the SOðnÞ indication of what would determine its curvature. In connection gives rise to a Yang-Mills field (trivial, since we the nonrelativistic case, the Wigner distribution extends only consider the flat case). The Riemannian curvature of the wave function to a distribution on phase space, but it the submanifolds was found to be constructible from is not obvious how this generalizes to the relativistic projection operators orthogonal to the time direction. case. Typically, phase space for field theory employs a Notice that the time direction emerges dynamically in natural symplectic structure on field space, but what accordance with the signature theorem of [13] and further occurs here seems to exist midway between the particle developed in [12]. and field cases. The symplectic base manifold allows By the definition of a fiber bundle, the SOðnÞ fibers are fields to acquire a momentum component automatically, in a direct product with the SOðn − 1; 1Þ symmetry of the and the field equations determine the structure of the base manifold. This direct product relation naturally entire phase space, apparently restricting these fields so satisfies the Coleman-Mandula theorem. that the only independent degrees of freedom are those We showed that the full biconformal space may be from the spacetime Lagrangian submanifold. given the structure of a cotangent bundle to the curved spacetime. V. CONCLUSION Preliminary work with curved biconformal spaces sug- gests that these results extend to those cases as well, though The Coleman-Mandula theorem shows that unifying the computations pose some interesting challenges. Since gauge theories that include gravity and are based on Lie these constructions depend only on the Lie algebra they groups require a direct product between the internal and will work in the same way with spinor representations. gravitational symmetries. This seemed to be an unnatural There is no obstruction to considering supersymmetric starting point for a unified theory and led to an increased generalizations [18], which among other benefits provide a emphasis on supersymmetric theories. These avoid being principled way of introducing spinor fields into bosonic direct products by allowing graded Lie groups, which in field theories. turn give improved quantum convergence and useful restrictions on possible models. APPENDIX A: VANISHING OF THE In the present work, we show that it is possible to write a SUBMANIFOLD YANG-MILLS FIELD unified theory as a gauge theory of a simple Lie group which dynamically enforces the Coleman-Mandula theo- The SOðnÞ Yang-Mills field strength is rem. Although the starting point is the conformal group Fa −Δad cf d ∧ d 2 d e ∧ d SOðn − 1; 1Þ of Euclidean n-space, the general solution of b ¼ cbb ð zd zf þ bde x zfÞ; the Maurer-Cartan structure equations shows that the connection retains its original SOðnÞ symmetry but the which vanishes on the za ¼ constant Lagrangian sub- Killing metric (which is nondegenerate in these models) manifold. To verify this explicitly, we substitute the a submanifold form of the connection, ω~ j 0 ¼ restricts to a Lorentzian signature on certain Lagrangian b za¼za − 4 Δac d d Fa dω~ a − ω~ c ∧ ω~ a submanifolds. x2 dbxc x into b ¼ b b c:

085002-8 DYNAMICAL SPACETIME SYMMETRY PHYSICAL REVIEW D 93, 085002 (2016) 4 4 4 Fa ¼ d − Δacx dxd − Δce x dxd ∧ Δafx dxg b x2 db c x2 db e x2 gc f 8 4 16 ac e d ac d ce af d g ¼ Δ x x dx ∧ dx − Δ dx ∧ dx − Δ Δgc x x dx ∧ dx ðx2Þ2 db e c x2 db c ðx2Þ2 db e f 8 4 4 ac e d ac e d c e ce a f af d g ¼ Δ x x dx ∧ dx − Δ δ dx ∧ dx − ðδ δ − δ δ Þðδ δc − δ δ Þx x dx ∧ dx ðx2Þ2 db e c x2 db ce ðx2Þ2 d b bd g gc e f 8 1 4 ¼ Δac x x − x2δ dxe ∧ dxd − ðδax x þ δ x xa − x2δ δaÞdxd ∧ dxe ðx2Þ2 db e c 2 ce ðx2Þ2 e b d bd e bd e 8 1 4 1 1 ¼ Δac x x − x2δ dxe ∧ dxd − x x − x2δ δa þ xax − x2δa δ dxd ∧ dxe ðx2Þ2 db e c 2 ce ðx2Þ2 b d 2 bd e e 2 e bd 8 1 4 1 ¼ Δac x x − x2δ dxe ∧ dxd − x x − x2δ ðδcδa − δacδ Þdxd ∧ dxe ðx2Þ2 db e c 2 ce ðx2Þ2 c d 2 cd b e be 4 4 ¼ Δacð2x x − x2δ Þdxe ∧ dxd − Δacð2x x − x2δ Þdxd ∧ dxe ðx2Þ2 db e c ce ðx2Þ2 eb c d cd 4 4 ¼ Δacð2x x − x2δ Þdxe ∧ dxd − Δacð2x x − x2δ Þdxe ∧ dxd ðx2Þ2 db e c ce ðx2Þ2 db c e cd ¼ 0:

If we write this instead in the form of an alternative basis, APPENDIX B: INVOLUTION OF THE 1 ORTHOGONAL SUBSPACES ~ a a ab ω ¼ ðκ − b λbÞ a 2 We find the involution conditions for the ðdx ; dubÞ ω~ λ κb basis and show that they are satisfied by the homogeneous a ¼ a þ bab solution. a The ðdx ; dubÞ basis is related to the original basis and substitute into the original structure equations, by ~ a ~ c ~ a ~ ~ a dω ¼ ω ∧ ω c þ ω ∧ ω 1 ω~ a d a − abd dω~ ω~ c ∧ ω~ ω~ ∧ ω~ ¼ 2 ð x b ubÞ a ¼ a c þ a ω~ d d b a ¼ ua þ bab x : we find the structure equations for κa and λa,

1 dκa κc ∧ ω~ a κa ∧ ω~ − acd κb ω~ ∧ κa aeω~ c ∧ κb ¼ 2 ð c þ b bcb þ þ b e bcb Þ 1 −bcbλ ∧ ω~ a − ω~ ∧ babλ dbabλ babω~ c ∧ λ babλ ∧ ω~ þ 2 ð b c b þ b þ b c þ b Þ 1 dλ cbλ ∧ ω~ d ω~ ∧ λ − d cbλ ω~ c ∧ λ λ ∧ ω~ a ¼ 2 ðbadb b c þ a bac b b þ a c þ a Þ 1 ω~ c ∧ κb κc ∧ ω~ − d κb − κc ∧ ω~ b − ω~ ∧ κb þ 2 ð a bcb þ bac bab bab c bab Þ:

a ae ~ c b a ~ ac b c ~ a Therefore, involution of the new basis requires Σ ≡ b ω e ∧ bcbκ þ κ ∧ ω − b dbcbκ − κ ∧ ω c − ω~ ∧ κa Λa ≡ − cbλ ∧ ω~ a − ω~ ∧ abλ d abλ abω~ c ∧ λ b b c b b þ b b þ b b c ab e λa þ b λb ∧ ω to be proportional to . The involution does not generically hold, but it only needs to hold for the solution. Expressing the solution in a a to be proportional to κ and terms of ðκ ; λbÞ,

085002-9 BENJAMIN C. LOVELADY and JAMES T. WHEELER PHYSICAL REVIEW D 93, 085002 (2016) 2 Dropping the irrelevant κa terms (since only λ ∧ λ terms ω~ a ¼ −2Δac xddu − b xddxf þ x dxd a b b db c cf x2 c violate the involution), 2 ¼ −2Δac xdλ − b xdκf þ x κd Λa ≅ babð−2Δce xdλ Þ ∧ λ þ bcbλ ∧ 2Δaeðxdλ Þ db c cf x2 c db e c b dc e − 2 eλ d 2 ∧ abλ d abλ 1 ðx e þ ðln x ÞÞ b b þ b b: ω~ a κa − abλ ¼ 2 ð b bÞ Also, since b depends only on xa, the derivatives db ω~ λ κb ab ab a ¼ a þ bab depend only on κa and may be dropped, ω~ ¼ xaðλ − b κbÞþdðln x2Þ a ab Λa ≅ bc2Δae d ab2Δce d 2 e ac λ ∧ λ ðb dbx þ b dbx þ x b Þ c e Λa bc δaδe − δaeδ d ab δcδe − δceδ d substitution into gives ¼ðb ð d b bdÞx þ b ð d b bdÞx 2xebac λ ∧ λ Λa ≡ abω~ c ∧ λ − cbλ ∧ ω~ a − 2ω~ ∧ abλ d abλ þ Þ c e b b c b b c b b þ b b ac e − bcδae λ ∧ λ 2 ¼ðb x xbb Þ c e ab −2Δce dλ − dκf κd ∧ λ ¼ b x e befx þ 2 xe c 2 a c e − 2 δac e δae c λ ∧ λ db x ¼ð x x x x ð x þ x ÞÞ c e 2 0: cb ae d d f d ¼ − b λb ∧ −2Δ x λe − befx κ þ xeκ κa dc x2 Therefore, is in involution for this solution. For the involution of λa we require Σa to be linear in λa. − 2 e λ − κf d 2 ∧ abλ d abλ ðx ð e bef Þþ ðln x ÞÞ b b þ b b: In fact, it vanishes identically:

Σa ≡ baeω~ c ∧ b κb þ κa ∧ ω~ − bacdb κb − κc ∧ ω~ a − ω~ ∧ κa e cb cb c 2 2 ¼ bae −2Δcg −b xdκf þ x κd ∧ b κb − κc ∧ −2Δab −b xdκf þ x κd de gf x2 g cb dc bf x2 b a a b 2 ac b þ 2κ ∧ ðx ð−babκ Þþdðln x ÞÞ − b dbcbκ 1 ¼ ðx2δax − xað4x x − 2x x − 2x x þ x2δ Þþ4xax x − 2x2δax Þκf ∧ κb ðx2Þ2 f b f b b f b f bf b f f b 1 2 1 2 þ ð2xax x − δax x2Þκc ∧ κf þ x κc ∧ κa − 2 x x2κa ∧ κb þ 2x κa ∧ dxc ðx2Þ2 c f f c x2 c ðx2Þ2 b x2 c 1 þ ð2x2x δa þ 2xa2x x − 2xaδ x2 − 4x x xa þ 2δax x2Þdxcκb ðx2Þ2 b c b c bc b c b c 1 1 2 2 1 ¼ − δax κf ∧ κc þ δax κf ∧ κc þ x κc ∧ κa þ 2x κa ∧ dxc − 2 x x2κa ∧ κb x2 f c x2 f c x2 c x2 c ðx2Þ2 b 1 þ ð2x2x δa þ 2xa2x x − 2xaδ x2 − 4x x xa þ 2δax x2Þdxcκb ðx2Þ2 b c b c bc b c b c 1 ¼ ð2δax þ 2δax Þκcκb x2 c b b c ¼ 0

b b where we use dx ¼ κ in the last step. When we hold ua constant The description of the spacetime and momentum 2 submanifolds follow from the solution, ω~ a ¼ 2Δac b xddxf − x dxd b db cf x2 c 1 2 ω~ a d a ω~ a ¼ −2Δac xddu − b xddxf þ x dxd ¼ x b db c cf x2 c 2 1 1 ω~ d 2 ω~ a d a − abd ¼ 2 ðln x Þ ¼ 2 ð x b ubÞ ~ b with ω~ b dxb. The submanifold is clearly Lagrangian, ωa ¼ dua þ babdx a ¼ ab the connection SOð4Þ, and inner product of the basis forms ω~ a d − d b d 2 ¼ x ð ua bab x Þþ ðln x Þ: is Lorentzian,

085002-10 DYNAMICAL SPACETIME SYMMETRY PHYSICAL REVIEW D 93, 085002 (2016) 1 ω~ ; ω~ du ; du hω~ a; ω~ bi¼ hdxa; dxbi h a bi¼h a bi 4 α 1 ¼ babðx0Þ: ab ¼ 4 b : Since the metric is now constant, the momentum submani- a a fold is flat. The Yang-Mills field strength has the form On the momentum submanifold, with x ¼ x0 constant,

ω~ a −2Δac dd Fa ¼ dω~ a − ω~ c ∧ ω~ a b ¼ dbx0 uc b b b c ~ −4Δec Δaf d gd ∧ d ωa ¼ dua ¼ db ge x0x0 uc uf ω~ ad −Δac dfd ∧ d ¼ x0 ua ¼ bdb0 uc uf;

~ a 1 ab α where ω ¼ − 2 b ðx0Þdub. This is again Lagrangian with which is of the form of a pure rescaling and can therefore, be α inner product made to vanish by a conformal transformation, ϕ ¼ −x0uα.

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