Gauge Theories of Spacetime Symmetries

Home , Spacetime symmetries

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Gauge theories of spacetime symmetries

Friedemann Brandt Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22-26, D-04103 Leipzig, Germany; Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, D-14476 Golm, Germany Gauge theories of conformal spacetime symmetries are presented which merge features of Yang- Mills theory and general relativity in a new way. The models are local but nonpolynomial in the gauge fields, with a nonpolynomial structure that can be elegantly written in terms of a metric (or vielbein) composed of the gauge fields. General relativity itself emerges from the construction as a gauge theory of spacetime translations. The role of the models within a general classification of consistent interactions of gauge fields is discussed as well.

I. INTRODUCTION AND CONCLUSION on conformal Killing vector fields, and cast the models in more conventional form. In particular, the stan- In this work new gauge theories of conformal space- dard formulation of general relativity arises in this way time symmetries are constructed which merge features of through field redefinitions which trade metric or vielbein Yang-Mills theories and general relativity in an interest- variables for gauge fields of translations. It is possible, ing way. This concerns both the Lagrangians and the and quite likely, that the (nonsupersymmetric version of) gauge transformations of these models. The Lagrangians models constructed in [1–13] can be reproduced analo- are local but nonpolynomial in the gauge fields, as gen- gously. However, it seems to be impossible to eliminate eral relativistic Lagrangians are local but nonpolynomial the dependence on conformal Killing vector fields in a in the gravitational (metric or vielbein) fields. In fact, generic model constructed here. they are formally very similar to general relativistic La- The models are not only interesting for their own sake, grangians, except that the metric and vielbein are poly- but also in the context of a systematic classification of nomials in the conformal gauge fields, cf. eqs. (25), (38). consistent interactions of gauge fields in general, which is Moreover, general relativity itself emerges from the con- quite a challenging problem and partly motivated this struction as a gauge theory of spacetime translations (see work. Such a classification was started in [16,17] us- section VI). ing the BRST cohomological approach to consistent de- The (infinitesimal) conformal gauge transformations formations of gauge theories [18]. The starting point contain a Yang-Mills type transformation and a general of that investigation was the free Maxwell Lagrangian (0) A µνA coordinate transformation, with the remarkable property L = (1/4) A Fµν F for a set of vector gauge A− that both parts are tied to each other by the fact that fields Aµ in flatP spacetime. In the deformation approach they involve the same gauge parameter fields, cf. eq. (23). one asks whether the action and its gauge symmetries This unites the symmetry principles of Yang-Mills the- can be nontrivially deformed, using an expansion in de- ory and general relativity in an interesting way and re- formation parameters. flects that the models are gauge theories of spacetime In [16,17] complete results were derived for Poincaré symmetries in a very direct sense. The latter also mani- invariant deformations of the free Maxwell Lagrangian fests itself in the explicit dependence of the Lagrangians to first order in the deformation parameters. The result and the gauge transformations on conformal spacetime is that the most general first order deformation which Killing vector fields. This, among other things, distin- is invariant under the standard Poincaré transforma- guishes the models presented here from gauge theories tions contains at most four types of nontrivial interac- of conformal symmetries constructed in the past, such tion vertices: (i) polynomials in the field strengths and as supergravity models [1–13], or, more recently, models their first or higher order derivatives; (ii) Chern-Simons presented in Refs. [14,15]. vertices of the form A F ... F (present only in ∧ ∧ ∧ At this point a comment seems to be in order. Partic- odd spacetime dimensions); (iii) cubic interaction ver- A B µνC ular models constructed in this work admit field redefi- tices fABCAµ Aν F where fABC = f[ABC] are anti- nitions (of fields that occur in the action, and of gauge symmetric constant coefficients; (iv) vertices of the form µ µ parameter fields) which completely remove the explicit Aµj where j is a gauge invariant Noether current of dependence of the Lagrangian and gauge transformations

1 the free theory1. First order deformations which are not standard Poincaré transformations because they involve required to be Poincaré invariant were also investigated. gauge invariant Noether currents of spacetime symme- The results are similar, apart from a few (partly unset- tries themselves. Such vertices exist in all spacetime di- tled) details (cf. comments at the end of section 13.2 in mensions because there is a gauge invariant form of the [17]). Noether currents of the Poincaré symmetries [27,28]. The Self-interacting theories for vector gauge fields with in- corresponding deformations of the gauge transformations teraction vertices (i), (ii) or (iii) are very well known. incorporate Poincaré symmetries in the deformed gauge Those of type (i) occur, for instance, in the Euler- transformations. This promotes global Poincaré symme- Heisenberg Lagrangian [19] or the Born-Infeld theory tries to local ones, yielding gauge theories of Poincaré [20]. Lately, vertices (i) which are not Lorentz invari- symmetries. In four-dimensional spacetime, the con- ant attracted attention in the context of so-called non- struction can be extended to the remaining conformal commutative U(1) gauge theory because the interactions transformations because dilatations and special confor- in that model can be written as an infinite sum of such mal transformations also give rise to vertices (iv).2 For vertices by means of a field redefinition (“Seiberg-Witten this reason I shall focus on models in four-dimensional map”) [21] (field redefinitions of this type are automati- spacetime; however, all formulas are also valid in all other cally taken care of by the BRST cohomological approach: dimensions when restricted to gauge theories of Poincaré two deformations related by such a field redefinition are symmetries, see section VI. equivalent in that approach). From the deformation The organization of the paper is the following. Sec- point of view, vertices (i) and (ii) are somewhat less in- tion II treats a relatively simple example with only one teresting because they are gauge invariant [in case (ii) gauge field and one vertex (iv) involving a Noether cur- modulo a total derivative] under the original gauge trans- rent of a conformal symmetry in four-dimensional space- formations of Maxwell theory. time. This results in a prototype model with just one In contrast, vertices (iii) and (iv) are not gauge invari- conformal gauge symmetry. In section III the prototype ant under the gauge transformations of the free model; model is rewritten by casting its gauge transformations rather, they are invariant only on-shell (in the free model) in a more suitable form and introducing a gauge field modulo a total derivative and therefore they give rise dependent “metric”. This paves the road for the gener- to nontrivial deformations of the gauge transformations. alization of the prototype model in section IV where four- This makes them particularly interesting. Interaction dimensional gauge theories of the full conformal algebra vertices (iii) are of course well known: they are encoun- or any of its subalgebras are constructed. These models tered in Yang-Mills theories [22,23] and lead to a non- involve not only first order interaction vertices (iv) but Abelian deformation of the commutator algebra of gauge in addition also Yang-Mills type interaction vertices (iii) transformations. But what about vertices (iv)? Such ver- because in general the involved conformal symmetries do tices are familiar from the coupling of vector gauge fields not commute. Then, in section V, the construction is fur- to matter fields, such as the coupling of the electromag- ther extended by including other fields (matter fields and µ ¯ µ netic gauge field Aµ to a fermion current j = ψγ ψ, but gauge fields). Section VI explains the relation to general what do we know about vertices involving gauge invari- relativity. ant currents made up of the gauge fields themselves? As a matter of fact, it depends on the spacetime dimension whether or not Poincaré invariant vertices (iv) II. PROTOTYPE MODEL are present at all. In three dimensions such vertices exist and occur in 3-dimensional Freedman-Townsend models Let us first examine deformations of the Maxwell ac- [24,25]. In contrast, they do not exist in four dimensions tion for only one gauge field Aµ, because Maxwell theory in four dimensions has no symmetry that gives rise to a Noether current needed for a 1 S(0) = d4xF F µν (1) Poincaré invariant vertex (iv) (this follows from the re- −4 Z µν sults of [26]). It is likely, though not proved, that this result in four dimensions extends to higher dimensions. where Fµν = ∂µAν ∂ν Aµ is the standard Abelian field − However, it must be kept in mind that this result on strength and indices µ are raised with the Minkowski vertices (iv) in four dimensions concerns only Poincaré invariant interactions. The new gauge theories constructed here contain vertices (iv) that are not invariant under the 2There are infinitely many additional vertices (iv) that are not Poincaré invariant because free Maxwell theory has infinitely many inequivalent Noether currents [29–32]. They 1Note the difference from vertices (iii): the latter are also of are not related to spacetime symmetries. I did not investi- A µ µ B µνC gate whether or not they also give rise to interesting gauge the form Aµ jA, but the currents jA = fABC Aν F are not gauge invariant. theories.

2 metric ηµν = diag(+, , , )[F µν = ηµρηνσF ]. Ac- view of (3), one defines a modified field strength Fˆ im- − − − ρσ µν tion (1) is invariant under the gauge transformations plicitly through the relations Fˆ = D A D A and µν µ ν − ν µ D A = ∂ A A ξρFˆ , solves these relations for Fˆ (0) µ ν µ ν − µ ρν µν δλ Aµ = ∂µλ (2) and finally constructs the action and gauge transforma- ˆ tions in terms of Fµν and Aµ. Both strategies work and and under global conformal transformations yield the same action and gauge transformations: ν δξAµ = ξ Fνµ (3) 1 ρ ˆ ˆµν L = (1 + ξ Aρ)Fµν F , (7) where ξµ is a conformal Killing vector field (no matter −4 ν ˆ which one) of flat four-dimensional spacetime,3 δλAµ = ∂µλ + λξ Fνµ (8)

1 with Fˆ given by ∂ ξ + ∂ ξ = η ∂ ξρ (ξ = η ξν ). (4) µν µ ν ν µ 2 µν ρ µ µν A ξρF A ξρF Fˆ = F µ ρν ν ρµ . (9) (3) is the gauge covariant form [27,28] of a conformal µν µν −σ − 1+ξ Aσ transformation and gives rise to the gauge invariant Noether current ˆ Fµν can be interpreted as the field strength for the gauge transformations (8) because its gauge transforma- µ ν µ µ 1 µ ρσ µρ j = ξ Tν ,Tν = δ FρσF + FνρF . (5) tion does not contain derivatives of λ: indeed, a straight- −4 ν forward, though somewhat lengthy, computation gives A first order deformation S(1) of action (1) that is of type λ (iv) and the corresponding first order deformation δ(1) of δ Fˆ = Fˆ (10) λ λ µν 1+ξσA ξ µν the gauge transformations (2) are σ L µ where ξ is the standard Lie derivative along ξ , (1) 4 µ (1) ν L S = d xAµj ,δλ Aµ = λξ Fνµ. (6) Z Fˆ = ξρ∂ Fˆ + ∂ ξρFˆ + ∂ ξρFˆ . Lξ µν ρ µν µ ρν ν µρ (1) (1) Indeed, it can be readily checked that S and δλ fulfill Using (10), as well as (4), it is easy to verify that the La- the first order invariance condition grangian (7) transforms under the gauge transformations

(0) (1) (1) (0) (8) into a total derivative, δλ S + δλ S =0. 1 µ ˆ ˆρν One may now proceed to higher orders. This amounts to δλL = ∂µ(λξ Fρν F ). (11) −4 (k) (k) looking for higher order terms S and δλ satisfying Furthermore, owing to (10), the algebra of the gauge k transformations (8) is obviously Abelian, i.e., two gauge (i) (k i) δλ S − =0,k=2, 3,... transformations with different parameter fields, denoted Xi=0 by λ and λ0, respectively, commute: It turns out that the deformation exists to all orders [δλ,δλ ]=0. (12) but that one obtains infinitely many terms giving rise 0 to a nonpolynomial structure. This calls for a more effi- I remark that, for notational convenience, I have sup- cient construction of the complete deformation. Let me pressed the gauge coupling constant (= deformation pa- briefly sketch two strategies, without going into details. rameter) in the formulas given above; it can be eas- The first one is a detour to a first order formulation: ily introduced in the usual way by substituting rescaled one casts the original free Lagrangian in first order form fields κAµ and κλ for Aµ and λ, respectively, and then µν (1/4)G (Gµν 2Fµν )whereGµν = Gνµ are auxil- dividing the Lagrangian by κ2. Expanding the result- − − iary fields, deforms this first order model analogously to ing action and gauge transformations in κ, one obtains (6), and finally eliminates the auxiliary fields. Another (0) (1) 2 (0) (1) 2 S = S + κS + O(κ )andδλ = δλ + κδλ + O(κ ) strategy is the use of a technique applied in [33,34]: in (1) (1) with S and δλ as in (6). This shows that (7) and (8) complete the first order deformation (6) to all orders. Note that the completion contains infinitely many terms

3 and is nonpolynomial but local in the gauge fields, as The construction is not restricted to flat spacetime but ap- promised. plies analogously to any fixed background metricg ˆµν with at least one conformal Killing vector field ξµ. Then (4) turns ˆ ρ into ξgˆµν =(1/2)ˆgµν Dρξ and subsequent formulas change accordingly.L

3 III. REFORMULATION OF THE PROTOTYPE δωAµ = ∂µω + εAµ. MODEL L Hence the gauge transformation of Aµ is the sum of a In the remainder of this work I shall first rewrite and standard Abelian gauge transformation with parameter then generalize the prototype model with the Lagrangian ω and a general coordinate transformation with parameters εµ [of course, these two transformations are related (7) and the gauge transformations (8). A surprising µ µ feature of the Lagrangian (7) is that its nonpolynomial because of ε = ωξ ]. As a consequence, the gauge transformation of Fµν is given just by the Lie derivative along structure can be written in terms of the “metric” µ ε , δωFµν = εFµν . (17) has the form of a general co- ρ ordinate transformationL of g with parameters εµ plus gµν = ηµν + ξµAν + ξν Aµ + ξρξ AµAν , (13) µν ρ a Weyl transformation with parameter (1/2)ω∂ρξ .As ν − where, again, ξµ = ηµν ξ . The inverse and determinant the Lagrangian is invariant under Weyl transformations of this metric are of gµν (we are still discussing the four-dimensional case), it transforms under gauge transformations δ just like a ξµAν + ξν Aµ A Aρξµξν ω gµν = ηµν + ρ , scalar density under general coordinate transformations 1+ξσA (1 + ξσA )2 µ µ − σ σ with parameters ε : δωL = ∂µ(ε L). This is exactly µ 2 µ µ µ ν det(gµν )= (1 + ξ Aµ) , equation (11), owing to ε = ωξ = λξ /(1 + ξ Aν ) − ν ˆ ˆµν and L/(1 + ξ Aν )= (1/4)Fµν F . A final remark on µ µν where A = η Aν . Using these formulas one readily the prototype model− is that the gauge transformations verifies that the Lagrangian (7) can be written as no longer commute when expressed in terms of ω rather than in terms of λ: 1 L = ggµρgνσF F (14) √ µν ρσ µ µ −4 [δ ,δ ]=δ ,ω00 = ω0ξ ∂ ω ωξ ∂ ω0. (19) ω ω0 ω00 µ − µ 1/2 where Fµν = ∂µAν ∂ν Aµ and √g = det(gµν ) = The reason for this is that the redefinition (16) of the 1+ξµA (assuming− 1 + ξµA > 0). Furthermore,| | it can µ µ gauge parameter field involves the gauge field Aµ. be easily checked that the gauge transformations (8) can be rewritten as

ν ν IV. GENERALIZATION δωAµ = ∂µω + ωξ ∂ν Aµ + ∂µ(ωξ )Aν (15)

µ The prototype model found above will now be general- where ω is constructed of λ, ξ and Aµ according to ized by gauging more than only one conformal symmetry λ in four-dimensional flat spacetime. Let be the Lie alge- ω = . (16) G µ bra of the full conformal group or any of its subalgebras. 1+ξ Aµ Let us pick a basis of and label its elements by an in- (15) is exactly the same transformation as (8), but writ- dex A [since the conformalG group in four dimensions is ten in terms of ω instead of λ. Since λ was completely 15-dimensional, we have A =1,...,N with 1 N 15]. arbitrary, ω is also completely arbitrary, and can thus be The corresponding set of conformal Killing vector≤ ≤ fields µ used as gauge parameter field in place of λ.Notethat is denoted by ξA . Since is a Lie algebra, one can (15) is polynomial in the gauge fields, in contrast to (8). choose the ξ’s such{ } that G To understand the gauge invariance of the model, and ξν ∂ ξµ ξν ∂ ξµ = f C ξµ (20) to generalize it subsequently, the following observation A ν B − B ν A BA C is crucial: under the gauge transformations (15) of the C gauge fields, the metric (13) transforms according to where fAB are the structure constants of in the cho- A G sen basis. I associate one gauge field Aµ and one gauge 1 A δ g = g g ω∂ ξρ, (17) parameter field ω with each element of and introduce ω µν Lε µν − 2 µν ρ the following generalization of the gaugeG transformations (15): where g is the Lie derivative of g along εµ = ωξµ: Lε µν µν δ AA = D ωA + ωBξν ∂ AA + ∂ (ωBξν )AA (21) g = ερ∂ g + ∂ ερg + ∂ ερg , ω µ µ B ν µ µ B ν Lε µν ρ µν µ ρν ν µρ εµ = ωξµ. (18) where

A A B A C In order to verify equation (17), one has to use the con- Dµω = ∂µω + Aµ fBC ω . (22) formal Killing vector equations (4). Equations (15) and A A (17) make it now easy to understand the gauge invari- The part Dµω of δωAµ is familiar from Yang-Mills the- A ance of the action with Lagrangian (14). Note that the ory; the remaining part is the Lie derivative of Aµ along last two terms on the right hand side of (15) are nothing a vector ﬁeld εµ containing the gauge parameter ﬁelds but the Lie derivative A of A along εµ: ωA, Lε µ µ

4 A A A µ B µ A δωAµ = Dµω + εAµ ,ε= ω ξB . (23) The second equation in (32) expresses that the EB are L the entries of a matrix E which inverts the matrix 1 + The commutator of two gauge transformations is µ A M where M is the matrix with entries ξB Aµ . E can thus be written as an infinite (geometric) series of matrix [δ ,δ ]=δ , ω ω0 ω00 products of M: A B C A B µ A B µ A ω00 = ω ω0 fBC + ω0 ξB ∂µω ω ξB∂µω0 . (24) − ∞ E = ( M)k,MA ξµ AA. (33) The crucial step for constructing an action which is in- − B ≡ B µ variant under these gauge transformations is the follow- kX=0 ing generalization of the prototype metric (13): A gauge coupling constant κ can be introduced as before A A A A A A ρ A B by means of the substitutions Aµ κAµ , ω κω , gµν = ηµν + ξAµAν + ξAν Aµ + ξAρξBAµ Aν , (25) 2 → C → C L L/κ . Equivalently, one may use fAB κfAB , µ→ µ → with ξ = η ξν . This metric behaves under gauge ξ κξ . Of course, the zeroth order Lagrangian is Aµ µν A A → A transformations (21) similarly as the prototype metric positive definite only for appropriate choices of .For G (13) under gauge transformations (15): instance, one may choose a that is Abelian or compact; then there is a basis ofG such that d = δ . G AB AB 1 A ρ The simplest case is a one-dimensional and reproduces δωgµν = εgµν gµν ω ∂ρξ , (26) G L − 2 A the prototype model. Choices such as = so(2, 4) (full G µ conformal algebra) or = so(1, 3) (Lorentz algebra) do with ε as in (23). To verify (26), one has to use (4) not give a positive definiteG zeroth order Lagrangian be- (which holds for each ξµ ) and (20). Note that (21) is A cause these algebras are not compact (one cannot achieve the sum of a Yang-Mills gauge transformation with pa- d = δ ). rameter fields ωB and a general coordinate transforma- AB AB µ B µ tion with parameter fields ε = ω ξB, while (26) has the form of a general coordinate transformation with pa- V. INCLUSION OF MATTER FIELDS AND µ rameters ε plus a Weyl transformation with parameter FURTHER GAUGE FIELDS A ρ (1/2)ω ∂ρξA. This immediately implies that the following− Lagrangian is invariant modulo a total derivative Using the metric (25), it is straightforward to extend under gauge transformations (21): the models of the previous section so as to include further 1 fields. First I discuss the case of just one (real) scalar field L = √ggµρgνσF A F B d , (27) −4 µν ρσ AB φ and introduce the gauge transformation 1 where dAB is a symmetric -invariant tensor, δ φ = ωAξµ ∂ φ + φωA∂ ξµ . (34) G ω A µ 4 µ A d = d ,f Dd + f Dd =0, (28) AB BA CA DB CB AD A contribution to the Lagrangian which is gauge invari- A ant modulo a total derivative is and the Fµν are field strengths familiar from Yang-Mills theory: 1 µν 1 2 Lφ = √gg ∂µφ∂ν φ √gRφ (35) A A A A B C 2 − 12 Fµν = ∂µAν ∂ν Aµ + fBC Aµ Aν . (29) − µν with gµν and g as before in (25) and (31), and R the Owing to (28), the Lagrangian (27) is invariant under Riemannian curvature scalar built from gµν , A Yang-Mills transformations of the Fµν . Furthermore it is µν ρ invariant under Weyl transformations of gµν . Hence, it R = g Rµρν , transforms under gauge transformations (21) just like a R σ = ∂ Γ σ +Γ σΓ λ (µ ν), scalar density under general coordinate transformations µνρ µ νρ µλ νρ − ↔ 1 with parameters εµ = ωAξµ : Γ ρ = gρσ(∂ g + ∂ g ∂ g ). A µν 2 µ νσ ν µσ − σ µν A µ δωL = ∂µ(ω ξAL). (30) Using (26), one easily derives the gauge variation of R:

Again, the Lagrangian is local but nonpolynomial in the µ 1 A µ µν δ R = ε ∂ R + Rω ∂ ξ gauge fields because it contains the inverse metric g . ω µ 2 µ A The latter is 3 µν ρ A σ g (∂µ∂ν Γµν ∂ρ)(ω ∂σξA). (36) µν µν µ Âν ν Âµ µ ν Â ˆBρ −2 − g = η ξAA ξAA + ξAξB Aρ A , (31) − − This makes it is easy to verify the gauge invariance of Âµ µν Â where A = η Aν ,with (35): Lφ transforms as a scalar density under standard general coordinate transformations of gµν and φ;there- Â B A C A µ A A Aµ = Aµ EB ,EB (δC + ξC Aµ )=δB. (32) fore the first term in (34) and the first term in (26) make a

5 µ νρ νσ ρ ρ contribution ∂µ(ε Lφ)toδωLφ; the second terms in (34) where C = η Cσ with Cσ as in (41). I denote a and (26) contribute a total derivative to δωLφ because Lφ fermion ﬁeld by ψ (without displaying its spinor indices), is invariant modulo a total derivative under Weyl trans- and introduce the gauge transformations formations of g and φ with weights of ratio 2 (in four µν − 1 3 dimensions). The complete transformation reads δ ψ = ωAξµ ∂ ψ Cµν σ ψ + ψωA∂ ξµ (45) ω A µ − 2 µν 8 µ A 1 δ L = ∂ ωAξµ L + √ggµν φ2∂ (ωA∂ ξρ ) . (37) where 4σ is the commutator of γ-matrices, using the ω φ µ A φ 8 ν ρ A µν h i conventions To include fermions, I introduce the “vierbein” γµγν + γν γµ =2ηµν , ν ν ν A 1 eµ = δµ + ξAAµ . (38) σ = (γ γ γ γ ),γ= η γν . µν 4 µ ν − ν µ µ µν ν The term vierbein is used because eµ is related to the A contribution to the Lagrangian which is invariant mod- “metric” (25) through ulo a total derivative under the gauge transformations ρ σ (21) and (45) is gµν = ηρσ eµ eν . (39) 1 Furthermore the vierbein transforms under the gauge L =i√g ψγ¯ ν E µ(∂ ψ + ω σρσ ψ). (46) ψ ν µ 2 µ σρ transformations (21) according to Lψ transforms under the gauge transformations like a ν ρ ν ρ ν δωeµ = ε ∂ρeµ + ∂µε eρ scalar density under general coordinate transformations 1 with parameters εµ = ωAξµ because the “Lorentz” and +C ν e ρ e ν ωA∂ ξρ (40) A ρ µ − 4 µ ρ A “Weyl” parts of the gauge transformation of the fermion, vierbein and spin connection cancel each other com- µ with ε as in (23) and pletely,

ν 1 A ν νσ ρ A µ C = ω (∂ ξ η η ∂ ξ ). (41) δωLψ = ∂µ(ω ξ Lψ). (47) µ −2 µ A − µρ σ A A The inclusion of standard Yang-Mills gauge fields AI is Note that (40) has indeed the familiar form of the trans- µ even simpler: the contribution to the Lagrangian is just formation of vierbein fields in general relativity: the ν the standard Yang-Mills Lagrangian in the metric (25), lower index of eµ transforms as a “world index” (it sees only the general coordinate transformation with param- 1 µρ νσ I J eters εµ) while the upper index transforms as a “Lorentz LYM = √gg g F F dIJ, (48) −4 µν ρσ index” (it sees only “Lorentz transformations with pa- I I I I J K ν µν Fµν = ∂µAν ∂ν Aµ + fJK AµAν (49) rameters Cµ ” – the Lorentz character is due to C = − νµ µν µρ ν C where C = η Cρ ). In addition (40) contains I − A ρ where fJK and dIJ are the structure constants and a Weyl transformation with parameter (1/2)ω ∂ρξA.I now define a “spin connection” ω νρ: − an invariant symmetric tensor of some Lie algebra YM. µ Note that the difference from (27) is that now theG field I νρ ν ρ σκ λτ strengths F involve the gauge fields of YM while the ωµ = Eσ Eλ η η ωµκτ , µν G metric gµν is composed of the gauge fields of . The con- ωµνρ = ω[µν]ρ ω[νρ]µ + ω[ρµ]ν , I G − formal gauge transformations of Aµ are just the standard 1 σ λ λ µ A µ ω = e η (∂ e ∂ e ) (42) Lie derivatives along ε = ω ξA, [µν]ρ 2 ρ σλ µ ν − ν µ I B ν I B ν I ν ν ρ ρ δωAµ = ω ξB ∂ν Aµ + ∂µ(ω ξB)Aν . (50) where Eµ is the inverse vierbein (Eµ eν = δµ), Since L is invariant under Weyl transformations of ν ν Â ν YM Eµ = δ A ξ (43) µ − µ A gµν , it transforms under the conformal gauge transformations (21) and (50) like a scalar density under general Â νρ ν with Aµ as in (32). Since ωµ is constructed of eµ coordinate transformations with parameters εµ, in exactly the same manner as one constructs the spin connection of the vierbein in general relativity, one infers A µ δωLYM = ∂µ(ω ξALYM). (51) νρ from (40) that ωµ transforms under the gauge transformations (21) according to In addition LYM is invariant under the usual Yang-Mills I I J I K gauge transformations δαAµ = ∂µα + AµfJK α for νρ νρ σν ρ σρ ν δωωµ = ∂µC ωµ Cσ + ωµ Cσ arbitrary gauge parameter fields αI . σ − νρ σ νρ +ε ∂σωµ + ∂µε ωσ It is straightforward to construct further interaction 1 terms, such as √gφ4 or Yukawa-interactions √gφψψ¯ ,and + (e ρE ν e ν E ρ)ησλ∂ (ωA∂ ξτ ) (44) 4 µ σ − µ σ λ τ A to extend the construction to scalar fields or fermions

6 µ transforming nontrivially under YM. In fact, it is even ∂µξA =0. (57) possible to construct models whereG the “matter fields” transform under according to a nontrivial representa- When (57) holds, the gauge transformations δω are lo- tion. I shall onlyG discuss the case of scalar fields trans- cal Poincaré transformations. This raises the question of forming under a nontrivial representation of ; the exten- whether there is a relation to general relativity. The an- sion to fermions is straightforward. Of course,G the notion swer to this question is affirmative and easily obtained “scalar fields” should be used cautiously when these fields from the following observation: when (57) holds, the sit in a nontrivial representation of as they may or may “Einstein-Hilbert action” constructed from the metric not transform nontrivially under LorentzG transformations (25), (depending on the choice of and its representation). I 1 denote these “scalar fields”G by φi. The corresponding S = dnx √gR, (58) EH Z representation matrices of are denoted by T and cho- −2 G A sen such that they represent with the same structure is invariant under gauge transformations (21) because C G constants fAB as in (20), i.e., equation (26) reduces to a general coordinate transforma- µ A µ tion of gµν with parameters ε = ω ξ . Now, consider T i T k T i T k = f C T i . (52) A Ak Bj − Bk Aj AB Cj the special case of an action given just by (58) (without any additional terms), and assume that AA contains Further properties of the representation will not matter µ (at least) the gauge fields of all spacetime{ translations.} to the construction. In place of (34), the gauge transfor- Then we may interpret (25) as a field redefinition which mations now read just substitutes new fields gµν for certain combinations of i A i j A µ i 1 i A µ the original field variables. Since the action depends on δωφ = ω TAjφ + ω ξ ∂µφ + φ ω ∂µξ . (53) − A 4 A the gauge fields only via the new fields gµν , it reproduces the standard theory of pure gravitation as described by Accordingly, one introduces covariant derivatives general relativity. In fact, the argument is even more transparent when D φi = ∂ φi + AAT i φj. (54) µ µ µ Aj one works with the vielbein (38) rather than with the These covariant derivatives transform under gauge trans- metric (25) [according to (39), the metric can be written formations (21), (53) according to in terms of the vielbein, and thus action (58) can also be written in terms of the vielbein, as usual]. That is, we i A i j i δωDµφ = ω TAjDµφ + εDµφ may label the translations by an index ν and choose the − L µ µ 1 1 corresponding Killing vector fields as ξν = δν . Accord- + (D φi)ωA∂ ξν + φi∂ (ωA∂ ξν ) ingly, the gauge fields of translations are denoted by Aν . 4 µ ν A 4 µ ν A µ (38) may then be interpreted as a field redefinition that i ν i ν i µ ν ν where εDµφ = ε ∂ν Dµφ + ∂µε Dν φ with ε as in substitutes eµ for Aµ. This field redefinition is clearly (23). TheL generalization of the Lagrangian (35) is simply local and invertible (at least locally), as (38) can obvi- ν ν ously be solved for Aµ in terms of eµ and the gauge ˜ 1 µν i j 1 i j fields of Lorentz transformations. Lφ = √g g Dµφ Dν φ Rφ φ dij (55) h2 − 12 i The same argument applies when we add to the in- tegrand of (58) the first term of the matter Lagrangian where d is a -invariant symmetric tensor, ij G (35) (the second term is not needed since we consider only gauged Poincaré transformations here), the fermion d = d ,dT k + d T k =0. (56) ij ji kj Ai ik Aj Lagrangian (46) or the Yang-Mills type Lagrangian (48). Using (56) and arguments analogous to those that led to Since these contributions also depend on the gauge fields A ν (37), one infers that Aµ only via the eµ , the same field redefinition implies the equivalence to general relativity coupled to matter ˜ A µ 1 µν i j A ρ fields in the standard way. δωLφ = ∂µ ω ξ Lφ + √gg φ φ dij ∂ν (ω ∂ρξ ) . h A 8 A i Acknowledgements. The author is grateful to Sergei Kuzenko for valuable discussions and for suggesting for- mula (35), and to Marc Henneaux for pointing out pre- VI. RELATION TO GENERAL RELATIVITY vious work on gauge theories of conformal symmetries.

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