A fresh perspective on gauging the conformal group
aa r X i v : . [ g r- q c ] A ug A fresh perspective on gauging the conformal group
M.P. Hobson a) and A.N. Lasenby
1, 2, b) Astrophysics Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK (Dated: 20 August 2020)
We consider the construction of gauge theories of gravity that are invariant under local conformal transfor-mations. We first clarify the nature of global conformal transformations, in both their infinitesimal and finiteforms, and the consequences of global conformal invariance for field theories, before reconsidering existingapproaches for gauging the conformal group, namely auxiliary conformal gauge theory and biconformal gaugetheory, neither of which is generally accepted as a complete solution. We then demonstrate that, providedany matter fields belong to an irreducible representation of the Lorentz group, the recently proposed extendedWeyl gauge theory (eWGT) may be considered as an alternative method for gauging the conformal group,since eWGT is invariant under the full set of local conformal transformations, including inversions, as wellas possessing conservation laws that provide a natural local generalisation of those satisfied by field theorieswith global conformal invariance, and also having an ‘ungauged’ limit that corresponds to global conformaltransformations. By contrast, although standard Weyl gauge theory also enjoys the first of these properties,it does not share the other two, and so cannot be considered a valid gauge theory of the conformal group.PACS numbers: 04.50.Kd, 11.15.-q, 11.25.Hf
I. INTRODUCTION
In the classical description of a physical system, anyproperty has meaning only relative to the same propertyof some other reference system, and not in any absolutesense . Thus, any measurement corresponds to calcu-lating the ratio of two quantities with the same units.By using ‘natural units’, all physical quantities can beexpressed in terms of length, and so the description ofphysical systems should be invariant under the groupof transformations that leave the ratios of lengths un-changed, namely the global conformal transformations.These include Poincar´e transformations, which preservelength, together with global scale changes and specialconformal transformations (SCTs), all of which are con-nected to the identity, and so may be considered in theirinfinitesimal forms. In addition, conformal transforma-tions also include inversions, which are both finite anddiscrete, and hence excluded from the infinitesimal trans-formations.The freedom to make an arbitrary choice of units atany point in space and time further suggests that the de-scription of physical systems should, in fact, be invariantunder local conformal transformations, which thereforemotivates the study of gauging the conformal group. Thisis usually performed by considering only the infinitesimaltransformations, hence excluding inversions, and allow-ing the constant group parameters to become arbitraryfunctions of position. Field theories constructed to beinvariant under these local transformations are known asconformal gauge theories, and have been widely studiedsince the 1970s as potential modified gravity theories. a) Electronic mail: [email protected] b) Electronic mail: [email protected]
One finds, however, that this standard approach togauging the conformal group and the resulting classof auxiliary conformal gauge theories (ACGT) sufferfrom serious theoretical difficulties. Most notably, SCTsare not represented in the final structure of AGCT, sincethe corresponding gauge field can be algebraically elimi-nated from the theory. More precisely, it may be shownthat for any self-consistent ACGT action, the resultingfield equation for the SCT gauge field can be solved andsubstituted back into the action to obtain an effective ac-tion that is independent of this gauge field, which is thustermed an auxiliary field (hence the name for this classof theories). Thus, in this approach, it appears that thesymmetry reduces back to the local Weyl group.These difficulties have motivated an alternative ap-proach, known as biconformal gauging , which is builton the observation that the reduction to the local Weylgroup that occurs in ACGT is associated with the break-ing of the symmetry that exists between the generatorsof translations and SCTs in the conformal algebra. Bi-conformal gauging preserves this symmetry by construc-tion, although again considers only transformations thatare connected to the identity. The resulting biconformalgauge theories (BCGT) are successful in circumventingmany of the problems encountered in the standard ap-proach and have some very interesting and promisingfeatures. Nonetheless, the resulting requirement of an8-dimensional base manifold complicates their physicalinterpretation.Neither of these approaches is thus currently generallyaccepted, and so the role of the conformal group in theconstruction of gauge theories of gravity remains uncer-tain. In this paper, we therefore consider an alternativeapproach to gauging the conformal group, which is moti-vated in part by consideration of finite conformal trans-formations, which are therefore not necessarily connectedto the identity and so include inversions.Our reasons for including inversions explicitly aretwofold. First, from a physical perspective, it is well-known that both the Faraday action for the electromag-netic field and the Dirac action for a massless spinorfield are invariant not only under the elements of theconformal group that are connected to the identity, butalso under inversions . Second, from a mathematicalviewpoint, if one considers finite conformal transforma-tions, rather than infinitesimal ones, then the inversionoperation effectively replaces the SCT as the fourth dis-tinct element of the conformal group, since the SCT ismerely the composition of an inversion, a finite transla-tion and second inversion. Indeed, this correspondenceextends to the action of the elements of the finite con-formal group on fields, provided the latter belong to anirreducible representation of the Lorentz group. More-over, the inversion is itself the composition of a scalingand reflection, both of which are position-dependent inprescribed ways. Thus, when one gauges the finite con-formal group, the only transformations to consider be-yond those of the local Weyl group are gauged reflec-tions, which have not been addressed previously, to ourknowledge. Since reflections are merely improper Lorentztransformations, however, they may be localised straight-forwardly by gauging the full Lorentz group, rather thanonly the restricted Lorentz group that is usually consid-ered. Once again, this approach extends to the action offinite conformal transformations on fields that belong toan irreducible representation of the Lorentz group.These considerations suggest an alternative means ofcircumventing the difficulties associated with the gaug-ing of SCTs discussed above. In particular, it followsthat both Weyl gauge theory (WGT) and the recentlyproposed extended Weyl gauge theory (eWGT) already accommodate all the gauged symmetries of the full finiteconformal group, without the need to introduce any moregauge fields, provided that each occurrence of a properLorentz transformation in the finite transformation lawsfor the covariant derivative and gauge fields, respectively,instead denotes an element of the full Lorentz group. Inthis way, both WGT and eWGT actions constructed inthe usual way are invariant under (finite) local conformaltransformations. As we will show, however, only eWGTalso possesses conservation laws that provide a naturallocal generalisation of those satisfied by field theorieswith global conformal invariance, and has an ‘ungauged’limit that corresponds to global conformal transforma-tions. This suggests that eWGT may be considered asa valid alternative gauge theory of the conformal group,whereas WGT cannot be considered as such.The remainder of this paper is arranged as follows. InSection II, we briefly outline global conformal invariance,and clarify the geometric nature of conformal transfor-mations, in both their infinitesimal and finite forms; inthe latter we focus particularly on the role played by in-versions. We also clarify the requirements that globalconformal invariance places on field theories, which issometimes unclear in the literature. In Section III, we briefly discuss the principles underlying the gauging of aspacetime symmetry group, and in particular compareKibble’s original method with the more recently em-ployed quotient manifold method. We then reconsiderprevious approaches to gauging the conformal group, fo-cussing first on applying Kibble’s approach directly toconstruct ACGT and summarising its theoretical short-comings mentioned above. We also discuss the concept of‘ungauging’ , by which one seeks to identify the sym-metry group underlying any given gauge theory, and pro-pose some modifications to the existing approach beforeapplying it to ACGT. We then give a very brief outline ofBCGT. In Section IV, we summarise eWGT, focussing inparticular on the forms of the covariant derivative, fieldstrengths, action, field equations and conservation laws,all of which illustrate its very different structure fromother gravitational gauge theories. We then demonstrateour central point that, with the above mentioned modestextension to the transformation laws of the gauge fields,both eWGT and WGT are invariant under the full set offinite local conformal transformations, including inver-sions, but that only eWGT also possesses conservationlaws that provide a natural local generalisation of thosesatisfied by field theories with global conformal invari-ance, and has an ‘ungauged’ limit that corresponds toglobal conformal transformations. We conclude in Sec-tion V. In addition, in Appendix A, we include a simplederivation of the finite forms for the action on the coor-dinates of every element of the conformal group (not justthose connected to the identity) directly from their defin-ing requirement, without integrating infinitesimal forms.Finally, in Appendix B, we present a brief outline of theconsequences of general global and local symmetries forfield theories, focussing in particular on Noether’s firstand second theorems, the latter being discussed surpris-ingly rarely in the literature. II. GLOBAL CONFORMAL INVARIANCE
In Minkowski spacetime, conformal coordinate trans-formations x µ → x ′ µ from some Cartesian inertial coor-dinate system x µ are those that leave the light-cone (andhence causal structure) invariant, such that ds = η µν dx µ dx ν = Ω ( x ) η µν dx ′ µ dx ′ ν , (1)for some (real) function Ω( x ). Indeed, more generally,conformal transformations preserve the ‘angle’ or innerproduct between any two vectors, which is equivalent tothe invariance of the ratio of their lengths. A. Infinitesimal global conformal transformations
For an infinitesimal coordinate transformation x ′ µ = x µ + ξ µ ( x ) to satisfy (1), it is readily established thatone requires ∂ ( α ξ β ) = n ( ∂ µ ξ µ ) η αβ , (2)which is the conformal Killing equation in n -dimensionalMinkowski spacetime, although hereinafter we will con-centrate exclusively on the case n = 4. One may show inthe usual (albeit rather intricate) manner that the mostgeneral solution for ξ µ ( x ) has the form ξ µ ( x ) = a µ + ω µν x ν + ρx µ + c µ x − c · x x µ , (3)where the 15 infinitesimal parameters a µ , ω µν = − ω νµ , ρ and c µ are constants , i.e. not functions of spacetime po-sition, and we use the shorthand notation x ≡ η µν x µ x ν and c · x ≡ η µν c µ x ν , in which η µν = diag(1 , − , − , −
1) isthe Minkowski metric in Cartesian inertial coordinates.If the 4 parameters c µ defining the so-called special con-formal transformation (SCT) vanish, then (3) reduces toan infinitesimal global Weyl transformation. Moreover,if the parameter ρ defining the dilation (or scale trans-formation) also vanishes, then (3) further reduces to aninfinitesimal global Poincar´e transformation, consistingof a Lorentz rotation defined by the 6 parameters ω µν and a spacetime translation defined by the 4 parame-ters a µ . The group of global Poincar´e transformationsis the isometry group of Minkowski spacetime, such thatΩ ( x ) = 1 in (1).A more intuitive geometric interpretation of an in-finitesimal global conformal transformation may be ar-rived at directly using (2), from which one may showthat ∂ β ξ α = ̟ αβ + ̺δ αβ , (4)where ̟ αβ = − ̟ αβ and ̺ represent (in general) aposition-dependent infinitesimal rotation and dilation,respectively, which must satisfy the conditions (assum-ing dimensionality n ≥ ∂ µ ̟ αβ − δ [ αµ ∂ β ] ̺ = 0 , (5a) ∂ α ∂ β ̺ = 0 . (5b)Successive integration of equations (4–5), from the lastto the first, then yields ̺ ( x ) = ρ − c · x , ̟ αβ ( x ) = ω αβ + 4 c [ α x β ] and the expression (3) for ξ α ( x ), as before,where ρ , ω αβ and c α are again constants.The action of an infinitesimal conformal transforma-tion on some field ϕ ( x ) defined on the spacetime may bedetermined by first considering the 11-parameter (little)subgroup, say H (1 , C (1 , a µ = 0, which leaves the origin x µ = 0 invariant. Its generator matrices Σ µν , ∆ and κ µ ,corresponding to Lorentz rotations, dilations and SCTs,respectively, satisfy the commutation relations[Σ µν , Σ ρσ ] = η µσ Σ νρ − η νσ Σ µρ + η νρ Σ µσ − η µρ Σ νσ , [Σ µν , κ ρ ] = η νρ κ µ − η µρ κ ν , [Σ µν , ∆] = 0 , [∆ , ∆] = 0 , [ κ µ , κ ν ] = 0 , [ κ µ , ∆] = κ µ . (6)Using the method of induced representations, one maythen show that the action of a full infinitesimal confor-mal transformation on some field ϕ ( x ) leads to a ‘form’variation δ ϕ ( x ) ≡ ϕ ′ ( x ) − ϕ ( x ) given by δ ϕ ( x ) = ( a µ P µ + ω µν M µν + ρD + c µ K µ ) ϕ ( x ) , (7) where the 15 generators of the conformal group C (1 , P µ = − ∂ µ , (8a) M µν = x µ ∂ ν − x ν ∂ µ + Σ µν , (8b) D = − x · ∂ + ∆ , (8c) K µ = (2 x µ x · ∂ − x ∂ µ ) + 2( x ν Σ µν − x µ ∆) + κ µ , (8d)which correspond to translations, Lorentz rotations, di-lations and SCTs, respectively. Note that the generatorof the SCT can be expressed in terms of (parts of) theother generators as K µ = x P µ + 2 x ν Σ µν − x µ D + κ µ .The generators (8) satisfy the commutation relations[ M µν , M ρσ ] = η µσ M νρ − η νσ M µρ + η νρ M µσ − η µρ M νσ , [ M µν , P ρ ] = η νρ P µ − η µρ P ν , [ M µν , K ρ ] = η νρ K µ − η µρ K ν , [ M µν , D ] = 0 , [ P µ , K ν ] = 2( M µν + η µν D ) , [ D, D ] = 0 , [ P µ , P ν ] = 0 , [ P µ , D ] = − P µ , [ K µ , K ν ] = 0 , [ K µ , D ] = K µ , (9)which define the Lie algebra of the conformal group. Notethat, as expected, one recovers the Lie algebra of theWeyl group W (1 ,
3) by ignoring commutators contain-ing K µ , and if one also ignores commutators containing D one recovers the Lie algebra of the Poincar´e group P (1 , δ ϕ ( x ) = [ ξ µ ( x ) P µ + ̟ µν ( x )Σ µν + ̺ ( x )∆ + c µ κ µ ] ϕ ( x ) , (10)where ξ µ ( x ) is given by (3), ̟ µν ( x ) = ω µν + 4 c [ µ x ν ] and ̺ ( x ) = ρ − c · x , which are clearly all functions ofspacetime position. The form variation (10) is of particu-lar interest when the field ϕ ( x ) belongs to an irreduciblerepresentation of the Lorentz group, since the action ofthe conformal group is considerably simplified becausethe matrix generators ∆ and κ µ have particularly sim-ple forms. First, according to Schur’s lemma, any matrixthat commutes with the generators Σ µν must be a mul-tiple of the identity. Indeed, one has ∆ = wI , where I is the identity and w is the Weyl weight (or scaling di-mension) of the field ϕ ( x ). Then, from [ κ µ , ∆] = κ µ , onefinds that κ µ = 0. In this case, with ∆ = wI and κ µ = 0,it is worth noting that the form variation (10) may beconsidered as a particular example of an infinitesimal lo-cal Weyl transformation, consisting of the combinationof particular forms of position-dependent translation, ro-tation and dilation; this is consistent with the geometricinterpretation of an infinitesimal global conformal trans-formation expressed in (4).Fields that transform according to (10) under a confor-mal transformation are called primary fields. There alsoexist non-primary fields, the most important example ofwhich is the derivative ∂ µ ϕ ( x ) of a primary field. It isstraightforward to show that δ ( ∂ µ ϕ ) = Θ ∂ µ ϕ − ( ̟ ν µ + ̺δ νµ ) ∂ ν ϕ + 2( c ν Σ νµ − c µ ∆) ϕ, (11)where Θ ≡ ξ α P α + ̟ αβ Σ αβ + ̺ ∆ + c α κ α is the quantityin square brackets on the RHS of (10), with generatorsappropriate to the nature of ϕ ( x ), and the final term onthe RHS of (11) shows that ∂ µ ϕ ( x ) is non-primary. In-deed, although the transformation law is linear, this finalterm also means that it is inhomogeneous. It is clear thatthis behaviour results solely from the SCT; if c µ = 0, andhence ̟ µν = ω µν and ̺ = ρ , one recovers a global Weyltransformation and (11) becomes homogeneous and ofan analogous form to the transformation law (10) of theoriginal field ϕ ( x ) with c µ = 0, once the generators havebeen augmented to accommodate the additional vectorindex on the partial derivative ∂ µ . B. Finite global conformal transformations
The finite forms for the action on the coordinates of theelements of the conformal group C (1 ,
3) corresponding totranslations, proper Lorentz rotations (jointly Poincar´etransformations) and dilations (jointly Weyl transforma-tions) are easily found by obtaining the integral curvesof the corresponding infinitesimal expressions. Again de-noting the 15 now finite constant parameters of the groupby a , ω , ρ and c , these finite forms are, respectively, x ′ µ = x µ + a µ , x ′ µ = Λ µν ( ω ) x ν , x ′ µ = e ρ x µ , (12)where Λ µν ( ω ) is a proper Lorentz transformation matrixsatisfying η µν Λ µρ Λ νσ = η ρσ and det Λ µν = 1.The same procedure can be used to find the finite formfor the action of a SCT, but more geometrical insight isobtained by first introducing the inversion transforma-tion, which may be taken to have the form x ′ µ = x µ x , (13)where x = η µν x µ x ν = 0. This discrete transforma-tion is clearly also an element of the full conformal group(although not one connected to the identity), since thenew (Minkowski spacetime) metric is given by γ ′ µν ( x ) = η µν / ( x ) . If one then considers the composite trans-formation consisting of an inversion, followed by a finitetranslation through c µ , followed by a second inversion,one finds x ′ µ = x µ + c µ x c · x + c x , (14)which reduces to the infinitesimal SCT in (3) for small c µ . Since every smooth conformal transformation of apseudo-Euclidean (Euclidean) space of dimension n ≥ , the expression (14) must repre-sent a finite SCT. It is worth mentioning that, althoughboth the numerator and denominator of (14) vanish for x µ = − c µ /c , this point is mapped to infinity and hencethe finite SCT is not defined globally. Indeed, in order to define finite SCTs globally, one must consider a confor-mally compactified Minkowski spacetime, which includesan additional special point at infinity and its null cone,but we will not consider this subtlety any further here.Although not usually presented in the literature, onecan in fact derive the finite forms for the action on thecoordinates of every element of the conformal group (notjust those connected to the identity) directly from thedefining requirement (1), without having to integrate in-finitesimal forms. As just mentioned, one may considerevery smooth conformal transformation as a compositionof isometry, dilation and inversion. The coordinate trans-formation matrix for the inversion (13) is given by X µν ≡ ∂x ′ µ ∂x ν = 1 x ( δ µν − x µ ˆ x ν ) ≡ x I µν (ˆ x ) , (15)which one may identify as the composition of a (posi-tion dependent) dilation 1 /x and reflection I µν (ˆ x ) inthe hyperplane perpendicular to the unit vector ˆ x ≡ x µ / √ x . It is straightforward to show that I µν (ˆ x ) isan improper Lorentz transformation matrix, satisfying η µν I µρ (ˆ x ) I νσ (ˆ x ) = η ρσ and det I µν (ˆ x ) = −
1. Thus, withno loss of generality, one may write the transformationmatrix of any smooth finite conformal transformation inthe form X µν = Ω( x )Λ µν ( x ) , (16)where, in general, Λ µν ( x ) represents a position-dependent finite Lorentz transformation (either properor improper) and Ω( x ) represents a position-dependentfinite dilation; indeed, one sees immediately that (16)satisfies the defining requirement (1). As shown in Ap-pendix A, one may further use (1) to derive conditionson Λ µν ( x ) and Ω( x ), which may be written as (for di-mensionality n ≥
3) Λ γα ∂ µ Λ γβ − δ [ αµ ∂ β ] ln Ω = 0 , (17a)2Ω ∂ α ∂ β Ω + η αβ ( ∂ γ Ω)( ∂ γ Ω) − ∂ α Ω)( ∂ β Ω) = 0 . (17b)It is straightforward to check that, on writing x ′ µ ≈ x µ + ξ µ ( x ), Λ µν ( x ) ≈ δ µν + ̟ µν ( x ) and Ω( x ) = e ̺ ( x ) ≈ ̺ ( x ),the expressions (16–17) reduce correctly in the infinitesi-mal limit to those given in (4–5). Moreover, as discussedfurther in Appendix A, the conditions (17) may be usedto determine directly the action on coordinates of the fourdistinct finite elements of the conformal group, namelyposition-independent translations, rotations and scalings,together with inversions (and hence also SCTs).In the space of fields, the action of the finite elementsof the conformal group that are connected to the iden-tity (thus excluding inversions) is formally given by theexponential of the operator appearing on the RHS of (7),such that for some field ϕ ( x ) one has ϕ ′ ( x ) = exp( a µ P µ + ω µν M µν + ρD + c µ K µ ) ϕ ( x ) , (18)where the group parameters a , ω , ρ and c are now finiteconstants.It is again of particular interest to consider the case ofsome physical field ϕ ( x ) belonging to an irreducible rep-resentation of the Lorentz group. Setting c µ = 0 for themoment, thereby neglecting the finite SCTs and consid-ering just the Weyl group, one may describe the actionof a finite transformation as ϕ ′ ( x ′ ) = e wρ S ( ω ) ϕ ( x ) , (19)where S ( ω ) is the matrix corresponding to the element ω of the proper Lorentz group (or SL(2,C) group) inthe representation to which ϕ ( x ) belongs (we have sup-pressed Lorentz indices on these objects for notationalsimplicity), and w is the Weyl (or conformal) weight ofthe field ϕ ( x ). Indeed, we have adopted the form (19)for the action of finite global Weyl transformations onphysical fields in our previous work .To determine the explicit form for the action of a fi-nite SCT on some such physical field ϕ ( x ), it is againconvenient to consider first the action of an inversion, forwhich the transformation matrix is given by (15). Thus,the action of an inversion on, for example, a vector field V µ ( x ) of Weyl (scaling) weight w , is given by V ′ µ ( x ′ ) = ( x ) − w I µν (ˆ x ) V ν ( x ) . (20)Analogous transformation laws hold for higher-rank ten-sor fields. One may also show that the action of an in-version on a spinor field ψ ( x ) of Weyl weight w is givenby ψ ′ ( x ′ ) = ( x ) − w ( γ · ˆ x ) ψ ( x ) , (21)where γ = { γ µ } denotes the set of Dirac matrices. It isworth noting that the quantities I µν (ˆ x ) and γ · ˆ x havepreviously been identified as matrices that decouple theLorentz indices on tensor and spinor fields, respectively,from SCTs, but surprisingly without giving their geo-metric interpretation as a reflection in the hyperplaneperpendicular to the unit vector ˆ x . It is straightforwardto show that these two decoupling matrices are relatedby the useful formula( γ · ˆ x ) γ µ ( γ · ˆ x ) = − I µν (ˆ x ) γ ν . (22)Recalling that a SCT is the composition of an inver-sion, followed by a translation c µ , followed by a secondinversion, it is now straightforward to show that the ac-tion of a finite SCT on, for example, a vector field orspinor field of Weyl weight w is given by, respectively, V ′ µ ( x ′ ) = [ σ ( x, c )] − w I µα (ˆ x ′ ) I αν (ˆ x ) V ν ( x ) , (23a) ψ ′ ( x ′ ) = [ σ ( x, c )] − w ( γ · ˆ x ′ )( γ · ˆ x ) ψ ( x ) , (23b)where we have defined the (inverse) scaling σ ( x, c ) ≡ c · x + c x that appears in the denominator ofthe coordinate transformation (14) resulting from a finiteSCT. The transformations (23) are thus the compositionof a reflection in the hyperplane perpendicular to ˆ x , areflection in the hyperplane perpendicular to ˆ x ′ and a scaling. Combining the two reflections, the action of afinite SCT on the fields thus consists of a rotation in thehyperplane defined by ˆ x and ˆ x ′ (through twice the anglebetween the two directions) and a scaling. It is worthnoting that, since the resulting rotation is composed oftwo reflections, it is of a special (or restricted) form; in4 dimensions not all (proper) Lorentz rotations can beconstructed in this way.The above geometrical interpretation of a finite SCTis consistent with the action of an infinitesimal SCT ona (primary) field that belongs to an irreducible represen-tation of the Lorentz group, which is given by (10) with∆ = wI , κ µ = 0, ω µν = 0 and ρ = 0, and describesthe combination of an infinitesimal scaling and rotation(both position dependent). Indeed, it is straightforwardto show that in the limit of small c µ the transforma-tions (23) yield precisely the forms ̟ µν ( x ) = 4 c [ µ x ν ] and ̺ ( x ) = − c · x for the infinitesimal parameters ap-pearing in (10). Moreover, having introduced the reflec-tion operator I µν (ˆ x ), it is worth noting that the gen-erator K µ for SCTs in (8d) can be written as K µ = − x I µν (ˆ x ) ∂ ν + 2( x ν Σ µν − x µ ∆) + κ µ .Finally, we note that since the action both of inver-sions and SCTs on fields that belong to an irreduciblerepresentation of the Lorentz group consist of a scalingand a Lorentz rotation, albeit an improper rotation forinversions, then the action of any element of the full fi-nite conformal group on such a field may be written inthe form (19), provided one extends the definition of S ( ω )to include matrices corresponding to elements of the fullLorentz group, and allows ρ and ω to become particularfunctions of spacetime position. C. Global conformal invariant field theory
We now discuss the consequences of global conformalinvariance for field theories, since this is sometimes un-clear in the literature. Consider a Minkowski space-time M , labelled using Cartesian inertial coordinates,in which the dynamics of some set of fields ϕ i ( x ) ( i =1 , , . . . ) is described by the action S = Z L ( ϕ i , ∂ µ ϕ i ) d x, (24)such as that considered in Appendix B 1. The index i here again merely labels different matter fields, ratherthan denoting the tensor or spinor components of indi-vidual fields (which are suppressed throughout). It isalso worth noting that these fields may include a scalarcompensator field (often denoted also by φ ) with Weylweight w = −
1, which may be used to replace mass pa-rameters in the standard forms of Lagrangians for mas-sive matter fields to achieve global conformal invariance(for example by making the substitution m ¯ ψψ → µφ ¯ ψψ in the action for a massive Dirac field ψ , where µ is adimensionless parameter but µφ has dimensions of massin natural units) .The consequences of invariance of the action (24) underan infinitesimal global coordinate transformation (con-nected to the identity) may be determined by substitut-ing the forms (3) and (7) into the general expressions(B2) and (B3). Recalling that the operators δ and ∂ µ commute, and equating to zero the coefficients multiply-ing the constant parameters a µ , ω µν , ρ and c µ , respec-tively, leads to the conditions ∂ µ L − ∂L∂ϕ i ∂ µ ϕ i − ∂L∂ ( ∂ α ϕ i ) ∂ µ ∂ α ϕ i = 0 , (25a) ∂L∂ϕ i Σ µν ϕ i + ∂L∂ ( ∂ α ϕ i ) [Σ µν ∂ α ϕ i + ( η αµ ∂ ν − η αν ∂ µ ) ϕ i ] = 0 , (25b) ∂L∂ϕ i ∆ ϕ i + ∂L∂ ( ∂ α ϕ i ) (∆ − I ) ∂ α ϕ i + 4 L = 0 , (25c) ∂L∂ϕ i κ µ ϕ i + ∂L∂ ( ∂ α ϕ i ) [ κ µ ∂ α ϕ i + 2(Σ µα − η µα ∆) ϕ i ] = 0 , (25d)which hold up to a total divergence of any quantity thatvanishes on the boundary of the integration region in(24). The first condition is equivalent to requiring that L has no explicit dependence on spacetime position x ,and this condition has been used to derive the secondand third conditions. Moreover, the first three conditionshave all been used to derive the final condition. In partic-ular, this means that for the action to be invariant underSCTs (which is necessary for conformal invariance), itmust be Poincar´e and scale invariant, in addition to sat-isfying the condition (25d). Conversely, an action thatis invariant under Poincar´e transformations and SCTs isnecessarily scale invariant.Again adopting the forms (3) and (7), one finds thatthe general expression (B8) for the Noether current be-comes J µ = − a α t µα + ω αβ M µαβ + ρD µ + c α K µα , (26)where the coefficients of the parameters of the conformaltransformation are defined by t µα ≡ ∂L∂ ( ∂ µ ϕ i ) ∂ α ϕ i − δ µα L, (27a) M µαβ ≡ x α t µβ − x β t µα + s µαβ , (27b) D µ ≡ − x α t µα + j µ , (27c) K µα ≡ (2 x α x β − δ βα x ) t µβ +2 x β ( s µαβ − η αβ j µ )+ k µα , (27d)which are the (total) canonical energy-momentum, an-gular momentum, dilation current and special conformalcurrent, respectively, of the fields ϕ i , and we have alsodefined the quantities s µαβ ≡ ∂L∂ ( ∂ µ ϕ i ) Σ αβ ϕ i , (28a) j µ ≡ ∂L∂ ( ∂ µ ϕ i ) ∆ ϕ i , (28b) k µα ≡ ∂L∂ ( ∂ µ ϕ i ) κ α ϕ i , (28c) which are the (total) canonical spin angular momentum,intrinsic dilation current and intrinsic special conformalcurrent of the fields. In (27d), it is worth noting that thefirst term on the RHS can be written as − x I αβ (ˆ x ) t µβ .If the field equations δL/δϕ i = 0 are satisfied, theninvariance of the action (24) reduces to the conservationlaw ∂ µ J µ = 0. Since the parameters of a global conformaltransformation in (26) are constants, one thus obtainsseparate conservation laws of the form (B9), given by ∂ µ t µα = 0 , (29a) ∂ µ s µαβ + 2 t [ αβ ] = 0 , (29b) ∂ µ j µ − t µµ = 0 , (29c) ∂ µ k µα + 2( s µαµ − j α ) = 0 , (29d)which again hold up to a total divergence, and may beconsidered as the “on-shell” specialisation of the condi-tions (25). As previously, the first condition has beenused to derive the second and third conditions, and thefirst three conditions have all been used to derive the fi-nal condition. It is worth noting that the conditions (25)and (29) in fact hold for any subset of terms in the La-grangian L for which the resulting action is conformallyinvariant.As mentioned earlier, if the fields ϕ i ( x ) belong to ir-reducible representations of the Lorentz group, which isusually the case for physical fields, then ∆ = wI and κ µ = 0, and hence k µα also vanishes. In this case, it isusual to define the “field virial” V α ≡ ∂L∂ ( ∂ β ϕ i ) (Σ αβ − η αβ w ) ϕ i = s βαβ − j α . (30)From (29d), one sees that for (any subset of) an actionthat is Poincar´e and scale invariant also to be conformallyinvariant, the field virial V α must vanish up to a totaldivergence. Remarkably, this last condition is found tohold for all renormalizable field theories involving parti-cles with spin 0, 1/2 and 1, even though scale invariance(and hence conformal invariance) is, in general, brokenin such theories .As is well known, the conservation laws (29a) and(29b), resulting from translational and Lorentz invari-ance, respectively, can both be expressed in terms of thesingle Belinfante energy-momentum tensor t µν B = t µν + ∂ λ ( s µνλ + s νµλ − s λνµ ) , (31)as the two properties ∂ µ t µν B = 0 and t µν B = t νµ B . More-over, for theories describing fields ϕ i ( x ) that belong to ir-reducible representations of the Lorentz group and wherethe field virial (30) vanishes up to a total divergence, scaleinvariance is thus equivalent to conformal invariance, andone may further define the improved energy-momentumtensor θ µν = t µν B − ( ∂ µ ∂ ν − η µν (cid:3) ) X i ϕ i , (32)such that the remaining conditions in (29) can be ex-pressed in terms of this single quantity as ∂ µ θ µν = 0, θ µν = θ νµ and θ µµ = 0.Since the discussion thus far relies on infinitesimaltransformations, it applies only to elements of the confor-mal group that are continuously connected to the iden-tity, and hence neglects invariance of the action (24) un-der inversions, which are intrinsically both finite and dis-crete. As mentioned in the Introduction, this issue is ofparticular interest since it is straightforward to show thatboth the Faraday action for the electromagnetic field andthe Dirac action for a massless spinor field are invariantnot only under the continuous elements of the conformalgroup considered above, but also under inversions.Additional conserved quantities can be generated bydiscrete symmetries. For example, theories invariant un-der spatial inversion x ′ i = − x i ( i = 1 , ,
3) conserve par-ity. It is far less straightforward, however, to determinedirectly the consequences of invariance of a general action(24) under the (conformal) inversion (13). Nonetheless,some insight may be gained by considering the large-parameter limit of the finite SCT (14), which is givenby x ′ µ = c µ c + 1 c I µν (ˆ c ) x ν x + O (cid:18) c (cid:19) . (33)Thus, a large- c SCT consists of the composition of aninversion (13), a reflection in the hyperplane perpendicu-lar to ˆ c , a scale transformation by 1 /c and a translationby c µ /c . An action that is invariant under translations,scale transformations and SCTs, as we considered above,must therefore also be invariant under the combinationof just an inversion and a reflection in the hyperplaneperpendicular to ˆ c . Hence, if the action is invariant un-der an inversion alone, it must also be invariant underreflection in an arbitrary hyperplane, which provides acovariant generalisation of the (one-dimensional) parityand time-reversal transformations . III. PREVIOUS APPROACHES TO CONFORMALGAUGING
As noted in the Introduction, there are strong theoreti-cal reasons to consider gauging the conformal group witha view to constructing theories of the gravitational inter-action that are invariant under local conformal transfor-mations. Previous approaches have focussed on infinites-imal transformations and hence considered gauging onlythe elements of the conformal group that are connectedto the identity (namely translations, Lorentz rotations,dilations and SCTs). This is achieved, in principle, byallowing the 15 parameters of the group a , ω , ρ and c tobecome independent arbitrary functions of position. A. Gauging a spacetime symmetry group
Some early approaches to constructing a gaugetheory of a spacetime symmetry group G (in our casethe conformal group) encountered complications arisingfrom attempting to draw too close an analogy with thegauge theories of internal symmetries . In modernterminology, the corresponding gauge fields (or Yang–Mills potentials) were introducd as the components of aconnection on a principal fiber bundle with spacetime asbase space and G as fiber. If the whole of a spacetimegroup G is gauged in the Yang–Mills sense, however, thegauged ‘internal translations’ prevent the identificationof the translational gauge fields with a vierbein in thegeometric interpretation of the gauge theory.It was the gauging of the Poincar´e group by Kibble that first revealed how to achieve a meaningful gaugingof groups that act on the points of spacetime as well ason the components of physical fields. The essense of Kib-ble’s approach was to note that when the parameters ofthe Poincar´e group become independent arbitrary func-tions of position, this leads to a complete decoupling ofthe translational parts from the rest of the group, and theformer are then interpreted as arising from a general co-ordinate transformation (GCT; or spacetime diffeomor-phisms, if interpreted actively). Thus the action of thegauged Poincar´e group is considered as a GCT x µ → x ′ µ ,together with the local action of its Lorentz subgroup H on the orthonormal tetrad basis vectors ˆ e a ( x ) that definelocal Lorentz reference frames, where we adopt the com-mon convention that Latin indices (from the start of thealphabet) refer to anholonomic local tetrad frames, whileGreek indices refer to holonomic coordinate frames. Thisapproach to gauging can be straightforwardly extendedto more general spacetime symmetry groups .The physical model envisaged in Kibble’s approachis an underlying Minkowski spacetime in which a setof matter fields ϕ i is distributed continuously. Thefield dynamics are described by a matter action S M = R L M ( ϕ i , ∂ µ ϕ i ) d x that is invariant under the global ac-tion of G . One then gauges the group G by demand-ing that the matter action be invariant with respect to(infinitesimal, passively interpreted) GCT and the localaction of the subgroup H , obtained by setting the trans-lation parameters of G to zero (which leaves the origin x µ = 0 invariant), and allowing the remaining group pa-rameters to become independent arbitrary functions ofposition. One is thus led to the introduction of new fieldvariables, which are interpreted as gravitational gaugefields. These are used to assemble a covariant derivative D a ϕ that transforms in the same way under the actionof the gauged group G as ∂ µ ϕ does under the global ac-tion of G . The matter action in the presence of grav-ity is then typically obtained by the minimal couplingprocedure of replacing partial derivatives in the special-relativistic matter Lagrangian by covariant ones, to ob-tain S M = R h − L M ( ϕ i , D a ϕ i ) d x , where the factor con-taining h ≡ det( h aµ ) (here h aµ is the translational gaugefield) is required to make the integrand a scalar densityrather than a scalar.In addition to the matter action, the total action mustalso contain terms describing the dynamics of the freegravitational gauge fields. Following the normal proce-dure used in gauging internal symmetries, Kibble firstconstructed covariant field strength tensors for the gaugefields by commuting covariant derivatives, i.e. by con-sidering [ D a , D b ] ϕ . The free gravitational action thentakes the form S G = R h − L G d x , where L G is someLagrangian that depends on the field strengths and issuch that S G is invariant under the action of the gaugedgroup G . The total action is taken as the sum of the mat-ter and gravitational actions, and variation of the totalaction with respect to the gauge fields leads to coupledgravitational field equations.Following Kibble’s work, several other approaches togauging a spacetime symmetry group have been pro-posed, in which, for example, the transformations areinterpreted actively, or one considers finite rather thaninfinitesimal transformations , but in terms of the fi-nal locally valid field equations that these formulationsreach, given an initial Lagrangian, they are equivalent toKibble’s original method.Finally, it is worth noting that Kibble’s gauge approachto gravitation is most naturally interpreted as a field the-ory in Minkowski spacetime , in the same way as thegauge field theories describing the other fundamental in-teractions, and this is the viewpoint that we shall adoptin this paper. It is more common, however, to reinter-pret the mathematical structure of such gauge theoriesgeometrically, where in particular the translational gaugefield h aµ is considered as the components of a vierbeinsystem in a more general spacetime . These issues arediscussed in more detail elsewhere .More recent approaches to gauging a spacetime sym-metry group adopt the geometric interpretation whole-heartedly, and are usually expressed in the language offiber bundles. In this view, it is clear from the discussionabove that only the subgroup H should act on the fibers,not the whole of G (i.e. no ‘internal translation’). Thesimplest and most natural translation of the scheme intofiber bundle language consists of expressing the gaugetheory of a spacetime symmetry group G in terms of thegroup manifold G ; specifically, in terms of the principalfiber bundle G ( G / H , H ), where the coset space G / H isspacetime .Indeed, this viewpoint is embodied in the so-calledquotient manifold method , which may be consideredas an inversion of Kibble’s approach, and is usually ex-pressed in the language of differential forms as follows.Consider some Lie group G possessing a Lie subgroup H .The Maurer–Cartan structure equations for G read d ω A − f BC A ω B ∧ ω C = 0 , (34)where f BCA are the structure constants of the algebra ofthe group G . These equations constitute an integrabilitycondition that give a 1-form ω A on G that carries the basic infinitesimal information about the group’s struc-ture. One may thus define the exponential map of thecorresponding Lie algebra and hence a local group ac-tion. One then takes the quotient G / H , which is neces-sarily a manifold M (usually interpreted as spacetime),and the 1-forms provide its connection. The result is aprincipal fiber bundle with local H symmetry and basemanifold M . This structure is then modified by gen-eralizing the manifold, and by changing the connection.Changing the manifold has no effect on the local struc-ture, but changing the connection modifies the Maurer–Cartan equations (to yield the Cartan equations), result-ing in curvature 2-forms R A = d ω A − f BCA ω B ∧ ω C . (35)Two restrictions are placed on these curvatures . First,the curvatures must characterize the manifold only,which requires them to be ‘horizontal’, i.e. bilinear inthe connections. Second, one requires integrability of theCartan equations, which leads to the Bianchi identitiessatisfied by the curvatures.Thus, once one has made the choice of G and H , thisapproach determines: the physical arena M , the localsymmetry group H , the relevant field strength tensors R A , and any structures inherited from G . While otherstructures may be imposed, such as additional (compen-sator) scalar fields, it is usual to consider only thosearising directly from properties of the gauge group. Tocomplete a gravity theory, one finally constructs a lo-cally H -invariant action from the available tensors R A ,together with the invariant metric η ab and Levi–Civitatensor ǫ abcd , and any desired matter fields. The remain-ing, broken, group transformations of G are replaced bydiffeomorphisms on M . Each gauge field in the connec-tion form ω A is then varied independently to find thefield equations. A key advantage of the approach is thatit keeps the curvatures and action expressed in terms ofthe gauge fields, making the variation straightforward.The quotient manifold method, although very power-ful, may appear somewhat rarified to most physicists.Fortunately, Kibble’s original approach may be used toarrive at precisely the same gauge theories as those ob-tained using the quotient manifold method, although thisis rarely demonstrated in the literature. We will thereforeprimarily adopt Kibble’s approach below, since it is morefamiliar to physicists. In particular, we will make use of itto illustrate the stucture of ACGT in a transparent man-ner, which can then be readily compared to our discus-sion of eWGT in Section IV A. To facilitate this compari-son further, we will also maintain the (more unorthodox)viewpoint of the gauge fields as fields in Minkowski space-time, without attaching any geometric interpretation tothem. Consequently, we will adopt a global Cartesian in-ertial coordinate system x µ in our Minkowski spacetime,which greatly simplifies calculations, but more generalcoordinate systems may be straightforwardly accommo-dated, if required . B. Auxiliary conformal gauging
In the standard approach to gauging the conformalgroup G = C (1 ,
3) (or, more precisely, the elements ofit that are connected to the identity), the subgroup H is that obtained by setting the translation parameters a µ = 0, which leaves the origin x µ = 0 invariant (i.e. thelittle group), and has the 11 generators Σ ab , ∆ and κ a that satisfy the commutation relations (6).Following Lord , under the simultaneous action ofan infinitesimal GCT x µ → x µ + ξ µ ( x ) and the infinites-imal local action of H , by analogy with (10), the formvariation of a (primary) field is given by δ ϕ ( x ) = − ξ µ ( x ) ∂ µ ϕ ( x ) + ε ( x ) ϕ ( x ) , (36)where ε ( x ) ≡ ̟ ab ( x )Σ ab + ̺ ( x )∆+ c a ( x ) κ a is an elementof the localised (little) subgroup H , and ̟ ab ( x ), ̺ ( x ), c a ( x ) and ξ µ ( x ) are now independent arbitrary functionsof position. Consequently, the transformation law of thederivative ∂ µ ϕ ( x ) is no longer given by (11).The construction of a covariant derivative that trans-forms like (11) under the gauged conformal group is typ-ically achieved in two steps. First, one defines the ‘ H -covariant’ derivative¯ D µ ϕ ( x ) ≡ [ ∂ µ + ¯Γ µ ( x )] ϕ ( x ) , (37)where ¯Γ µ ( x ) is a linear combination of the generators of H that depends on the gauge fields corresponding to localLorentz rotations, local dilations, and local special con-formal transformations, respectively (the bars appearingin these definitions are to distinguish the correspondingquantities from others to be defined later). In the sec-ond step, we define a ‘generalised H -covariant’ derivative,linearly related to ¯ D µ ϕ , by¯ D a ϕ ( x ) ≡ h aµ ( x ) ¯ D µ ϕ ( x ) , (38)where we have introduced the translational gauge field h aµ ( x ). It is assumed that h aµ ( x ) has an inverse, usuallydenoted by b aµ ( x ), such that h aµ b aν = δ µν and h aµ b cµ = δ ca (where, for brevity, we henceforth typically drop theexplicit x -dependence).Under the action of an infinitesimal GCT x ′ µ = x µ + ξ µ and the infinitesimal local action of H , with the associ-ated field transformation law (36), we require ¯ D a ϕ to have an analogous transformation law to (11), namely δ ( ¯ D a ϕ ) = − ξ µ ∂ µ ¯ D a ϕ + ε ¯ D a ϕ − ( ̟ ba + ̺ δ ba ) ¯ D b ϕ +2( c b Σ ba − c a ∆) ϕ, (39)but where ̟ ab , ̺ , c a and ξ µ are independent arbitraryfunctions of position. This requirement leads uniquely tothe transformation laws δ h aµ = − ξ ν ∂ ν h aµ + h aν ∂ ν ξ µ − ( ̟ ba + ̺ δ ba ) h bµ , (40a) δ ¯Γ µ = − ξ ν ∂ ν ¯Γ µ − ¯Γ ν ∂ µ ξ ν − ∂ µ ε − [¯Γ µ , ε ]+2 b aµ ( c b Σ ba − c a ∆) . (40b)One sees that h aµ transforms as a GCT vector, a localLorentz four-vector, and has Weyl weight w = − b aµ transforms in an analogousway, but has w = 1). Similarly, the quantity ¯Γ µ trans-forms as a covariant GCT vector, but the last term onthe RHS of (40b) shows that ¯Γ µ is not the connectionfor the gauge group H . Indeed, it was already apparentfrom the corresponding final term in (39) that ¯ D a ϕ is notan H -covariant derivative in the usual sense; its transfor-mation law is linear but inhomogeneous. As mentionedearlier, this behaviour originates in the final term of thetransformation law (11) for ∂ µ ϕ under the action of aglobal conformal transformation, and can be traced tothe fact that translations do not form an invariant sub-group of the conformal group (whereas they do for theWeyl group, obtained by setting c µ = 0). Nonetheless,since ¯ D a ϕ was constructed to have an analogous transfor-mation law to that of ∂ µ ϕ in (11), one can still constructa matter action that is fully invariant under the gaugedconformal group from one that is invariant under globalconformal transformations by employing the usual mini-mal coupling procedure of replacing partial derivatives bycovariant ones to obtain S M = R h − L M ( ϕ i , ¯ D a ϕ i ) d x .As discussed in Section II C, the set of fields ϕ i may in-clude a scalar compensator field (denoted also by φ ).It is usual to assume the linear combination ¯Γ µ of thegenerators of H to have the form¯Γ µ ≡ A abµ Σ ab + B µ ∆ + f aµ κ a , (41)where A abµ ( x ), B µ ( x ) and f aµ ( x ) are the gauge fieldscorresponding to local Lorentz rotations, local dilations,and local special conformal transformations, respectively.It is worth pointing out that this assumed form for ¯Γ µ constitutes a choice of how to include the gauge fields,and leads directly to their required transformation laws δ A abµ = − ξ ν ∂ ν A abµ − A abν ∂ µ ξ ν − ̟ [ ac A b ] cµ − ∂ µ ̟ ab − b [ aµ c b ] , (42a) δ B µ = − ξ ν ∂ ν B µ − B ν ∂ µ ξ ν − ∂ µ ̺ − b aµ c a , (42b) δ f aµ = − ξ ν ∂ ν f aµ − f aν ∂ µ ξ ν − ( ̟ ba + ̺ δ ab ) f bµ − ( ∂ µ c a − B µ c a − A abµ c b ) , (42c)which are obtained by substituting (41) into (40b) and equating coefficients of Σ ab , ∆ and κ a , respectively.0We note that if one sets c a = 0, the transformationlaws of h aµ , A abµ and B µ in (40) and (42) are preciselythe infinitesimal versions (i.e. first-order in the group pa-rameters) of those obtained for these gauge fields underfinite transformations in Weyl gauge theory (WGT) . Inthe case c a = 0, one also sees that f aµ has similar trans-formation properties to h aµ in (40a), since it transformsas a (covariant) GCT vector, a local Lorentz four-vector,and has Weyl weight w = − c a = 0, however, the transformation laws ofall the gauge fields A abµ , B µ and f aµ involve terms con-taining c a . Indeed, these terms lead to the inclusion ofthe (inverse) translational gauge field b aµ in the transfor-mation laws of A abµ and B µ , and also cause the trans-formation law of f aµ to depend on A abµ and B µ . Hence,the action of (local) SCTs leads to considerable differ-ences between the gauge theory of the conformal groupand those of its Weyl or Poincar´e subgroups.The total action must also contain terms describing thedynamics of the free gravitational gauge fields. Theseterms are constructed from the gauge field strengths,which are usually defined in terms of the commutator ofcovariant derivatives. Considering first the H -covariantderivative, one finds[ ¯ D µ , ¯ D ν ] ϕ = ( R abµν Σ ab + H µν ∆ + S ∗ aµν κ a ) ϕ, (43)where we have defined the ‘ H -rotational’, ‘ H -dilational’and the ‘ H -special conformal’ field strength tensors, re-spectively (the reason for notating the last of these withan asterisk will become clear shortly). In terms of thegauge fields A abµ , B µ and f aµ , the field strengths havethe forms R abµν ≡ ∂ [ µ A abν ] + η cd A ac [ µ A dbν ] ) , (44a) H µν ≡ ∂ [ µ B ν ] , (44b) S ∗ aµν ≡ ∂ [ µ f aν ] + A ac [ µ f cν ] − B [ µ f aν ] ) = 2 D ∗ [ µ f aν ] . (44c)For the sake of brevity, in the final expression we have in-troduced the derivative operator D ∗ µ ≡ ∂ µ + A abµ Σ ab + wB µ , familiar from WGT , where w is the Weylweight of the field on which it acts. All three fieldstrengths in (44) transform covariantly under GCT andlocal Lorentz rotations in accordance with their respec-tive index structures, and also under local dilations withthe Weyl weights w ( R abµν ) = 0, w ( H µν ) = 0 and w ( S ∗ aµν ) = −
1, respectively, but none of them trans-forms covariantly under local SCTs, as we discuss furtherbelow.Before doing so, however, we next consider the commu-tator of two ‘generalised H -covariant’ derivatives. Since¯ D a ϕ = h aµ ¯ D µ ϕ , this commutator differs from (43) byan additional term containing the derivatives of h aµ , andreads[ ¯ D c , ¯ D d ] ϕ = ( R abcd Σ ab + H cd ∆ + S ∗ acd κ a − T ∗ acd ¯ D a ) ϕ, (45)where R abcd ≡ h cµ h dν R abµν , H cd = h cµ h dν H µν and S ∗ acd ≡ h cµ h dν S ∗ aµν , and the ‘ H -translational’ field strength of the gauge field h aµ is given by T ∗ acd ≡ h cµ h dν T ∗ aµν ≡ h cµ h dν D ∗ [ µ b aν ] , (46)which clearly has the same form as S ∗ acd , but with f aµ replaced by b aµ . It is worth noting that R abcd , H cd and T ∗ acd have the same functional forms of the gauge fieldsas the rotational, dilational and translational gauge fieldstrengths, respectively, in WGT .It is straightforward to show that R abcd , H cd , S ∗ acd and T ∗ acd are GCT scalars and transform covariantlyunder local Lorentz rotations and under local dilations,with weights w ( R abcd ) = w ( H cd ) = − w ( S ∗ acd ) = − w ( T ∗ acd ) = −
1, respectively. As one might expect,however, the transformation laws under local SCTs aremore complicated, and are given by δ R abcd = 4 c [ a T ∗ b ] cd + 8 δ [ a [ c D ∗ d ] c b ] (47a) δ H cd = − c a T ∗ acd + 4 δ a [ c D ∗ d ] c a (47b) δ S ∗ acd = − c b ( R abcd + 8 δ [ a | [ c h d ] µ f | b ] µ )+2 c a ( H cd + 4 f [ c | µ h | d ] µ ) , (47c) δ T ∗ acd = 0 , (47d)where the action of D ∗ a assumes that w ( c a ) = −
1. Thus,there is a ‘mixing’ of the transformation laws of the fieldstrengths, which arises from mixing of the transforma-tion laws of the gauge fields themselves, as describedabove. Moreover, one sees that the transformation laws(47) also depend on the gauge fields directly, rather thanjust through the field strengths. Indeed, it is only the H -translational field strength T ∗ acd that transforms co-variantly (indeed, invariantly) under local SCTs.The transformations laws (47) mean that the only com-bination of terms containing field strengths that maybe included in the total Lagrangian to obtain an actionthat is invariant under local conformal transformations is φ ( β T ∗ abc T ∗ abc + β T ∗ abc T ∗ bac + β T ∗ a T ∗ a ), where the β i are dimensionless parameters and φ is some (compen-sator) scalar field with Weyl weight w ( φ ) = −
1. This be-haviour differs markedly from that encountered in WGTor PGT, in which all the fields strengths transform co-variantly under all the localised transformations. It istherefore necessary to adapt Kibble’s approach slightly ,as we now outline, to reduce the complications arisingfrom the gauging of SCTs.The above complications arise, in part, from the factthat the quantity ¯Γ µ is not the connection for the gaugegroup H , as is apparent from its transformation law(40b). The nature of ¯Γ µ can be better understood by con-sidering a purely internal SO (2 ,
4) symmetry, to whichthe conformal group C (1 ,
3) is isomorphic, with genera-tors π a , Σ ab , ∆ and κ a satisfying a set of commutationrules analogous to (9), where π a is the translational gen-erator. One then defines the quantity e Γ µ ≡ b aµ π a + ¯Γ µ , (48)1which transforms under the simultaneous local action of H and a GCT as δ e Γ µ = − ξ ν ∂ ν e Γ µ − e Γ ν ∂ µ ξ ν − ∂ µ ε − [ e Γ µ , ε ] , (49)and is hence a connection for the group SO (2 , e Γ µ correspond precisely those found in (40) (where we recallthat b aµ is the inverse of h aµ ). Thus, b aµ and ¯Γ µ togetherconstitute a connection for the group SO (2 , G -covariant’ derivative op-erator e D µ ≡ ∂ µ + e Γ µ , the resulting commutator reads[ e D µ , e D ν ] ϕ = ( e R abµν Σ ab + e H µν ∆+ S ∗ aµν κ a + T ∗ aµν π a ) ϕ, (50)where the new rotational and dilational ‘ G -covariant’ fieldstrengths are given in terms of those defined in (44) as e R abµν = R abµν + 8 b [ a [ µ f b ] ν ] , (51a) e H µν = H µν + 4 b a [ µ | f a | ν ] . (51b)One may define the corresponding GCT scalar fieldstrengths e R abcd and e H cd , in an analogous manner to thatused above. The resulting set of field strengths againtransform covariantly under local Lorentz rotations andlocal dilations (with the same weights as given previ-ously), but now transform under local SCTs as δ e R abcd = 4 c [ a T ∗ b ] cd (52a) δ e H cd = − c a T ∗ acd (52b) δ S ∗ a cd = − c b e R abcd + 2 c a e H cd , (52c) δ T ∗ acd = 0 . (52d)Once again, there is ‘mixing’ between these transforma-tion laws, although now they depend only on the fieldstrengths (and on the parameters of the local SCT, asexpected).
1. Auxiliary conformal gauge theory with non-zero torsion
Given the transformation laws (52), the most general,parity-even free-gravitational action (containing no com-pensator scalar fields) that is invariant under local con-formal transformations is uniquely determined (up to anoverall multiple), and given by S G = α Z h − ( e R abcd e R abcd + 4 T ∗ abc S ∗ abc + 2 e H ab e H ab ) d x. (53)The phenomenology of the resulting gravity theory re-mains to be fully explored, but one can show that thegauge field f aµ corresponding to local SCTs acts as an auxiliary field (hence the name for this approach), sinceits field equation may be used to eliminate it from theaction (53). Hence it appears that the symmetry re-duces back to the local Weyl group. In principle, the to-tal Lagrangian could again also include the combination φ ( β T ∗ abc T ∗ abc + β T ∗ abc T ∗ bac + β T ∗ a T ∗ a ), where the β i are dimensionless parameters and φ is some (compen-sator) scalar field with Weyl weight w ( φ ) = −
1, togetherpossibly with an additional kinetic term for φ , but theseadditional terms appear not to have been considered pre-viously.
2. Auxiliary conformal gauge theory with vanishing torsion
One sees from (52) that, since T ∗ acd transforms co-variantly under the full gauged conformal group, one canconsistently set it to zero, if desired. In this case, e R abcd and e H cd then also become fully covariant, and so thenumber of terms that may be included in a total actionthat remains invariant under local conformal transforma-tions is considerably increased. Moreover, the condition T ∗ acd = 0 can be used to eliminate the rotational gaugefield A abµ by writing it in terms of the translational anddilational gauge fields h aµ and B µ . This torsionless spe-cial case of auxiliary conformal gauge theory has beenstudied more extensively . The only admissible La-grangian term that is linear in the gauge field strengthsis φ e R , but it may be shown that variation of the re-sulting action with respect to the gauge field f aµ leadsto inconsistencies . Attention has therefore focussed onLagrangians consisting of (arbitrary) linear combinationsof terms quadratic in the field strengths e R abcd and e H cd and their contractions (and without compensator scalarfields). It may be shown, however, that in every suchcase, the gauge field f aµ may again be eliminated fromthe original action by its own field equation , suchthat the resulting action depends only on h aµ and B µ .Indeed, every such action is found to be equivalent to S G = Z h − ( α C abcd C abcd + β H ab H ab ) d x, (54)where C abcd is the conformal tensor, defined by C abcd = R abcd − η c [ a R b ] d + η d [ a R b ] c + η c [ a η b ] d R , (55)in which R abcd = 2 h cµ h dν ( ∂ [ µ A abν ] + A ac [ µ A cbν ] ) isthe gauge theory equivalent of the Riemann tensor, obey-ing all the usual symmetries and identities, and the quan-tities A abµ are the Ricci rotation coefficients A abµ = h aν ∂ [ µ | b b | ν ] − h bν ∂ [ µ | b a | ν ] − b cµ h [ aλ h b ] ν ∂ λ b cν , (56)which depend entirely on the translational gauge field h µa (and its inverse) . Thus, with the elimination of f aµ ,the symmetry appears once again to have reduced backto the local Weyl group.We conclude this section by noting that the auxiliaryconformal gauge theories that we have constructed us-ing (a slight generalisation of) Kibble’s approach areidentical to those obtained using the quotient manifold2method of gauging, in which the Lie group G is the con-formal group (more precisely, the elements of it thatare connected to the identity) with the 15 generators { P a , M ab , D, K a } given in (8), H is the inhomogeneousWeyl group with 11 generators { M ab , D, K a } , and thequotient G / H is thus a homogeneous 4-dimensional man-ifold M (interpreted as spacetime). In particular, thisapproach leads to field strength tensors that agree pre-cisely with those used to construct the action (53) . C. Ungauging the conformal group
According to Lord , in order to justify that the localaction of the (little) subgroup H , together with generaldiffeomorphisms (or GCT) on M , does indeed constitutea true gauge theory of a spacetime group G , one mustshow that the limiting case of ‘ungauged’ transformationsdoes in fact correspond to the correct global action of G on M and on fields in M . Lord demonstrates that thisholds for the auxiliary conformal gauge theory (ACGT)described above, and also for analogous gauge theoriesbased on the de Sitter group and (by Wigner–Inonii con-traction of the de Sitter case) the Poincar´e group; the‘ungauged’ limit of Poincar´e gauge theory is also consid-ered by Hehl .The ‘ungauged’ limit corresponds to vanishing gaugefield strengths. For ACGT, one thus requires e R abcd , e H cd , S ∗ a cd and T ∗ acd to vanish. In this limit, the coordinatesystem and H -gauge can be chosen such that h aµ ( x ) = δ µa , A abµ ( x ) = 0 , B µ ( x ) = 0 , f aµ ( x ) = 0 . (57)In this reference system, the first condition means thatthe distinction between Latin and Greek indices is lost.It is important, however, to retain this distinction (andthat between calligraphic and non-calligraphic quanti-ties) when considering behaviour under any subsequentGCT and H -gauge transformation.From the transformation laws (40) and (42) of thegauge fields, Lord notes simply that in order for any sub-sequent GCT and H -gauge transformation to preserve the relations (57), one requires ∂ β ξ α = ̟ αβ + ̺δ αβ , (58a) ∂ µ ̟ αβ = 4 c [ α δ β ] µ , (58b) ∂ µ ̺ = − c µ , (58c) ∂ µ c α = 0 . (58d)Successive integration of these equations (from the lastto the first) is straightforward and yields an expressionfor ξ α ( x ) of the form (3) for an infinitesimal global con-formal transformation. Similarly, the transformation law(36) reduces to that given in (10) for the action of aninfinitesimal global conformal transformation on a (pri-mary) physical field. Hence, Lord concludes that ACGThas the correct ‘ungauged’ limit. It is not clear from Lord’s discussion, however, why the‘ungauged’ limit should be derived by requiring that therelations (57) be preserved. Although preserving theserelations ensures that the covariant derivative remainsequal to the simple partial derivative, it is certainly un-necessary for the field strength tensors to remain zero,to which the ‘ungauged’ limit corresponds. Indeed, sincethese tensors are GCT scalars and transform covariantlyunder local Lorentz rotations and local dilations, and ac-cording to (52) under local SCTs, they will remain zerounder any subsequent GCT and H -gauge transformation,which in general will not preserve the relations (57).Moreover, as we now show, the final three relationsin (57) are superfluous for identifying global conformaltransformations as the ‘ungauged’ limit of ACGT. Re-quiring only that the first relation in (57) be preserved(which ensures the equivalence of Latin and Greek in-dices before and after the transformation) leads imme-diately to the first equation in (58), which is simply aconsequence of demanding that δ h aµ = 0. It is straight-forward to show, however, that the first relation in (58)is both a necessary and sufficient condition for ξ α ( x ) tosatisfy the 4-dimensional conformal Killing equation inMinkowski spacetime given in (2), from which it followsthat the most general solution for ξ α ( x ) has the form (3)of an infinitesimal global conformal transformation. Theremaining three conditions in (58) then follow automat-ically, which in turn means that the final three relationsin (57) are also preserved. Thus, for ACGT, the require-ment that these further relations be preserved is superflu-ous, and the correct ‘ungauged’ limit can be identified byrequiring only that the first relation in (57) is preserved.It is clear, however, that imposing this reduced require-ment to other gauge theories will not in general isolatethe correct ‘ungauged’ limit. Consider WGT, for ex-ample, for which G is the inhomogeneous Weyl groupand H is the homogeneous Weyl group. The structureof WGT is easily obtained from that of ACGT by set-ting c a ≡ f aµ ≡ c α = 0. This furthercondition can be obtained only by requiring the relations A abµ = 0 and B µ = 0 in (57) are also preserved (recallthat f aµ ≡ A abµ and B µ in WGT, which are given by (42) with c a = 0.Given the lack of a clear rationale for identifying the‘ungauged’ limit of a gravitational gauge theory by im-posing Lord’s condition that (the appropriate subset of)the relations (57) should be preserved, it is of interest toinvestigate an alternative prescription. This may be mo-3tivated most naturally by considering more closely theidentification of the ‘ungauged’ limit with the require-ment that field strength tensors should vanish.To this end, let us consider the ACGT covariant deriva-tive of some matter field ϕ ( x ), which from (37), (38) and(41) is given by¯ D c ϕ = h cµ ( ∂ µ + A abµ Σ ab + B µ ∆ + f aµ κ a ) ϕ. (59)It is clear that the dynamics of the matter field will besensitive to the translational gauge field h cµ , irrespec-tive of the nature of ϕ . This is not the case, however,for the other gauge fields A abµ , B µ and f aµ . Depend-ing on the nature of ϕ , the dynamics of the matter fieldmay be insensitive to one or more of these gauge fields.To establish the ‘ungauged’ limit, one should thereforeconsider the ‘subsidiary’ field strength tensors obtainedfrom e R abcd , e H cd , S ∗ a cd and T ∗ acd by including only thoseterms that depend on A abµ , B µ or f aµ , respectively (or,equivalently, by setting the other gauge fields in each caseto be identically zero).Thus, starting from the relations (57), one should de-mand that under subsequent GCT and H -gauge trans-formations that preserve the first relation (such that δ h aµ = 0), all ‘subsidiary’ field strength tensors re-main zero. Such tensors are still covariant under GCTs,but (typically) not so under general H -gauge transfor-mations, and hence will not automatically remain zero,even if one starts from the set of relations (57). Thus, de-manding that they do so can impose further constraintson the allowed nature of the subsequent GCT and H -gauge transformations, beyond the requirement imposedby δ h aµ = 0 that the most general solution for ξ α ( x ) isthe infinitesimal global conformal transformation (3).A straightforward way of imposing this requirement isto demand that, under subsequent GCT and H -gaugetransformations satisfying δ h aµ = 0, the change in each‘full’ field strength e R abcd , e H cd , S ∗ a cd and T ∗ acd aris-ing from the change in each gauge field A abµ , B µ or f aµ should vanish separately . It is clear that the condi-tion δ h aµ = 0 guarantees that the variation in the fieldstrength tensors vanishes if the variation in their non-calligraphic counterparts does. The transformation lawsof the latter (assuming δ h aµ = 0) are given in terms ofthe transformations of the other gauge fields by δ e R ab µν = 2 ∂ [ µ δ A abν ] + 8 δ [ a [ µ δ f b ] ν ] , (60a) δ e H µν = 2 ∂ [ µ δ B ν ] + 4 η a [ µ δ f aν ] , (60b) δ S ∗ aµν = 2 ∂ [ µ δ f aν ] , (60c) δ T ∗ aµν = 2 δ A ab [ µ δ bν ] + 2 δ B [ µ δ aν ] . (60d)We thus require that each term on the RHS of theseequations should vanish separately. From the transfor-mation laws (42) of these gauge fields, this requirementis satisfied only if δ A abµ , δ B µ and δ f aν each vanish.This is, however, equivalent merely to the final three re-lations in (57) being preserved, which follows automati-cally from our initial requirement that the first relation in (57) is preserved. Thus, no further conditions applyand one correctly deduces that the most general solu-tion for ξ α ( x ) has the form (3) of an infinitesimal globalconformal transformation.Let us now repeat the above process for WGT. In thiscase, the field strengths are R abcd , H cd and T ∗ acd , wherethe first two may be obtained from their counterparts inACGT by setting f aµ ≡
0, as is clear from (51). Thus,if one again starts from the conditions (57) (again recall-ing that f aµ ≡ H -gauge transformations, the transformation laws of theWGT field strengths are given in terms of the transfor-mations of the WGT gauge fields A abµ and B µ by thecorresponding expressions in (60) with δ f aµ ≡
0. Fromthe transformation laws of the WGT gauge fields A abµ and B µ , which may be obtained from (42) by setting c a ≡
0, one may show that our additional requirementis satisfied only if δ A abµ and δ B µ each vanish. Thesefurther conditions correspond to the second and third re-lations in (57) being preserved, which in turn requires c a = 0. Thus, one correctly deduces that the most gen-eral solution for ξ α ( x ) has the form of a global Weyltransformation.Finally, it is a simple matter to verify that an analo-gous procedure applied to PGT leads to the correct iden-tification of the corresponding ‘ungauged’ limit as globalPoincar´e transformations. Indeed, for PGT, WGT andACGT, this procedure is found to be equivalent to requir-ing that (the appropriate subset of) the relations (57) arepreserved, as Lord originally suggested. As we will seein Section IV D, however, these two approaches are notalways equivalent. D. Biconformal gauging
Before moving on to discuss our new approach forgauging the conformal group in Section IV, we concludethis section with a brief discussion of an existing alterna-tive scheme, known as biconformal gauging, which leadsto a very different conformal gauge theory to ACGT, withseveral interesting properties.As discussed in Section III B, in the standard approachto gauging the conformal group, one may eliminate thegauge field f aµ corresponding to local SCTs, which im-plies that the symmetry has reduced back to the localWeyl group. As pointed out by Wheeler , however, thisreduction is rather curious, since in addition to the elim-ination of f aµ , the symmetry between the generators P a and K a in the Lie algebra of the conformal group has alsobeen lost. Indeed, the elimination of f aµ occurs becauseone chooses to identify the translational gauge field h aµ with the vierbein (at least in the geometrical interpreta-tion), as is done in PGT. In gauging the conformal group,however, one can make the alternative choice of identify-ing the SCT gauge field f aµ with the vierbein, in whichcase the translational gauge field h aµ may be eliminated4instead. Thus, in the case of the conformal group, thereis an additional symmetry between the two generators P a and K a , which is broken by one’s (arbitrary) choicefor identifying the vierbein.An alternative approach to gauging the conformalgroup, which preserves the symmetry between P a and K a by construction, is the so-called biconformal gauging .Expressed in terms of the quotient manifold method, inbiconformal gauging the Lie group G is again the con-formal group (only the elements connected to the iden-tity) with the 15 generators { P a , M ab , D, K a } , but theLie subgroup H is now the homogeneous Weyl groupwith the 7 generators { M ab , D } . The resulting quotient G / H is thus an 8-dimensional manifold, called bicon-formal space, which has a number of very interestingproperties , as we now briefly describe.Biconformal space is spanned by the basis 1-forms h a and f a , derived from the translations and special confor-mal transformations, respectively. If the dilational cur-vature vanishes, then the non-degenerate 2-form h a ∧ f a is also closed (i.e. an exact differential), and hence sym-plectic. Moreover, the Killing metric is non-degeneratewhen restricted to the base manifold, so that the groupstructure determines a metric, rather than imposing oneby hand. If the basis 1-forms h a and f a are separatelyin involution and orthogonal, then biconformal space canbe considered as a form of relativistic phase space, con-sisting of separate configuration and momentum metricsubmanifolds. Moreover, the signatures of these subman-ifolds are severely limited and, in particular, the notionof time emerges naturally, since the configuration spacemust be Lorentzian and is therefore interpreted as space-time. It is usually argued that the full biconformal spaceshould be interpreted as representing ‘the world’, sinceboth classical and quantum mechanics take their most el-egant forms in phase space and, moreover, a phase spaceis required to formulate the uncertainty principle.To define a dynamical theory, one writes down an ac-tion on the full biconformal space. Its symplectic struc-ture means that the volume element is dimensionless, andso an action linear in the curvatures can be conformally invariant, without introducing any additional (compen-sating) fields. If one assumes the torsion to vanish andthe momentum subspace is flat, the theory reduces togeneral relativity on the spacetime tangent bundle.It is clear that biconformal gauge theory (BCGT) hassome very interesting properties and is worthy of contin-ued investigation, but we will not pursue it further here. IV. NEW APPROACH TO CONFORMAL GAUGING
In an earlier paper , we introduced a novel alternativeto standard Weyl gauge theory, in which we proposed an‘extended’ form for the transformation law of the rota-tional gauge field under finite local dilations, given by A ′ abµ = A abµ + θ ( b aµ Q b − b bµ Q a ) , (61)where Q a ≡ h aµ Q µ , Q µ ≡ ∂ µ ̺ , and θ is an arbitrary pa-rameter that can take any value. We noted further therethat this extended transformation law implements Weylscaling in a novel way that may be related to gauging ofthe full conformal group. We now discuss this issue inmore detail.The proposal (61) was motivated by the observationthat the WGT (and PGT) matter actions for the mass-less Dirac field and the electromagnetic field are invariantunder local dilations even if one assumes this ‘extended’transformation law for the rotational gauge field, whichincludes its ‘normal’ transformation law in WGT as thespecial case θ = 0. Moreover, under a global scale trans-formation, the two transformation laws clearly coincide.A complementary motivation for introducing the ex-tended transformation law (61) is that under local dila-tions it places the transformation properties of the PGTrotational gauge field strength (or ‘curvature’) R abcd andtranslational gauge field strength (or ‘torsion’) T abc ona more equal footing with one another than is the caseunder the standard WGT transformation law with θ = 0.Indeed, assuming the extended transformation law, theytransform under local dilations as R ′ abcd = e − ̺ {R abcd + 2 θδ [ ad ( D c − θ Q c ) Q b ] − θδ [ ac ( D d − θ Q d ) Q b ] − θ Q [ a T b ] cd − θ δ [ ac δ b ] d Q e Q e } , (62a) T ′ abc = e − ̺ {T abc + 2(1 − θ ) Q [ b δ ac ] } . (62b)Thus, for general values of θ , neither R abcd nor T abc transforms covariantly. For θ = 0, however, one recov-ers the ‘normal’ transformation law for the A -field, suchthat R abcd transforms covariantly under local dilations,but T abc transforms inhomogeneously. By contrast, for θ = 1 one obtains a covariant transformation law for T abc , but an inhomogeneous one for R abcd . The extended A -field transformation law (61) accommodates these ex-treme cases in a balanced manner. We therefore developed our so-called ‘extended’ Weylgauge theory (eWGT), which is based on the construc-tion of a new form of covariant derivative D † a ϕ that trans-forms in the same way under local Weyl transformationsas ∂ µ ϕ does under the global Weyl transformations, butwhere the rotational gauge field introduced is assumedto transform under local dilations as (61). The resultingtheory has a number of interesting features, which wesummarise briefly below.5 A. Extended Weyl gauge theory (eWGT)
In eWGT, the spacetime group G under considerationis the inhomogeneous Weyl group and its subgroup H is the homogeneous Weyl group, as in standard WGT.It is also assumed that any physical (matter) fields ϕ ( x )belong to an irreducible representation of the Lorentzgroup (as is usually the case in physical theories). Con-sequently, as discussed in Sections II A and II B, the gen-erator of dilations takes the simple form ∆ = wI , where w is the Weyl weight of ϕ . Adopting Kibble’s generalmethodology, the gauged action of G on such a field isconsidered as a GCT x µ → x ′ µ , together with the localaction of H , such that (in finite form) one obtains a localversion of (19), namely ϕ ′ ( x ′ ) = e w̺ ( x ) S ( ω ( x )) ϕ ( x ) . (63)
1. Covariant derivative
Following the usual approach, the construction of thecovariant derivative in eWGT is achieved in two steps.First, one defines the ‘ H -covariant’ derivative D † µ ϕ ( x ) ≡ [ ∂ µ + Γ † µ ( x )] ϕ ( x ) , (64)where Γ † µ ( x ) is a linear combination of the generators of H that depends on the gauge fields. Second, one con-structs the ‘generalised H -covariant’ derivative, linearlyrelated to D † µ ϕ by D † a ϕ ( x ) ≡ h aµ ( x ) D † µ ϕ ( x ) , (65) where h aµ ( x ) is the translational gauge field, again as-sumed to have the inverse b aµ ( x ).In eWGT, however, one does not adopt the standardapproach of introducing each gauge field in Γ † µ ( x ) as thelinear coefficient of the corresponding generator, such asin (41), since this would lead directly to the standardWGT transformation laws for the rotational and dila-tional gauge fields (given in infinitesimal form by the firsttwo relations in (42) with c a ≡ A abµ ( x ) and the ‘dilational’ gaugefield B µ ( x ) in a very different way, so thatΓ † µ = A † abµ Σ ab + ( B µ − T µ )∆ , (66)in which T µ = b aµ T a , where T a ≡ T bab is the trace ofthe PGT torsion, and we have introduced the modified A -field A † abµ ≡ A abµ + ( b aµ B b − b bµ B a ) , (67)where B a = h aµ B µ . It is worth noting that A † abµ isnot considered to be a fundamental field, but merely ashorthand for the above combination of the gauge fields h aµ (or its inverse), A abµ and B µ . Similarly, T µ is merelya shorthand for the corresponding function of the gaugefields h aµ (or its inverse) and A abµ .It is straightforward to show that, if ϕ has Weyl weight w , then (65) does indeed transform covariantly with Weylweight w −
1, as required, under the gauged (finite) actionof G , such that D †′ a ϕ ′ ( x ′ ) = e ( w − ̺ ( x ) Λ ab ( x ) S ( ω ( x )) D † b ϕ ( x ) , (68)if the gauge fields transform according to h ′ aµ ( x ′ ) = X µν e − ̺ ( x ) Λ ab ( x ) h bν ( x ) , (69a) A ′ abµ ( x ′ ) = X µν [Λ ac ( x )Λ bd ( x ) A cdν ( x ) + Λ [ ac ( x ) ∂ ν Λ b ] c ( x ) + 2 θb [ aν ( x ) Q b ] ( x )] , (69b) B ′ µ ( x ′ ) = X µν [ B ν ( x ) − θQ ν ( x )] , (69c)where X µν ≡ ∂x ′ µ /∂x ν are the elements of the GCTtransformation matrix and X µν ≡ ∂x ν /∂x ′ µ are the el-ements of its inverse. Hence, we have achieved our goalof accommodating the A -field transformation (61) underlocal dilations, while recovering the full transformationlaw in WGT for the special case θ = 0. By contrast,the transformation law for B µ reduces to that in WGTfor the special case θ = 1. Unlike the transformationlaws for A abµ and B µ , the covariant derivative (65) doesnot explicitly contain the parameter θ . Consequently, itdoes not reduce to the standard WGT covariant deriva-tive D ∗ µ ϕ in either special case θ = 0 or θ = 1, whileretaining the transformation law (68) for any value of θ . It is clear that the structure of the ‘connection’ Γ † in(66) is very different to that normal adopted, such as (41)in ACGT. In particular, whereas each generator in (41) ismultiplied purely by the corresponding gauge field, eachgenerator in (66) is multiplied by a (non-linear) functionof all the gauge fields, including the translational gaugefield h aµ (or its inverse), which is completely absent from(41). This results in eWGT having a fundamentally dif-ferent structure to standard gauge theories. In particu-lar, the eWGT field strengths depend very differently onthe gauge fields from their counterparts in other gaugetheories, as we now describe.6
2. Field strengths
The eWGT gauge field strengths are defined in theusual way in terms of the commutator of the covari-ant derivatives. Considering first the eWGT H -covariantderivative, one finds that[ D † µ , D † ν ] ϕ = ( R † abµν Σ ab + H † µν ∆) ϕ, (70)which is of an analogous form to the corresponding resultin WGT (which may be obtained from (43) by setting f aµ = 0 and hence S ∗ aµν = 0), but the eWGT fieldstrengths have very different dependencies on the gaugefields. In particular, one finds R † abµν ≡ ∂ [ µ A † abν ] + η cd A † ac [ µ A † dbν ] ) ,H † µν ≡ ∂ [ µ ( B ν ] − T ν ] ) , (71)both of which transform covariantly under GCT and lo-cal Lorentz rotations in accordance with their respectiveindex structures, and are invariant under local dilations.Considering next the commutator of two ‘generalised H -covariant’ derivatives, one finds[ D † c , D † d ] ϕ = ( R † abcd Σ ab + H † cd ∆ − T † acd D † a ) ϕ, (72)where R † abcd = h cµ h dν R † abµν and H † cd = h cµ h dν H † µν ,and the translational field strength is given by T † abc ≡ h bµ h cν T † aµν ≡ h bµ h cν D † [ µ b aν ] . (73)We note that R † abcd and T † abc are given in terms of theircounterparts R abcd and T abc in PGT by R † abcd = R abcd +4 δ [ b [ c ( D d ] −B d ] ) B a ] − B δ [ ac δ b ] d − B [ a T b ] cd , T † abc = T abc + δ a [ b T c ] , (74)where B ≡ B a B a and for brevity we have introduced thederivative operator D a ≡ h aµ D µ ≡ h aµ ( ∂ µ + A abµ Σ ab )familiar from PGT. It is particularly important to notethat the trace of the eWGT torsion vanishes identically,namely T † b ≡ T † aba = 0, so that T † abc is completelytrace-free (contraction on any pair of indices yields zero). R † abcd , H † cd and T † acd are GCT scalars and transform co-variantly under local Lorentz transformations and underlocal dilations with weights w ( R † abcd ) = w ( H † cd ) = − w ( T † acd ) = −
3. Action
As in other gravitational gauge theories, the total ac-tion in eWGT consists typically of kinetic terms for anymatter field(s) ϕ , terms describing the coupling of thematter field(s) to the gravitational gauge fields (and pos-sibly to each other), and (kinetic) terms describing thedynamics of the free gravitational gauge fields. Since D † a ϕ is constructed to have an analogous trans-formation law under extended local Weyl transformationsto that of ∂ µ ϕ under global Weyl transformations, onemay immediately construct a matter action that is fullyinvariant under the extended gauged Weyl group fromone that is invariant under global Weyl transformationsby employing the usual minimal coupling procedure of re-placing partial derivatives by covariant ones to obtain S M = Z h − L M ( ϕ i , D † a ϕ i ) d x. (75)As mentioned previously, the set of fields ϕ i may alreadyinclude a scalar compensator field (denoted also by φ )with Weyl weight w = −
1, for example in a Yukawacoupling term of the form µφ ¯ ψψ with a massless Diracfield ψ (since this allows for the Dirac field to acquirea mass dynamically upon adopting the Einstein gauge φ = φ ) , together perhaps with kinetic and quarticpotential terms for φ of the form ν D † a φ D † a φ − λφ (where µ , ν and λ are dimensionless parameters).The terms in the total action that describe the dy-namics of the free gravitational gauge fields are con-structed from the gauge field strengths. In contrast toACGT, the eWGT field strengths all transform covari-antly under the full group of localised transformations,and so may be used straightforwardly to construct thefree-gravitational action. The requirement of local scaleinvariance requires the free-gravitational Lagrangian L G to be a relative scalar with Weyl weight w ( L G ) = − L R † of the six distinct terms quadratic in R † abcd and its contractions, and a term L H † ∝ H † ab H † ab . Inprinciple, one could also include quartic terms in T † abc (which has no non-trivial contractions, unlike its coun-terparts in PGT, WGT and ACGT), or cross-terms suchas R † [ ab ] H † ab , but these are not usually considered. Thus,one typically has S G = Z h − ( L R † + L H † ) d x, (76)where any parameters in the action are dimensionless.In particular, L G cannot contain the linear Einstein–Hilbert analogue term L R † ≡ − a R † (where R † ≡R † abab and the factor of − / L T † ≡ β T † abc T † abc + β T † abc T † bac . Nonetheless, such terms canbe included in the total Lagrangian if they are multipliedby a compensator scalar field term φ . Such combina-tions are therefore usually considered not to belong tothe free gravitational Lagrangian and are instead addedto the matter Lagrangian L M48 . Thus, the matter La-grangian may have an extended form, including all inter-actions of the matter fields with the gravitational gaugefields, which is given by L M + ≡ L M + φ ( L R † + L T † )(in which the parameters a , β i are again dimensionless),such that the corresponding action has the functional de-7pendencies S M = Z h − L M + ( ϕ i , D † a ϕ i , R † , T † abc ) d x, (77)where the set of fields ϕ i includes the scalar compensator.In any case, it is only the form of the total Lagrangian L T = L M + + L G that is relevant for the field equations.Finally, it is worth noting that terms containing co-variant derivatives of (contracted) field strengths, suchas D † a D † b R † ab or D † a D † a R † , are of Weyl weight w = − L G7 . In eWGT,however, such terms contribute only surface terms to theaction, as a consequence of the trace T † a of the eWGTtorsion vanishing identically. Thus, such terms have noeffect on the resulting field equations, and so may omit-ted (at least classically); this is not true in general forother gauges theories, such as PGT, WGT and ACGT.
4. Field equations
The eWGT field equations are obtained by varying thetotal action S T with respect to the gravitational gaugefields h aµ , A abµ and B µ , together with the matter fields ϕ i (which may include a scalar compensator field φ ).Defining τ aµ ≡ δ L T /δh aµ , σ abµ ≡ δ L T /δA abµ and ζ µ ≡ δ L T /δB µ , where L T ≡ h − L T , the set of gravi-tational field equations are most naturally expressed interms of their counterparts carrying only Latin indices τ ab ≡ τ aµ h bµ , σ abc ≡ σ abµ b cµ and ζ a ≡ ζ µ b aµ , as τ ab = 0 , (78a) σ abc = 0 , (78b) ζ a = 0 . (78c)The quantities τ ab , σ abc and ζ a are clearly scalars un-der GCT, and it is straightforward to show that eachof them also transforms covariantly under local Lorentzrotations and local dilations, as expected, with Weylweights w = 0, w = 1 and w = 1 respectively. More-over, with one exception, these transformation propertiesalso hold for the corresponding quantities obtained from any subset of the terms in L T that transforms covariantlywith weight w = − L M , L M + or L G separately).The exception relates to quantities corresponding to τ ab ,which transform covariantly under local dilations only ifone considers all the terms in L T , and then only by virtueof the A -field equation (78b). This unusual feature is aresult of the extended transformation law (61) for A abµ containing one of the other gauge fields, namely b aµ , andleads one to introduce the related quantities τ † ab ≡ τ ab − σ cba B c − σ cac B b . (79) These do transform covariantly when one considers onlysome subset of the terms in L T that themselves transformcovariantly and with weight w = − τ † ab = 0 . (80)Indeed, this field equation emerges naturally if oneadopts an alternative variational principle, in which A abµ is replaced by A † abµ in the set of field variables; thisapproach also considerably shortens the calculations in-volved in deriving all the gravitational field equations.Another unusual feature of the eWGT field equations,which also emerges most naturally from the alternativevariational principle, is that for any total Lagrangian L T in which the gravitational gauge fields appear onlythrough eWGT covariant derivatives or field strengths(which is usually the case), one may show that ζ a ≡ σ bab . (81)Consequently, in this generic case, the B -field equation(78c) is no longer independent, but merely the relevantcontraction of the A -field equation (78b). Moreover, therelation (81) also holds for the corresponding quanti-ties obtained from any subset of the terms in L T thattransforms covariantly with weight w = − S T with respect to the fields ϕ i (whichmay include a scalar compensator field φ ). In the (usual)case in which L T is a function of the matter fields onlythrough ϕ i and D † a ϕ i , these may be shown to have thesimple (and manifestly covariant) forms¯ ∂L T ∂ϕ i − D † a ∂L T ∂ ( D † a ϕ i ) ! = 0 , (82)where ¯ ∂L T /∂ϕ i ≡ [ ∂L T ( ϕ i , D † a u ) /∂ϕ i ] u = ϕ i , so that ϕ i and D † a ϕ i are treated as independent variables.
5. Conservation laws
Invariance of S T under (infinitesimal) GCTs, localLorentz rotations and extended local dilations, respec-tively, lead to conservations laws of the general form(B12), as discussed in Appendix B 2. These can be writ-ten in the following manifestly covariant form:8 D † c ( hτ † cd ) + h ( τ † cb T † bcd − σ abc R † abcd − ζ † c H cd ) + δL T δϕ i D † d ϕ i = 0 , (83a) D † c ( hσ abc ) + 2 hτ † [ ab ] + δL T δϕ i Σ ab ϕ i = 0 , (83b) hτ † cc − δL T δϕ i w i ϕ i = 0 , (83c) D † c ( hζ † c ) = 0 , (83d)where we have defined the quantities ζ † a ≡ ζ a − σ bab and H ab = 2 h aµ h bν ∂ [ µ B ν ] , which are both easily verified tobe GCT scalars and to transform covariantly under localLorentz rotations and local dilations.These conservation equations have a very differentform to those in WGT. In particular, invariance of S T under local dilations leads to both of the last two conser-vation laws. The third conservation law (83c) is unusualin being an algebraic condition on the trace τ † aa in termsof the field equations of the matter fields ϕ i . Indeed, as-suming the field equations of all non-compensator mat-ter fields to hold, one thus finds that the field equationfor the scalar compensator field φ is no longer indepen-dent, but simply related to the trace of the (alternative) h -field equation (80). Also, as mentioned above, for theusual forms of S T , the condition (81) holds, in which case ζ † a ≡ any subset of terms in L T that is covariant under localLorentz transformations and under local dilations withweight w = − L M , L M + or L G separately). B. Finite local conformal invariance
As we noted in Sections II A and II B, for physicalfields ϕ i ( x ) that belong to irreducible representations ofthe Lorentz group, as is assumed in eWGT, the action(both infinitesimal and finite) of a general element ofthe conformal group that is connected to the identitycorresponds to a combination of a translation, (proper)Lorentz rotation and dilation; in particular, a SCT corre-sponds merely to a Lorentz rotation and dilation that de-pend on spacetime position x in a prescribed way. Sincetranslations, (proper) Lorentz rotations and dilations arealready gauged in eWGT (and WGT), then so too areSCTs and hence any element of the conformal group thatis connected to the identity.As discussed in Section II B, however, the full con-formal group also includes the inversion operation (13),which is finite and discrete, and hence not connected tothe identity. Moreover, it is worth recalling that a SCTis merely the composition of an inversion, a translationand a second inversion. In Section II B, we demonstratedthat an inversion, together with its action on physical fields that belong to an irreducible representation of theLorentz group, consists of the composition of a dilation1 /x and a reflection I µν (ˆ x ) in the hyperplane perpendic-ular to ˆ x , both of which are clearly position dependentin a prescribed way. In particular, under an inversion,physical fields are acted upon by I µν (ˆ x ) for each tensorindex and by γ · ˆ x for each 4-spinor index.Since dilations are already gauged in eWGT (andWGT), the only new operation to consider is the reflec-tion. To our knowledge, the gauging of reflections hasnot been addressed previously, but the most natural ap-proach is to generalise the reflection in the hyperplaneperpendicular to ˆ x at each point to a reflection in thehyperplane perpendicular to some unit vector ˆ n ( x ) thatcan vary arbitrarily with spacetime position x . As usual,this gauged transformation should be completely decou-pled from GCTs, and so we denote the reflection matrixat each spacetime point by I ab (ˆ n ( x )), which operates oneach Latin tensor index carried by a field (or, equiva-lently, γ · ˆ n ( x ) for each spinor index).From the discussion in Section II B, however, I ab (ˆ n ( x ))corresponds to a finite improper Lorentz transformationmatrix at each spacetime point. Thus eWGT already accommodates gauged reflections, without the need tointroduce any more gauge fields, provided that each oc-currence of the proper Lorentz transformation matrixΛ ab ( x ) in the finite transformation laws (68) and (69)for the covariant derivative and the existing gauge fields,respectively, is extended to denote a general transforma-tion matrix of the full Lorentz group (which consists ofproper Lorentz rotations and spacetime reflections) and,in particular, is given by I ab (ˆ n ( x )) under gauged reflec-tions. Indeed, the same holds true for WGT, for whichthe finite transformation laws of the gauge fields are givenby (69), with θ = 0 in (69b) and θ = 1 in (69c).Thus, provided all matter fields ϕ i ( x ) are assumed tobelong to irreducible representations of the Lorentz groupand with the above modest extension to the transfor-mation laws of the gauge fields, both WGT and eWGTaccommodate all the gauged symmetries of the full con-formal group, such that actions constructed in the usualway in each theory are invariant under (finite) local con-formal transformations. As we now demonstrate below,however, WGT cannot be considered as a true gauge the-ory of the conformal group in the usual sense, whereaseWGT can be interpreted as such.9
C. Local conformal conservation laws
As discussed in Section II C, if one considers a field the-ory in Minkowski spacetime that describes the dynamicsof a set of fields ϕ i ( x ) that belong to irreducible repre-sentations of the Lorentz group, then for the action (24)to be invariant under global conformal transformations(that are connected to the identity), one requires the firstthree conservation laws in (29) to hold ‘on-shell’ (whichtogether ensure Poincar´e and scale invariance) and thefield virial (30) to vanish (which ensures the additionalinvariance under SCTs), up to a total divergence.We now consider in more detail the forms of the con-servation laws in WGT and eWGT, both of which wehave just demonstrated have actions that are invariantunder local conformal transformations. As we will see,eWGT has very different conservation laws to WGT, which arises primarily from the unconventional form ofthe eWGT covariant derivative as compared with othergravitational gauge theories.
1. WGT
Let us first consider a matter action in WGTcontaining all the terms in the total action exceptthose that depend only on the gauge fields and theirderivatives. This typically has the form S M = R h − L M + ( ϕ i , D ∗ a ϕ i , R , T ∗ abc ) d x , by analogy with ourdiscussion leading to (77) in the context of eWGT. As-suming that the matter equations of motion are satis-fied (including that of the compensator field), such that δL M + /δϕ i = 0, invariance of S M under local Weyl trans-formations leads to three conservation laws of the generalform (B12), which can be written as the manifestly co-variant conditions ( D ∗ c + T ∗ c )( hτ cd ) + h ( τ cb T ∗ bcd − σ abc R abcd − ζ c H cd ) = 0 , (84a)( D ∗ c + T ∗ c )( hσ abc ) + 2 hτ [ ab ] = 0 , (84b)( D ∗ c + T ∗ c )( hζ c ) − hτ cc = 0 , (84c)where τ aµ ≡ δ L M + /δh aµ , σ abµ ≡ δ L M + /δA abµ and ζ µ ≡ δ L M + /δB µ (in which L M + ≡ h − L M + ), and theircounterparts carrying only Latin indices τ ab ≡ τ aµ h bµ , σ abc ≡ σ abµ b cµ and ζ a ≡ ζ µ b aµ are most naturally con-sidered as the (total) dynamical energy-momentum, spin-angular-momentum and dilation current, respectively, ofthe matter fields. The above conservation laws are clearlyinvariant under GCTs and transform covariantly underthe local action of the subgroup H of homogeneous Weyltransformations, as expected.The conservation laws (84) provide a natural gener-alisation for localised Weyl transformations of the firstthree conservation laws in (29). There is not, however,any further conservation law corresponding to the gener-alisation of the condition that the field virial (30) shouldvanish up to a total divergence, which was necessary toensure that the original action (24) be invariant underSCTs, in addition to global Weyl transformations, andhence invariant under global conformal transformations(connected to the identity). The absence of such a fur-ther conservation law in WGT demonstrates that it doesnot constitute a gauge theory of the conformal group inthe usual sense.One should note, however, that the quantities in (84)are dynamical currents, whereas those in (29) are canon-ical. It is therefore of interest to compare the formsof these two types of current. This comparison is fa-cilitated by first separating the contributions to the dy-namical matter currents resulting from each of the termsin L M + ≡ L M + φ L R + φ L T ∗ , which we denote by τ ab = ( τ M ) ab + ( τ R ) ab + ( τ T ) ab , and similarly for σ abc and ζ a . We then introduce the following covariant canon-ical currents of the matter fields t ∗ ab ≡ ∂L M ∂ ( D ∗ a ϕ i ) D ∗ b ϕ i − δ ab L M , (85a) s ∗ cab ≡ ∂L M ∂ ( D ∗ c ϕ i ) Σ ab ϕ i , (85b) j ∗ a ≡ ∂L M ∂ ( D ∗ a ϕ i ) w i ϕ i , (85c)which provide a natural generalisation of the standardcanonical currents t µa , s µαβ and j µ in (27) and (28).By considering the form of the WGT covariant deriva-tive D ∗ a ϕ i , one may show directly that the covariantcanonical currents and the dynamical currents derivedfrom L M alone are essentially equivalent in WGT, since h ( τ M ) ab ≡ t ∗ ab , h ( σ M ) abc ≡ s ∗ cab and h ( ζ M ) a ≡ j ∗ a .These equivalences may also be derived by demandingthe coincidence of the currents J µ and S µ derived from L M , which are discussed in Appendix B 2.
2. eWGT
Let us now repeat the above analysis for a matter ac-tion of the form S M = R h − L M + ( ϕ i , D † a ϕ i , R † , T † abc ) d x in eWGT, as given in (77). In this case, from the combi-nation of (81) and (83) (applied only to S M and assumingall the matter equations of motion to hold), one instead0obtains the conditions D † c ( hτ † cd ) + h ( τ † cb T † bcd − σ abc R † abcd ) = 0 , (86a) D † c ( hσ abc ) + 2 hτ † [ ab ] = 0 , (86b) hτ † cc = 0 , (86c) h ( ζ a − σ bab ) = 0 , (86d)where τ † ab is defined in (79). The conditions (86) havea somewhat different form from their WGT counterpartsin (84). In particular, (86c) shows that the trace of themodified dynamical energy-momentum tensor vanishes.This is reminiscent of the vanishing trace of the improvedenergy-momentum tensor (32), which encodes the invari-ance of theories under global scale transformations. In(86), however, one has not used the Belinfante procedureto combine the translational and rotational currents, butinstead retained the distinction between them. Thus, τ † ab remains non-symmetric, which is appropriate whenworking in terms of the tetrad rather than the metric,and also allows one straightforwardly to accommodatetorsion. Most important in eWGT, however, is the addi-tional final condition (86d), which is analogous to a co-variant generalisation of the condition that the field virial(30) should vanish. Thus the eWGT conservations laws(86) provide a natural local generalisation of all of theusual conservation laws (29) for theories that are invari-ant under global conformal transformations (and containonly fields that belong to irreducible representations ofthe Lorentz group).As was the case in our consideration of the WGT con-servation laws, however, it is also of interest to considerthe relationship between the dynamical currents in (86)and their canonical counterparts. This comparison isagain facilitated by first separating the contributions tothe dynamical matter currents resulting from each of theterms in L M + ≡ L M + φ L R † + φ L T † , in a similarmanner to that used for WGT. We also define a set ofeWGT covariant canonical currents t † ab , s † cab and j † a inan analogous manner to their WGT counterparts in (85),but with each occurence of the WGT covariant derivative D ∗ a ϕ i replaced by the eWGT covariant derivative D † a ϕ i .By considering the form of the latter, one may again di-rectly relate the dynamical and covariant canonical cur-rents, but in eWGT these relationships are somewhatmore complicated than those in WGT. In particular, onefinds (after a lengthy calculation) that h ( τ M ) † ab ≡ t † ab + ( D † b j † a − δ ab D † c j † c ) , (87a) h ( σ M ) abc ≡ s † cab + ( δ ca j † b − δ cb j † a ) , (87b) h ( ζ M ) a ≡ j † a − s † bab . (87c)Once again, these equivalences may also be derived bydemanding the coincidence of the currents J µ and S µ asderived from L M , as discussed in Appendix B 2. Substi-tuting the above expressions into (86), for the restrictedcase in which L M is the full matter Lagrangian density, yields the covariant canonical conservation laws D † c t † cd + t † cb T † bcd − s † cab R † abcd − j † c H † cd = 0 , (88a) D † c s † cab + 2 t † [ ab ] = 0 , (88b) D † c j † c − t † cc = 0 , (88c)and the final condition (86d) is satisfied identically.The expressions (88) clearly represent a natural lo-cal generalisation of the first three conservation laws in(29). Moreover, one sees from (87c) that, provided ( ζ M ) a vanishes up to a total divergence, then so too should s † bab − j † a , which provides a replacement additional con-dition that is a natural generalisation of the analogousrequirement on the field virial (30) for globally confor-mal invariant theories. This requirement is indeed satis-fied not only by ( ζ M ) a , but also by ζ a evaluated from thefull matter Lagrangian density L M + , which is given by hζ a = j † a − s † bab − ( ν + 3 a ) D † a φ , (89)provided the terms in L M + corresponding to the non-compensator matter fields ϕ i do not contain the dila-tion gauge field B µ . This occurs naturally if the ϕ i cor-respond to the Dirac field and/or the electromagneticfield . Thus, in terms of the covariant canonical cur-rents, the eWGT conservation laws once again provide anatural local generalisation of all of the usual conserva-tion laws (29) for theories that are invariant under globalconformal transformations. D. Ungauging eWGT
In Section III C, we considered the process of ‘ungaug-ing’ ACGT, and obtained the correct limit of global con-formal transformations. We also considered ‘ungauging’WGT and found the limit to correspond to global Weyltransformations, which again shows that WGT cannotbe considered as a true gauge theory of the conformalgroup. In this section, we consider the ‘ungauged’ limitof eWGT.One begins by requiring the field strength tensors R † abcd , H † cd and T † acd to vanish in the ‘ungauged’ limit.Similarly to WGT, in this limit, the coordinate systemand H -gauge can be chosen such that h aµ ( x ) = δ µa , A abµ ( x ) = 0 , B µ ( x ) = 0 . (90)It is worth noting that, in this reference system, theeWGT covariant derivative reduces to a partial deriva-tive. Thus, the eWGT covariant canonical currents t † ab , s † cab and j † a reduce to the standard ones in (27) and (28),and the conditions (88) and (89) reduce, respectively, tothe first three conservation laws in (29) and the vanishingof the field virial (30) up to a total divergence. Indeed,we note further that all these reductions also occur un-der the less restrictive set of conditions h aµ ( x ) = δ µa and A † abµ ( x ) = 0, which also ensure that R † abcd , H † cd and T † acd vanish.1Let us now apply an analogous approach to that dis-cussed in Section III C to ‘ungauge’ eWGT. We thusbegin by considering the eWGT covariant derivative ofsome matter field ϕ ( x ) (here belonging to some irre-ducible representation of the Lorentz group), which from(64), (65) and (66) is given by D † c ϕ = h cµ [ ∂ µ + A † abµ Σ ab + ( B µ − T µ )∆] ϕ, (91)where T a ≡ T bab is the trace of the PGT torsion andthe modified A -field is A † abµ ≡ A abµ + ( b aµ B b − b bµ B a ),as defined in (67). As previously, the dynamics of thematter field will be sensitive to the translational gaugefield h cµ , irrespective of the nature of ϕ . In eWGT, how-ever, depending on the nature of ϕ , the dynamics of thematter field may be insensitive to one or both of the com-binations A † abµ and B µ − T µ of the gauge fields. Thus,following the reasoning presented in Section III C, to es-tablish the ‘ungauged’ limit one should consider the ‘sub-sidiary’ field strength tensors obtained from R † abcd , H † cd and T † acd by including only those terms that depend onthe combinations A † abµ or B µ − T µ of the gauge fields,respectively.
1. Infinitesimal transformations
We first consider infinitesimal transformations, as wedid in Section III C. Thus, starting from the relations(90), one should demand that under subsequent GCTand H -gauge transformations that preserve the first re-lation (such that δ h aµ = 0 and so ensuring the equiv-alence of Latin and Greek indices before and after thetransformation), all ‘subsidiary’ field strength tensors re-main zero. The infinitesimal form of the eWGT trans-formation law for h aµ is easily obtained from its finiteform in (69), and is identical to that given in (40) forACGT (and WGT). Thus, following the argument givenin Section III C, the most general solution for ξ α ( x ) hasthe form (3) of an infinitesimal global conformal trans-formation. One now has to check, however, if any furtherconditions apply to this solution by our requirement onthe behaviour of the ‘subsidiary’ field strength tensors.By analogy with the discussion in Section III C, astraightforward way of imposing this requirement is todemand that, under subsequent GCT and H -gauge trans-formations satisfying δ h aµ = 0, the change in each ‘full’field strength R † abcd , H † cd and T † acd arising from thechange in either combination of gauge fields A † abµ or B µ − T µ should vanish separately . Unlike the cases con-sidered in Section III C, however, the quantities A † abµ and B µ − T µ are not independent. Indeed, startingfrom (67) and assuming δ h aµ = 0, it is easily shownthat δ ( B µ − T µ ) = δ A † aµa . Thus, one need consideronly the changes in R † abcd , H † cd and T † acd arising fromthe change in the combination A † abµ of the gauge fields.As previously, the condition δ h aµ = 0 guarantees thatthe variation in the field strength tensors vanishes if the variation in their non-calligraphic counterparts does so.The transformation laws of the latter (assuming δ h aµ =0) are given very simply in terms of the transformationsof the quantities A † abµ by δ R † abµν = 2 ∂ [ µ δ A † abν ] , (92a) δ H † µν = ∂ [ µ δ A † bν ] b , (92b) δ T † aµν = 2 δ A † ab [ µ δ bν ] − δ a [ µ δ A † bν ] b . (92c)Since these expressions depend solely on δ A † abµ , ourprocedure is equivalent to demanding only that the vari-ation in each field strength tensor vanishes, but this issatisfied by construction. Alternatively, one may showthis directly by making use of the transformation law δ A † abµ = − δ [ aµ ∂ b ] ̺ , which may be derived by takingthe infinitesimal limits of (69) and assuming δ h aµ = 0.Substituting this form for δ A † abµ into (92) one finds that(92b) and (92c) vanish identically, and (92a) vanishes byvirtue of the condition (5b). Hence no further conditionsapply to this solution, which is sufficient to show that the‘ungauged’ limit of eWGT corresponds to global confor-mal transformations; this differs markedly from WGT,for which we showed in Section III C that the ‘ungauged’limit corresponds to global Weyl transformations.It is worth noting that under the global conformaltransformation (3), the second and third conditions in(90) are not preserved. Indeed, one finds δ A abµ = 4(1 − θ ) δ [ aµ c b ] , δ B µ = 2 θc µ , (93)which in turn lead to δ A † abµ = 4 δ [ aµ c b ] . Thus, as antici-pated in Section III C, although applying our ‘ungauging’approach to WGT is equivalent to requiring that all theconditions in (90) are preserved, this equivalence doesnot hold when it is applied to eWGT. Indeed, as is clearfrom (93), the latter requirement in eWGT would leadto the condition c µ = 0, which corresponds to a globalWeyl transformation.
2. Finite transformations
We may extend our discussion to finite transforma-tions, which also include inversions. Starting again fromthe relations (90), one should demand that under sub-sequent finite GCT and H -gauge transformations thatpreserve the first relation (such that h ′ aµ ( x ′ ) = δ µa ), all‘subsidiary’ field strength tensors remain zero.It is straightforward to show that h ′ aµ ( x ′ ) = δ µa is a nec-essary and sufficient condition for the coordinate trans-formation matrix to satisfy (16); this in turn satisfies(1), from which it follows that the most general formfor the transformation is a finite global conformal trans-formation satisfying the conditions (17), as described inSection II B. It therefore remains to check if any furtherconditions apply arising from our requirement on the be-haviour of the ‘subsidiary’ field strength tensors.2By analogy with the infinitesimal case, we demandthat, under subsequent GCT and H -gauge transfor-mations satisfying h ′ aµ ( x ′ ) = δ µa , the change in each‘full’ field strength R † abcd , H † cd and T † acd arising fromthe change in either combination of gauge fields A † abµ or B µ − T µ should vanish separately . Starting from(67) and assuming h ′ aµ ( x ′ ) = δ µa , one may show that B ′ µ ( x ′ ) − T ′ µ ( x ′ ) = A †′ aµa ( x ′ ), and so again one needonly consider changes in R † abcd , H † cd and T † acd arisingfrom the change in the combination A † abµ of the gaugefields.The condition h ′ aµ ( x ′ ) = δ µa guarantees that thetransformed field strength tensors vanish if their non-calligraphic counterparts do so. The transformation lawsof the latter (assuming h ′ aµ ( x ′ ) = δ µa ) are given in termsof A †′ abµ ( x ′ ) by R †′ abµν = 2 ∂ ′ [ µ A †′ abν ] + 2 A †′ ae [ µ A †′ ebν ] , (94a) H †′ µν = ∂ ′ [ µ A †′ bν ] b , (94b) T †′ aµν = 2 A †′ ab [ µ δ bν ] − δ a [ µ A †′ bν ] b . (94c)As in the infinitesimal case, since these expressions de-pend solely on A †′ abµ , our procedure is equivalent to de-manding only that each transformed field strength tensorvanishes, but this is again satisfied by construction. Al-ternatively, one may show this directly by making use ofthe transformation law A †′ abµ = − δ [ aµ ∂ b ] ̺ , which maybe derived from (69) with the assumption h ′ aµ ( x ′ ) = δ µa .Substituting this form for A †′ abµ into (94) one finds that(94b) and (94c) vanish identically, and (94a) vanishes byvirtue of the condition (17b). Hence no further condi-tions apply to the solution, so that the ‘ungauged’ limitof eWGT corresponds to finite global conformal transfor-mations, including inversions. V. CONCLUSIONS
We have reconsidered the process of gauging of theconformal group and the resulting construction of gravi-tational gauge theories that are invariant under local con-formal transformations. The standard approach leads toauxiliary conformal gauge theories (ACGT), so called be-cause they suffer from the problem that the gauge fieldcorresponding to special conformal transformations canbe eliminated from the theory using its own equation ofmotion, so that the symmetry appears to reduce backto the local Weyl group. Such theoretical difficultieswith AGCT have led to the development of an alterntivebiconformal gauging and the construction of its associ-ated biconformal gauge field theories (BCGT). Althoughthese theories possess some very interesting and promis-ing properties, their physical interpretation is compli-cated by their requirement of an 8-dimensional base man-ifold. Thus, the role played by local conformal invariancein gravitational gauge theories remains uncertain. We have therefore revisited the recently proposed ex-tended Weyl gauge theory (eWGT), which was previouslynoted to implement Weyl scaling in a novel way thatmay be related to gauging of the full conformal group.We demonstrated this relationship here by first showingthat, provided any physical matter fields belong to an ir-reducible representation of the Lorentz group, eWGT isindeed invariant under the full set of (finite) local confor-mal transformations, including inversions. This propertyis, however, also shared by standard WGT, as might beexpected from the theoretical shortcomings of ACGT.Nonetheless, we also show that eWGT has two furtherproperties not shared by WGT. First, the conservationlaws of eWGT provide a natural local generalisation ofthose satisfied by field theories with global conformal in-variance, in particular that the field virial should van-ish; this is the key criterion for an action to be invari-ant under SCTs, in addition to the remainder of theglobal conformal group (connected to the identity). Sec-ond, we show that the ‘ungauged’ limit of eWGT corre-sponds to global conformal transformations, rather thanglobal Weyl transformations. These findings suggest thateWGT can be regarded as a valid alternative gauge the-ory of the conformal group, despite not having been de-rived by direct consideration of the localisation of itsgroup parameters. Therefore, eWGT might be consid-ered as a ‘concealed’ conformal gauge theory (CCGT).
ACKNOWLEDGMENTS
The authors thank Will Barker for useful comments.The data that support the findings of this study are avail-able within the article.
Appendix A: Direct derivation of finite global conformaltransformations
For a finite coordinate transformation x ′ µ = f µ ( x ) in n -dimensional Minkowski spacetime to satisfy the defin-ing condition (1) to be conformal, one immediately re-quires ( ∂ α f γ )( ∂ β f γ ) = n ( ∂ ν f µ )( ∂ ν f µ ) η αβ , (A1)where ( ∂ ν f µ )( ∂ ν f µ ) = n Ω and ∂ ν f µ = ∂x ′ µ /∂x ν = X µν is the coordinate transformation matrix. Equation(A1) is the finite version of the conformal Killing equation(2), to which it reduces in the infinitesimal limit x ′ µ ≈ x µ + ξ µ ( x ).Acting on (A1) with ∂ λ , cyclically permuting the in-dices λ , α and β to obtain two further equivalent equa-tions and subtracting the first equation from the sum ofthe other two, one obtains2( ∂ α ∂ β f γ )( ∂ λ f γ ) = n ( η λα ∂ β + η βλ ∂ α − η αβ ∂ λ )Ω . (A2)of which we will make use shortly.3Another useful equation may be obtained by first act-ing on (A1) with ∂ β , then acting on the resulting equa-tion with ∂ β , symmetrising on α and β and finally using(A1) again. This yields[ η αβ (cid:3) + ( n − ∂ α ∂ β ]Ω + 2( ∂ α ∂ β f γ ) (cid:3) f γ − ∂ α ∂ λ f γ )( ∂ β ∂ λ f γ ) = 0 , (A3) from which one may derive two further useful equations.First, contracting (A3) with η αβ , one obtains( n − (cid:3) Ω + ( (cid:3) f γ )( (cid:3) f γ ) − ( ∂ τ ∂ λ f γ )( ∂ τ ∂ λ f γ ) = 0 . (A4)Then, multiplying (A4) by η αβ and subtracting the resultfrom ( n −
1) times (A3) gives( n − n − ∂ α ∂ β Ω + 2( n − ∂ α ∂ β f γ ) (cid:3) f γ − ( ∂ α ∂ λ f γ )( ∂ β ∂ λ f γ )] − η αβ [( (cid:3) f γ )( (cid:3) f γ ) − ( ∂ τ ∂ λ f γ )( ∂ τ ∂ λ f γ )] = 0 . (A5)As discussed in Section II B, one may write the trans-formation matrix of a smooth conformal transformationin the form (16), such that ∂ ν f µ = Ω( x )Λ µν ( x ) , (A6)where Λ µν ( x ) is, in general, a position-dependent Lorentzrotation matrix (either proper or improper). First, sub-stituting (A6) into (A2), one finds that Ω( x ) and Λ µν ( x ) must satisfy the relation ∂ µ Λ γβ = (Λ γµ ∂ β − η µβ Λ γλ ∂ λ ) ln Ω , (A7)from which one may straightforwardly obtain the result(17a), namelyΛ γα ∂ µ Λ γβ − δ [ αµ ∂ β ] ln Ω = 0 . (A8)Then, substituting (A6) into (A4) and (A5), respectively,and using the result (A7), one finds( n − (cid:3) Ω + ( n − ∂ γ Ω)( ∂ γ Ω)] = 0 , (A9)( n − n − ∂ α ∂ β Ω + η αβ ( ∂ γ Ω)( ∂ γ Ω) − ∂ α Ω)( ∂ β Ω)] = 0 , (A10)where the second result matches (17b) for n ≥ µν , it is in factmore convenient work in terms of the reciprocal dilation σ ≡ / Ω, for which (A8–A10) becomeΛ γα ∂ µ Λ γβ + 2 δ [ αµ ∂ β ] ln σ = 0 , (A11)( n − n ( ∂ γ σ )( ∂ γ σ ) − σ (cid:3) σ ] = 0 , (A12)( n − n − η αβ ( ∂ γ σ )( ∂ γ σ ) − σ∂ α ∂ β σ ] = 0 . (A13)Assuming n ≥
3, (A13) immediately implies that ∂ α ∂ β σ = 0 for α = β . One thus requires σ ( x ) = q µ ( x µ ),i.e. the sum of n functions, each of which is a functiononly of the corresponding coordinate (and possibly a con-stant). Substituting this form back into (A13) with α = β and adopting the signature η αβ = diag(1 , − , . . . , − q ′′ = − q ′′ i = σ ( ∂ γ ln σ )( ∂ γ ln σ ) , (A14)where primes denote differentiation with respect to thefunction argument and the index i runs from 1 to n − q ′′ µ must be equal to the sameconstant. Consequently, σ ( x ) must have the form σ ( x ) = a + 2 c µ x µ + bx , (A15)where a , b and c µ are constants. Finally, substitutingthis form back into (A14) or (A12), yields the conditionthat ab = c . Let us first assume that the vector c µ is non-null. Inthis case, there are three non-trivial possibilities:(i) a = 0 and b = 0 = c µ , so that σ = a ;(ii) b = 0 and a = 0 = c µ , so that σ = bx ;(iii) a , b and at least one component of c µ are non-zero,so that σ = a (1 + 2¯ c µ x µ + ¯ c x ), where ¯ c µ ≡ c µ /a .Turning then to the case where the vector c µ is non-zerobut null, one requires at least one of a and b to be zero.Hence, there are three further possibilities:(iv) a = 0 and b = 0, so that σ = a + 2 c µ x µ ;(v) a = 0 and b = 0, so that σ = 2 c µ x µ + bx ;(vi) a = 0 = b , so that σ = 2 c µ x µ .For each of the above possible forms for the recip-rocal scale factor σ ( x ), one may now use (A11) to de-termine the form of the corresponding (proper or im-proper) Lorentz transformation Λ µν ( x ), and hence thefull transformation matrix X µν ( x ) = [1 /σ ( x )]Λ µν ( x ) ineach case. Before proceeding, however, it should be notedthat (A11) is insufficient to determine Λ µν ( x ) fully, sinceif Λ µν ( x ) satisfies (A11), then so too does Λ µλ ( x )Λ λν ,where Λ λν may be any position-independent Lorentz4transformation matrix. With this caveat in mind, wenow consider each of the possible forms for σ ( x ) listedabove.(i) For σ = a , one requires Λ µν = constant. Thus X µν corresponds to a combination of a position-independent scaling 1 /a , (proper or improper)Lorentz transformation and translation (namely aglobal Weyl transformation).(ii) For σ = bx , (A11) is solved by Λ µν ( x ) = I µν (ˆ x ),which corresponds to a reflection in the hyperplaneperpendicular to the unit vector ˆ x . Thus X µν corresponds an inversion followed by a position-independent scaling 1 /b .(iii) For σ = a (1 + 2¯ c µ x µ + ¯ c x ), (A11) is solved byΛ µν ( x ) = I µλ (ˆ x ′ ) I λν (ˆ x ), where x ′ µ is given by (14)with c µ replaced by ¯ c µ . Thus X µν correspondsto a SCT, in which the intermediate translationis through the vector ¯ c µ , followed by a position-independent scaling 1 /a (see Section II B).(iv) For σ = a + 2 c µ x µ = a (1 + 2¯ c µ x µ ), (A11) is solvedby Λ µν ( x ) = I µλ (ˆ x ′ ) I λν (ˆ x ), where x ′ µ is given by(14) with c µ replaced by ¯ c µ , and for which ¯ c =0. Thus X µν corresponds to a SCT, in which theintermediate translation is through the null vector¯ c µ , followed by a position-independent scaling 1 /a .(v) For σ = 2 c µ x µ + bx = b (2˜ c µ x µ + x ), where ˜ c µ ≡ c µ /b , (A11) is solved by Λ µν ( x ) = I µν [ˆ n ( x )], wherethe unit vector ˆ n ( x ) has componentsˆ n µ ( x ) = x µ + ˜ c µ √ x + 2˜ c · x . (A16)It is straightforward to show that the resulting X µν corresponds to a translation through ˜ c µ , followedby an inversion, followed by a position-independentscaling 1 /b .(vi) For σ = 2 c µ x µ , (A11) is solved by Λ µν = I µν (ˆ c ).In a similar way to case (v), it is straightforward toshow that the resulting X µν corresponds to a trans-lation through c µ in the limit c µ → ∞ , followed byan inversion.In deriving the above solutions, we have made use ofthe following results. First, if Λ µν ( x ) = I µν [ˆ n ( x )], whichcorresponds to a reflection in the hyperplane perpendic-ular to a position-dependent unit vector ˆ n ( x ), the firstterm in (A11) may be written asΛ γα ∂ µ Λ γβ = 4ˆ n [ α ∂ µ ˆ n β ] . (A17)By then considering the identity ˆ n [ γ ˆ n α ∂ µ ˆ n β ] = 0, onequickly finds that (A11) implies that ˆ n µ ∝ ∂ µ σ . Second,if Λ µν ( x ) = I µλ [ˆ n ( x )] I λν [ ˆ m ( x )], which corresponds to areflection in the hyperplane perpendicular to a position-dependent unit vector ˆ m ( x ), followed by a reflection in the hyperplane perpendicular to ˆ n ( x ) (which togetherconstitute a local rotation in the hyperplane defined byˆ m ( x ) and ˆ n ( x ), through twice the angle between them),then the first term in (A11) may be written asΛ γα ∂ µ Λ γβ = 4( ˆ m [ α ∂ µ ˆ m β ] + ˆ n [ α ∂ µ ˆ n β ] + 2 ˆ m [ α ˆ n β ] ˆ m γ ∂ µ ˆ n γ − m · ˆ n ˆ m [ α ∂ µ ˆ n β ] ) . (A18)As expected, cases (i)-(vi) contain only the four dis-tinct finite elements of the conformal group, namelyposition-independent translations, rotations and scalings,together with inversions. Appendix B: Global and local symmetries in field theory
Consider a Minkowski spacetime M , labelled usingCartesian inertial coordinates, in which the dynamics ofsome set of fields χ ( x ) = { χ i ( x ) } ( i = 1 , , . . . ) is de-scribed by the action S = Z L ( χ, ∂ µ χ ) d x. (B1)It should be understood here that the index i merelylabels different matter fields, rather than denoting thetensor or spinor components of individual fields (whichare suppressed throughout). It is worth noting that, ingeneral, the fields χ i ( x ) may include matter fields ϕ i ( x )and gauge fields g i ( x ).Invariance of the action (B1) under an infinitesimal co-ordinate transformation x ′ µ = x µ + ξ µ ( x ) and form varia-tions δ χ i ( x ) in the fields (where the latter do not neces-sarily result solely from the coordinate transformation),implies that, up to a total divergence of any quantity thatvanishes on the boundary of the integration region, onehas δ L + ∂ µ ( ξ µ L ) = 0 , (B2)where the ‘form’ variation of the Lagrangian is given by δ L = ∂ L ∂χ i δ χ i + ∂ L ∂ ( ∂ µ χ i ) δ ( ∂ µ χ i ) , (B3)and, according to the usual summation convention, thereis an implied sum on the index i .The invariance condition (B2) can alternatively berewritten as δ L δχ i δ χ i + ∂ µ J µ = 0 , (B4)where δ L /δχ i denotes the standard variational derivativeand the Noether current J µ is given by J µ = ∂ L ∂ ( ∂ µ χ i ) δ χ i + ξ µ L . (B5)If the field equations δL/δχ i = 0 are satisfied, then (B4)reduces to the (on-shell) conservation law ∂ µ J µ = 0,which is the content of Noether’s first theorem and ap-plies both to global and local symmetries.5
1. Global symmetries
Let us first consider an action invariant under a globalsymmetry. In the context of constructing gauge theories,it is usual first to consider an action of the form S = Z L ( ϕ, ∂ µ ϕ ) d x, (B6)where the Lagrangian density L and the Lagrangian L coincide and depend only on a set of matter fields ϕ ( x ) = { ϕ i ( x ) } and their first derivatives. Moreover, we willconsider only the case for which the action of the globalsymmetry on the coordinates and fields can be realisedlinearly.In this case, the coordinate transformation and the re-sulting form variations of the fields that leave the actioninvariant can be written as, respectively, ξ µ ( x ) = λ j ξ µj ( x ) , δ ϕ i ( x ) = λ j G j ϕ i ( x ) (B7)where λ j are a set of constant parameters, ξ µj ( x ) are givenfunctions and G j are the generators of the global sym-metry corresponding to the representation to which ϕ i belongs. Note that, for each value of j , the param-eter λ j typically represents a set of infinitesimal con-stants carrying one or more coordinate indices; for ex-ample, if one considers global conformal invariance, then { λ , λ , λ , λ } = { a α , ω αβ , ρ, c α } .The Noether current (B5) then takes the form J µ = λ j (cid:18) ∂L∂ ( ∂ µ ϕ i ) G j ϕ i + ξ µj L (cid:19) . (B8)Since the parameters λ j are constants, the (on-shell) con-servation law ∂ µ J µ = 0 hence leads to a separate condi-tion for each value of j , given by ∂ µ (cid:18) ∂L∂ ( ∂ µ ϕ i ) G j ϕ i + ξ µj L (cid:19) = 0 , (B9)which again hold up to a total divergence of any quantitythat vanishes on the boundary of the integration regionof the action (B6).
2. Local symmetries
We now consider an action of the form (B1) that isinvariant under a local symmetry. In particular, we focuson the (usual) case in which the form variations of thefields can be written as δ χ i = λ j f ij ( χ, ∂χ ) + ( ∂ µ λ j ) f µij ( χ, ∂χ ) , (B10)where now λ j = λ j ( x ) are a set of independent arbi-trary functions of position, and f ij ( χ, ∂χ ) and f µij ( χ, ∂χ )are two sets of given functions that, in general, may de-pend on all the fields and their first derivatives. The general form (B10) typically applies only when χ i = g i is a gauge field, whereas if χ i = ϕ i is a mat-ter field, then f ij ( χ, ∂χ ) = G j ϕ i and f µij ( χ, ∂χ ) =0, as for a global symmetry. By analogy with ourdiscussion above, for each value of j , the function λ j ( x ) typically represents a set of infinitesimal func-tions carrying one or more coordinate or local Lorentzframe indices; for example, if one considers local con-formal invariance, then { λ ( x ) , λ ( x ) , λ ( x ) , λ ( x ) } = { a α ( x ) , ω ab ( x ) , ρ ( x ) , c a ( x ) } , where a α ( x ) is interpretedas an infinitesimal general coordinate transformation(GCT) and is usually denoted instead by ξ α ( x ).Using the expression (B10), and after performing anintegration by parts, the corresponding variation of theaction (B1) is given by (suppressing functional depen-dencies for brevity) δS = Z λ j (cid:20) f ij δ L δχ i − ∂ µ (cid:18) f µij δ L δχ i (cid:19)(cid:21) + ∂ µ ( J µ − S µ ) d x, (B11)where we define the new current S µ ≡ − λ j f µij δ L /δχ i .Since the λ j are arbitrary functions, for the action to beinvariant one requires the separate conditions f ij δ L δχ i − ∂ µ (cid:18) f µij δ L δχ i (cid:19) = 0 , (B12) ∂ µ ( J µ − S µ ) = 0 , (B13)where the former hold for each value of j separately andthe latter holds up to a total divergence of a quantitythan vanishes on the boundary of the integration region.The first set of conditions (B12) are usually inter-preted as conservation laws, which are covariant underthe local symmetry, although not manifestly so in theform given above. The condition (B13) implies that J µ = S µ + ∂ ν Q νµ , where Q νµ = − Q µν , so the twocurrents coincide up to a total divergence. By contrastwith the case of a global symmetry, if the field equa-tions δ L /δχ i = 0 are satisfied, then the conservation laws(B12) hold identically and S µ vanishes. Thus, the con-ditions (B12–B13) effectively contain no information on-shell, which is essentially the content of Noether’s secondtheorem .Nonetheless, the on-shell condition that all the fieldequations δ L /δχ i = 0 are satisfied can only be imposedif L is the total Lagrangian density, and not if L corre-sponds only to some subset thereof (albeit one for whichthe corresponding action should still be invariant underthe local symmetry). In particular, suppose one is con-sidering a field theory for which the total Lagrangiandensity L T = L M + L G , where L G contains every termthat depends only on the gauge fields g i and/or theirderivatives, and L M contains all the remaining terms.Thus, if L = L M , then only the matter field equations δ L /δϕ i = 0 can be imposed, whereas if L = L G none ofthe field equations can be imposed. In either case, thesurviving terms in (B12–B13) do contain information .6 REFERENCES This may not hold for the quantum description in the presenceof quantum anomalies. J. Crispim-Romao, A. Ferber, P.G.O. Freund, Nucl. Phys. B ,429 (1977) J. Crispim-Romao, Nuc. Phys. B , 535 (1978) M. Kaku, P.K. Townsend, P. van Nieuwenhuizen, Phys. Lett. B , 304 (1977) M. Kaku, P.K. Townsend, P. van Nieuwenhuizen, Phys. Rev. D , 3179 (1978) E.A. Lord, P. Goswami, Pramana J. Phys. , 635 (1985) J.T. Wheeler, Phys. Rev. D , 1769 (1991) E.A. Ivanov, J. Niederle, Phys. Rev. D , 976 (1982) E.A. Ivanov, J. Niederle, Phys. Rev. D , 988 (1982) J.T. Wheeler, J. Math. Phys. , 299 (1998) A. Wehner, J.T. Wheeler, Nuc. Phys. B , 380 (1999) J.S. Hazboun, J.T. Wheeler, J. Phys. Conf. Ser. , 012013(2012) J.T. Wheeler, J. Phys. Conf. Ser. , 012059 (2013) J.T. Wheeler, Phys. Rev. D , 025027 (2014) J.T. Wheeler, Nucl. Phys. B , 114624 (2019) E. Cunningham, Proc. London Math. Soc. , 77 (1910) H. Bateman, Proc. London Math. Soc. , 223 (1910) H. Bateman, Proc. London Math. Soc. , 469 (1910) J.A. Schouten, J. Haantjes, Kon. Ned. Akad. Wet. Proc. , 1059(1936) A.N. Lasenby, M.P. Hobson, J. Math. Phys. , 092505 (2016) F.W. Hehl, in
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