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IMPROVED ENERGY-MOMENTUM CURRENTS

arXiv:gr-qc/9210009v1 15 Oct 1992file con7.tex, 1992-10-14IMPROVED ENERGY-MOMENTUM CURRENTSIN METRIC-AFFINE SPACETIMEbyRalf Hecht∗, Friedrich W. Hehl∗$, J. Dermott McCrea∗∗,Eckehard W. Mielke∗$, and Yuval Ne’eman∗∗∗⋄$+)∗) Institute for Theoretical Physics, University of Cologne, D(W)-5000 K¨oln 41, Ger-many∗∗) Department of Mathematical Physics, University College, Dublin 4, and DublinInstitute for Advanced Studies, Dublin 4, Ireland∗∗∗) Institut des Hautes Etudes Scientifiques, F-91400 Bures-sur-Yvette, France, andRaymond and Beverley Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, IsraelAbstractIn Minkowski spacetime it is well-known that the canonical energy-momentumcurrent is involved in the construction of the globally conserved currents of energy-momentum and total angular momentum. For the construction of conserved currentscorresponding to (approximate) scale and proper conformal symmetries, however, animproved energy-momentum current is needed.

By extending the Minkowskian frame-work to a genuine metric-affine spacetime, we find that the affine Noether identitiesand the conformal Killing equations enforce this improvement in a rather naturalway. So far, no gravitational dynamics is involved in our construction.

The result-ing dilation and proper conformal currents are conserved provided the trace of theenergy-momentum current satisfies a (mild) scaling relation or even vanishes.⋄) On leave from the Center for Particle Physics, University of Texas, Austin, Texas78712, USA$) Supported by the German-Israeli Foundation for Scientific Research & Development(GIF), Jerusalem and Munich.+)Supported in part by DOE Grant DE-FG05-85-ER40200.1

1. IntroductionIf a spacetime admits symmetries, we can construct a set of invariant conservedquantities, one for each symmetry.

Consider the Riemannian spacetime of generalrelativity (GR). By a Killing symmetry we understand a diffeomorphism of the Rie-mannian spacetime under which the metric is invariant.

Let the vector field ξ = ξα eαbe the generator of such a 1-parameter group of diffeomorphisms of the spacetime.Then it obeys the Killing equation(1)Lξ g = 0⇒e(α⌋oDξβ) = 0 ,(1.1)with the Riemannian exterior covariant derivativeoD.Following the standard procedure of GR, see Penrose and Rindler [1], it isstraightforward to construct an energy-momentum current which is a closed form,provided the matter field equation is satisfied.Let tαβ be the symmetric energy-momentum tensor of matter and tα the corresponding covariantly conserved energy-momentum 3-form(2) . Then the energy current ε R represents a weakly closed 3-form:ε R := ξα tα ,d ε R ∼= 0 .

(1.2)Weakly means that the last equation is only valid, if the matter field equation holds.Accordingly, for a timelike Killing vector field, ε R yields, after integration over aspacelike hypersurface S, the conserved energyRS ε R.For a bosonic description of gravitational interactions, the metric-affine gaugetheory of gravity (MAG) is a very general framework, see Ref.[2]. The spacetimecontinuum of MAG is represented by a differentiable manifold (L4, g) with a coframeϑα, an arbitrary linear connection Γαβ, and a second rank symmetric tensor field, themetric g = gαβ ϑα ⊗ϑβ.

In extending the previous notion, we will now understandby a Killing symmetry a diffeomorphism under which the metric and the connectionare invariant.The aim of this paper is twofold: First we want to generalize the notion of aweakly conserved energy current `a la (1.2) to the metric-affine spacetime of MAG. InSec.4, we will derive such a current ε MA, which generalizes the ε R of (1.2).Moreover, in Sec.5 we will relax the Killing constraints on the vector field ξ andconsider conformal Killing vector fields in order to obtain also conserved dilation andproper conformal currents.

In Minkowski spacetime, this construct is well-known. (1) Here eα is an arbitrary vector basis of the tangent space and gαβ are the com-ponents of the metric.

The Lie derivative with respect to ξ is denoted by Lξ, thecovariant exterior derivative by D, and the interior product by ⌋. (2) If ϑα is the 1-form basis dual to eα, we have tα = tαβ ηβ where ηα = ∗ϑα and ∗denotes the Hodge star.

Moreover, ηαβ = ∗(ϑα ∧ϑβ).2

Here we are generalizing it for the first time to MAG, in the framework of which itfinds its most natural embedding.In Minkowski spacetime (cf. Jackiw[3])Dα = xγ ˜tγα ,(1.3)Kαβ = [2xβxγ −gβγx2] ˜tγα(1.4)are the dilation current and the proper conformal current, respectively, if ˜tαβ repre-sents the improved energy-momentum tensor.

From the divergence of these currentswe see that both, scale and proper conformal invariance, are broken by the trace ˜tµµof the improved energy tensor. Since MAG is formulated in a locally dilation invariantway, we expect that an appropriately constructed conformal current ε C emerges interms of a conformal Killing vector.

This expression is displayed in Eq.(6.10). Wewill prove that it is ‘weakly’ conserved, provided the energy-momentum is tracefree.Therefrom, in Sect.6, we will construct an ‘improved’ energy–momentum current ˜ε MAwhich has a ‘soft’ trace also for scalar fields, as exemplified by the dilaton, providedthe mild constraint (6.11) holds.

All the previously discussed energy–momentum ex-pressions can be recovered as special cases of our new expressions (5.10) or (6.10). Weconclude our paper in Sec.7 with a relevant application of the Ogievetsky theorem.2.

Metric-affine spacetime in briefThe geometrical variables of such a metric-affine spacetime are the forms (gαβ, ϑα,Γαβ) with an appropriate transformation behavior under the local GL(n, R) group inn dimensions. The components gαβ of the metric g are 0-forms, the coframe and theconnection are 1-forms.

In MAG, the nonmetricityQαβ := −Dgαβ⇒Qαβ = Dgαβ ,(2.1)besides torsion T α and curvature Rαβ, enters the spacetime arena as a new fieldstrength. The gauge covariant derivativeDΨ = dΨ + †Γαβ ∧ρ(Lβα)Ψ + w2 Q ∧Ψ(2.2)contains the volume–preserving SL(n, R) connection†Γαβ and a volume–changingpiece depending on the Weyl 1-form Q := (1/n) Qαα.

The matter fields are allowedto be densities, that is, under the scale part Λαβ = δβα Ωof the GL(n, R) gaugetransformation the matter field will transform as Ψ′ = Ωw Ψ. The weight w dependson the SL(n, R) index structure and the density character.

Our notation is the sameas in Refs. [2].3

potentialfield strengthBianchi identitymetric gαβQαβ = −DgαβDQαβ = 2R(αβ)coframe ϑαT α = DϑαDT α = Rµα ∧ϑµconnection ΓαβRαβ = dΓαβ + Γµβ ∧ΓαµDRαβ = 0Following Wallner [4], we use Lξ to denote the usual Lie derivative of a generalgeometric object and ℓξ to denote the restriction of the Lie derivative to differentialforms, for which one can show thatℓξ := ξ⌋d + d ξ⌋. (2.3)The “gauge covariant Lie derivative” of a form is given by L := ξ⌋D + D ξ⌋.

(2.4)As an example, we obtain for the connectionLξΓαβ = (ℓξ −δξ) ΓαβwithδξΓαβ = −Deα⌋(ℓξϑβ). (2.5)Later on, we will employ the covariant exterior derivative⌢D with respect to thetransposed connection⌢Γαβ := Γαβ + eα⌋T β .

(2.6)Its somewhat unclear role becomes more transparent by the following observation:If applied to the vector components ξα, the transposed derivative is identical to thegauge-covariant Lie derivative of the coframe, i.e. Lξ ϑα ≡⌢D ξα .

(2.7)3. Lagrange-Noether machineryThe external currents of a matter field are those currents which are related to localspacetime symmetries.

On a fundamental level, we adopt the view that fundamentalmatter is described in terms of infinite-dimensional spinor or tensor representationsof SL(4, R), the manifields Ψ, see Ne’eman [5].In a first order formalism (cf. Nester [6] and Kopczy´nski [7]) we assume that thematerial Lagrangian n-form for these manifields depends most generally on Ψ, dΨ,and the potentials gαβ, ϑα, Γαβ.

According to the minimal coupling prescription,derivatives of these potentials are not permitted. We usually adhere to this principle.However, Pauli type terms and the Jordan-Brans-Dicke term ¯ΦΦ Rαβ ∧ηαβ may occur4

in conformal models with ‘improved’ energy-momentum tensors or in the context ofsymmetry breaking. Therefore, we have developed [2] our Lagrangian formalism insufficient generality in order to cope with such models by including in the Lagrangianalso the derivatives dgαβ, dϑα, and dΓαβ of the gravitational potentials:L = L(gαβ , dgαβ , ϑα , dϑα , Γαβ , dΓαβ , Ψ, dΨ) .

(3.1)As a further bonus, we can then easily read offthe Noether identities for the gravita-tional gauge fields in n > 2 dimensions (the restriction is related to the introductionof conformal symmetry in Sec. 5).For such a gauge-invariant Lagrangian L, the variational derivative δL/δΨ be-comes identical to the GL(n, R)-covariant variational derivative of L with respect tothe q-form Ψ:δLδΨ := ∂L∂Ψ −(−1)qD∂L∂(DΨ) .

(3.2)The matter currents are the metric stress-energy σαβ, the canonical energy-momen-tum Σα, and the hypermomentum ∆αβ, which is asymmetric in α and β. They aregiven byσαβ := 2 δLδgαβ= 2 ∂L∂gαβ+ 2D ∂L∂Qαβ,(3.3)Σα := δLδϑα = ∂L∂ϑα + D ∂L∂T α ,(3.4)and∆αβ :=δLδΓαβ = Lαβ Ψ ∧∂L∂(DΨ)+ 2gβγ∂L∂Qαγ+ ϑα ∧∂L∂T β + D ∂L∂Rαβ .

(3.5)The explicit form of the canonical energy-momentum current readsΣα = eα⌋L −(eα⌋DΨ) ∧∂L∂DΨ −(eα⌋Ψ) ∧∂L∂Ψ−(eα⌋Qβγ)∂L∂Qβγ−(eα⌋T β) ∧∂L∂T β + D ∂L∂T α −(eα⌋Rβγ) ∧∂L∂Rβγ . (3.6)The first line in (3.6) represents the result known in the context of special relativisticclassical field theory.

The last Ψ-dependent term in (3.6) vanishes for a 0-form, asis exemplified by a scalar or the Dirac field. The second line in (3.6) accounts forpossible Pauli terms and is absent in the case of minimal coupling.In the following, our Noether currents, aside from corresponding to the gaugeaction of GL(n, R), involve local translations as well.In Minkowski spacetime,5

we could restrict the transformations to constant parameters and apply A(n, R) =Rn ⊂× GL(n, R) globally, still obtaining the familiar conservation laws, including thoseof the dilations and conformal transformations (1.3, 1.4). In our present paper, how-ever, we treat the whole A(n, R) locally, even though one should observe that the“gauging” of Rn transcends the definition of a Lie algebra and the correspondingbundle Maurer-Cartan equations.The first Noether identity in MAG takes the formDΣα ≡(eα⌋T β) ∧Σβ + (eα⌋Rβγ) ∧∆βγ −12(eα⌋Qβγ) σβγ+ (eα⌋DΨ) δLδΨ + (−1)p(eα⌋Ψ) ∧D δLδΨ∼=(eα⌋T β) ∧Σβ + (eα⌋Rβγ) ∧∆βγ −12(eα⌋Qβγ) σβγ.

(1st)(3.7)Usually, our first result is given in the strong form, where no field equation is in-voked. The weak identity, which is denoted by ∼=, holds only provided the matterfield equation δL/δΨ = 0 is satisfied.The second Noether identity in MAG readsD∆αβ + ϑα ∧Σβ −gβγ σαγ ≡−LαβΨ ∧δLδΨ∼= 0 .

(2nd)(3.8)The dilational part of the second Noether identity can be easily extracted directlyfrom (3.8) by sheer contraction:D∆+ ϑα ∧Σα −σαα ≡−LγγΨ ∧δLδΨ∼= 0 . (3.9)Only the trace piece ϑα ∧Σα of the energy-momentum current contributes to thisdilational identity, cf.

[8, 9]. This identity plays an important role for the approxi-mate scale invariance in the high-energy-limit of particle physics (see [10]) and in theconstruction of the improved current.4.

Conserved currents in MAGIn MAG, the connection is an independent field variable.The correspondingmatter current, coupled to it, will be the hypermomentum ∆αβ. We require ξ = ξα eαto be a Killing vector field for metric and connection,Lξ g = Lgαβ + 2gγ(αeβ)⌋ Lϑγϑα ⊗ϑβ = 0 ,LξΓαβ = 0 .

(4.1)According to (2.3), (2.5), and (2.7), these conditions can be recast into the formgγ(α eβ)⌋⌢D ξγ −12ξ⌋Qαβ = 0 ,D(eα⌋⌢D ξβ) + ξ⌋Rαβ = 0 . (4.2)6

Note that the first equation of (4.2) can be written alternatively, in terms of theRiemannian derivative, as e(α⌋oDξβ) = 0, compare the second equation of (1.1).Let us define the current (n −1)-formε MA := ξα Σα + (eβ⌋⌢Dξγ) ∆βγ. (4.3)We compute its exterior covariant derivative, substitute the two Noether identities(3.7) and (3.8), and reshuffle the emerging expressions:d ε MA = (Dξα) ∧Σα + ξα DΣα + D(eβ⌋⌢Dξγ) ∧∆βγ + (eβ⌋⌢Dξγ) D∆βγ∼= (Dξα) ∧Σα + (ξ⌋T β) ∧Σβ + (ξ⌋Rβγ) ∧∆βγ −12(ξ⌋Qβγ) σβγ+ D(eβ⌋⌢Dξγ) ∧∆βγ + (eβ⌋⌢Dξγ) (σβγ −ϑβ ∧Σγ)=h⌢Dξα −ϑβ (eβ⌋⌢Dξα)i∧Σα +h(Deβ⌋⌢Dξγ) + ξ⌋Rβγi∧∆βγ+hgα(β eγ)⌋⌢D ξα −12ξ⌋Qβγiσβγ .

(4.4)While transforming the exterior derivative into the gauge covariant derivative (2.2),we assumed that ε has zero weight wε, which is in accordance with the weight zerofor the Lagrangian and a usual vector field ξ. Moreover, we recognize that the firstsquare bracket vanishes identically because of the relation pψ = ϑα ∧(eα⌋ψ), which isvalid for any p-form.

In view of the generalized Killing equations (4.2), also the otherexpressions in the square brackets vanish. Thus, the current (4.3) is weakly conservedd ε MA ∼= 0 .

(4.5)For the Riemann-Cartan spacetime of the Einstein-Cartan-Sciama-Kibble theorya similar result has been obtained by Trautman [11] and, for the linearized case, byTod [12]. The corresponding current readsε RC := ξα Σα + (eβ⌋⌢Dξγ) τ βγ ,(4.6)where the spin current is defined according to τ βγ := ∆[βγ].

In Audretsch et al. [13],this current was used to construct a Hamiltonian for the Dirac field.Provided a timelike Killing vector field exists, we have obtained, via (4.5), a glob-ally conserved energyRS ε MA.

Our deduction of this expression follows the patternlaid out in GR, but generalizes it to a metric-affine spacetime. Some steps of this de-duction resemble Penrose’s recent local mass construction [14], except that we refrainfrom using spinor or twistor methods at this stage.7

5. Conserved dilation and proper conformal currentsIf the metric-affine spacetime admits a conformal symmetry, an important gener-alization of (4.3) can be constructed as follows: Let ξ = ξα eα be a conformal Killingvector field such that the Lie derivative of the metric g and the connection Γαβ read(3)Lξ g = ω g ,LξΓαβ = 12δβα dω .

(5.1)The same algebra as in Sect.4 yieldsgγ(α eβ)⌋⌢D ξγ −12ξ⌋Qαβ = 12gαβ ω ,D(eα⌋⌢D ξβ) + ξ⌋Rαβ = 12δβα dω , (5.2)compare with (4.2), which we recover for ω = 0. It follows from (5.1)1 that ξ generatesa transformation, parametrized by λ, of the spacetime mainfold such that the metricundergoes the special [15] conformal change g →˜g = eλω g. For a given geometry,the scalar function ω = ω(x) is determined by the trace of (5.2)1, i.e.

byω = 2neγ⌋⌢Dξγ −ξ⌋Q . (5.3)Thus, in metric-affine spacetime the conformal Killing equation for the metric readsgγ(α eβ)⌋⌢D ξγ −1ngαβ eγ⌋⌢Dξγ = 12ξ⌋Qրαβ ,(5.4)where Qրαβ:= Qαβ −gαβ Q is the tracefree part of the nonmetricity.Again we compute the exterior derivative of (4.3), but now under the assumptionthat the vector field ξ is a conformal Killing field obeying (5.1) or (5.2), respectively.Then the expressions in the last two square brackets in (4.4) do not vanish any more.Rather we find with the aid of the Noether identity (3.9) the relationd ε MA ∼= 12 Dω ∧∆αα + 12ω σαα ∼= d (12ω ∆) + 12ω (ϑα ∧Σα) .

(5.5)In the case of vanishing ω, we recover (4.5). If ω does not vanish, eq.

(5.5) yieldsd (ε MA −12ω ∆) ∼= 12ω (ϑα ∧Σα) . (5.6)For conformally invariant gauge theories, such as the Maxwell or the Yang-Mills vac-uum theory, the trace ϑα ∧Σα of the energy-momentum current vanishes and (5.6)provides already the conserved quantityε C := ε MA −12ω ∆.

(5.7)(3) Eq. (5.1)2 implies the vanishing of the trace free part of (4.2)2.

The same wouldhold, too, by requiring (5.1)2 for the volume–preserving connection instead.8

Since the traces of the energy-momentum current and the hypermomentum cur-rent (n −1)–forms are crucial here, we decompose these currents into their tracefreepieces and their traces, respectively:Σα = րΣα + 1n eα⌋(ϑγ ∧Σγ) ,∆αβ = ր∆αβ + 1n δβα ∆. (5.8)We substitute first (4.3) and then (5.8) into (5.7).

This yieldsε C = ξαրΣα + 1n ξ⌋(ϑα ∧Σα) + (eβ⌋⌢Dξγ) ր∆βγ + 1nh(eα⌋⌢Dξα) −n2 ωi∆. (5.9)If we apply the trace (5.3) of one of the conformal Killing relations, we obtainε C = ξαրΣα + (eβ⌋⌢Dξγ) ր∆βγ + 1n ξ⌋(ϑα ∧Σα) + 12(ξ⌋Q) ∆.

(5.10)Thus we have found generalizations of the well-known dilation and proper confor-mal currents in Minkowski spacetime (cf. Ref.

[3]) to a metric-affine spacetime. Sucha spacetime provides the most natural gravitational background for these currents.6.

Improved currents with a ‘soft’ energy–momentum trace for scalar fieldsFollowing Isham et al. [16], we consider the so-called dilaton field σ(x) beingimmersed into our MAG framework.On the classical level, we can assume that the scalar field carries canonical di-mensions , i.e.

(length)−1 in n = 4 dimensions. With respect to a generic conformalchange g →˜g = Ω(x)g of the underlying metric structure, the scalar field will thentransform according toσ(x) →˜σ(x) = Ω(x)−(n−2)/4σ(x) ,(6.1)i.e.

as a scalar density of weight wσ = (2 −n)/2. Then the gauge–covariant exteriorderivative (2.2) is given byDσ :=d −n −24Qσ(6.2)which, due to the inhomogeneous transformation of the Weyl 1–form under conformalchanges, is conformally covariant.The dilaton Lagrangian with a polynomial self–interaction n–form V (|σ|) is givenbyLσ = L + V (|σ|),L:= 12 Dσ ∧∗Dσ .

(6.3)9

Eq. (6.2) implies that its kinetic part is conformally invariant in any dimensions.

Byvariation of (6.3) with respect to σ, we obtain the scalar wave equationδLσδσ =σ + ∂V (|σ|)∂σ= 0,(6.4)whereσ := −D ∗(Dσ)(6.5)is the d’Alembertian.The trace of the dilaton’s energy–momentum current, i.e.ϑα ∧Σα(σ) = (n −2) L + nV (|σ|)(6.6)does not vanish. Therefore, we need to “improve” Σα in this respect.

Since the kineticpart of the dilaton Lagrangian (6.3) depends explicitly on the Weyl 1–form Q, thescalar field does also provide an intrinsic dilation current. According to (3.5), thelatter is dynamically defined by∆:= ∆γγ =δLδΓγγ = 2nδLδQ = 2 −n2nσ ∗Dσ .

(6.7)In our formalism we may define a “new improved” energy–momentum currentfor scalar fields byΘα := Σα + eα⌋D∆. (6.8)After insertion of the field equation (6.4), we find for its trace the “weak” relationϑα ∧Θα(σ) = ϑα ∧Σα(σ) + nD∆∼= nV (|σ|) −n −22σ ∂V (|σ|)∂σ.

(6.9)Compared to (6.6), the kinetic term Lis absent in (6.9). Moreover, due to Euler’stheorem for homogeneous functions, the σ2n/(n−2) piece in the potential drops out.For a polynomial potential V (|σ|) of degree p ≤2n/(n −2), the operator dimension-ality is then smaller than n (for n ≥4).

Therefore, the new trace is indeed “soft” ina momentum representation in the sense of Jackiw ([3], p. 213; cf. Kopczy´nski et al.[17]).

Note that, for n = 4, a pure σ2n/(n−2) model is known to be renormalizableaccording to the criteria of power counting.The necessary modification of our globally conserved current (5.10) is the follow-ing: We replace Σα by Θα and define the new ‘improved’ current by˜ε MA := ξαΘα + (eβ⌋⌢Dξγ) ր∆βγ + 12(ξ⌋Q)∆= ξαΣα + ξ⌋D∆+ (eβ⌋⌢Dξγ) ր∆βγ + 12(ξ⌋Q)∆= ε C + ξ⌋D∆. (6.10)10

Provided the scaling relationℓξ(D∆) = n2 ω D∆(6.11)holds, we finally obtaind˜ε MA = 12ω(ϑα ∧Θα) . (6.12)This means that the divergence of the ‘new improved’ current (6.10) has indeed asoft trace also for scalar fields, as is required quantum–theoretically.

If ϑα ∧Θα isvanishing, the scaling property (6.11) converts into one for the trace of the canonicalenergy-momentum current. In future we hope to find further physical motivations forthis hypothetical scaling property.7.

Emergence of the infinite-dimensional group of active diffeomorphismsIn (3.5) we defined the hypermomentum current in metric-affine spacetime. InMinkowski spacetime we can expand the hypermomentum so as to exhibit the hyper-surface (n −1)-form ηα.

By using the improved energy-momentum tensor, we obtainan orbital representation:˜∆αβ = ∆αβγηγ,∆αβγ = xα˜tβγ . (7.1)The antisymmetric piece ∆[αβ]γ is the angular momentum tensor, whereas the shearcomponents are given by the traceless part of the symmetric piece ∆(αβ)γ.Theintegrated total ‘charges’ are given by (cf.

the introduction)D =Zxγ ˜tγαηα(dilation charge) ,(7.2)Kβ =Z[2xβxγ −gβγ x2] ˜tγαηα(proper conformal charges) ,(7.3)M[βγ] =Zx[β˜tγ]αηα(angular momentum charges) ,(7.4)Sր(βγ)=Z[x(β˜tγ)α −1ngβγ xδ˜tδα]ηα(shear charges) . (7.5)Using Bohr’s principle of correspondence, Ogievetsky [18] has shown that thequantized system of shear Sր(βγ) and proper conformal Kβ charges does not closealgebraically and generates the infinite algebra of diffeomorphism charges in n dimen-sions:nLγδǫξηθ···α=Zn−1Yi=0(xi)pi ˜tαβηβ .

(7.6)11

The algebraic relations are preserved anholonomically. Thus, a metric-affine space-time which admits a conformal symmetry will have its frames locally invariant (in theactive operational sense of [19],[20]) under the group of analytical diffeomorphisms.This result overlaps with the fact that we have included in our affine gauge approachlocal translations, i.e.

active diffeomorphisms, except that whereas the latter are onlyinfinitesimal (their generators do not form a Lie algebra anyhow), the Ogievetskytransformations can be integrated to finite diffeomorphisms, provided we restrict toconstant parameters and thus do not require an infinite set of commutators. Theemergence of an explicit infinite Lie algebra may make it possible to treat conformalfields in n-dimensions similarly to what is done in the special case of two dimensions.In n = 2, there is an infinite–dimensional conformal algebra which is isomorphic tothe algebra of analytic two–dimensional diffeomorphisms [21].

In two dimensions thisfeature constrains the fields and leads to the highly restrictive ‘fusion’ rules [21], whichhave recently put 2-dimensional conformal field theory into the focus of interest ofstatistical mechanics and string theory.The Ogievetsky algebra in n dimensions is conceptually the analog of the deWitt algebra in two dimensions and should possess a quantum extension with centralcharges as in the Virasoro algebra.Neither this extension nor the representationtheory have been investigated to date.AcknowledgmentOne of us (Y.N.) would like to thank Prof. M. Berger for support and hospitalityat the IHES.References[1] R. Penrose and W. Rindler: Spinors in spacetime, Vol.

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