General Relativity as the effective theory of GL(4,R) spontaneous symmetry breaking
aa r X i v : . [ h e p - t h ] M a y General Relativity as the effective theory ofGL(4,R) spontaneous symmetry breaking
E. T. Tomboulis Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547
Abstract
We assume a GL (4 , R ) space-time symmetry which is spontaneously broken to SO (3 , GL (4 , R ). The coset construction makes no assumptionsabout any underlying theory that might be responsible for the assumed symmetrybreaking. We give a brief discussion of the possibility of field theories with GL (4 , R )rather than Lorentz space-time symmetry providing the underlying dynamics. e-mail: [email protected] Introduction
The discovery of broken chiral symmetry in the sixties and the subsequent developmentof chiral Lagrangians apparently first gave rise to the idea among particle physiciststhat General Relativity (GR) might also be an effective theory of the same sort. Thesimilarity of the structure of GR to that of effective Lagrangians of spontaneouslybroken symmetries is indeed rather obvious. Such effective Lagrangians assume ahighly geometrical appearance being field theories on homogeneous (coset) spaces, andare formulated in terms of covariant derivatives, curvatures etc. In this connection,recall also that in GR both the metric field and its inverse occur in the action, ahallmark of effective Lagrangians. An inverse for a field quantized as an elementaryfield makes of course no sense in general, but inverse fields occurs naturally in effectiveactions for broken symmetries with their exponential parametrizations of Goldstonefields.The possibility that this analogy holds for GR is attractive since it obviates quanti-zation of the gravitational field as an elementary field and, as in the case of broken chiralsymmetry and QCD, might point to a more fundamental, presumably UV complete,underlying theory responsible for the symmetry breaking.The broken symmetry in the case of GR must apparently be a space-time symmetry.Starting with the work of Bjorken, it was indeed suggested that the graviton is theGoldstone boson of spontaneously broken Lorentz invariance [1]. This has been revivedin recent years [2] - [3]. A gravity theory along these lines, though approximating GRat low energies, has to fundamentally differ from GR since it contains actual Lorentzsymmetry breaking.In this paper we pursue the same idea but consider symmetry breaking leavingLorentz symmetry intact. Specifically, we assume a GL (4 , R ) space-time symmetrywhich is spontaneously broken to its Lorentz subgroup SO (3 , SO (3 , GL (4 , R ) rather than only the un-broken SO (3 , GL (4 , R ) tensor related to the existence of aninvariant tensor, the Minkowski η µν , in the unbroken SO (3 , GL (4 R ).It should perhaps be pointed out that, though some mathematical manipulationsare similar, this construction has nothing to do with the many attempts to recast2R as a gauge theory of sorts - in this extensive and contorted literature the term‘Goldstone boson’ is occasionally invoked in various guises. In this paper the pointof view is the complete opposite: no fundamental fields or connections gauging somesymmetry groups are introduced. Instead, spontaneous breaking (by some unknownunderlying dynamics) is assumed to have occurred, and the effective theory of theresulting Goldstone degrees of freedom is seen, when expressed in the right variables,to assume the form of gravitational interactions. There are in fact only two works [6],[7] the author is aware of where GL (4 , R ) spontaneous breaking in connection to gravityas a Goldstone boson is considered in a similar vein. We comment on the relation tothese earlier works in more detail below.A salient feature of the coset construction [4] - [5] is that only a symmetry breakingpattern is postulated and no assumptions whatsoever are made concerning the nature ordynamics of an underlying theory responsible for the symmetry breaking. An obviousquestion then is whether such an underlying theory can be envisioned in the case of GL (4 , R ) spontaneous breaking. Since this symmetry is here taken as a space-timesymmetry, a natural suggestion would be to consider a field theory model where theLorentz SO (3 ,
1) group is replaced by GL (4 , R ) (or SL (4 , R )) as the global symmetryof space-time. Such a GL (4 , R )-invariant theory will be physically very different fromLorentz-invariant theories, the familiar features of the latter, in particular the usualnotion of single-particle states, emerging only after symmetry breaking to the Lorentzsubgroup. We give some discussion in the last section below. SL (4 , R ) and General Rel-ativity The generators of GL (4 , R ) can be grouped in the six antisymmetric J αβ generators ofits Lorentz subgroup, and the remaining 10 symmetric linear generators T αβ comprisingnine (shape changing, volume preserving) shear generators and one (volume changing)dilatation generator. For brevity, we refer to all linear symmetric transformations asshears. The Lie algebra is given by:[ J αβ , J γδ ] = − i (cid:0) η αγ J βδ − η αδ J βγ − η βγ J αδ + η βδ J αγ (cid:1) (2.1)[ J αβ , T γδ ] = − i (cid:0) η αγ T βδ + η αδ T βγ − η βγ T αδ − η βδ T αγ (cid:1) (2.2)[ T αβ , T γδ ] = i (cid:0) η αγ J βδ + η αδ J βγ + η βγ J αδ + η βδ J αγ (cid:1) . (2.3)Adjoining the translations[ J αβ , P γ ] = − i ( η αγ P β − η βγ P α ) (2.4)[ T αβ , P γ ] = − i ( η αγ P β + η βγ P α ) , (2.5)we have the algebra of the Affine group. The Euclidean version of these commutationrelations are obtained by the replacement η αβ → − δ αβ .3 .1 Coset construction for GL (4 , R ) breaking We now consider the spontaneous breaking of the group G = GL (4 , R ) to its subgroup H = SO (3 , g ∈ G are uniquely decomposed as g = γh where h ∈ H and γ ( ξ ) = exp[ i ξ · T ] ∈ G/H . (2.6)We use the obvious notations ξ · T = ξ αβ T αβ , u · J = u γδ J γδ . This decompositionamounts to the ‘canonical’ parametrization of the left cosets G/H by means of the pa-rameters (‘preferred fields’) ξ αβ . (Any other parametrization leads of course to equiv-alent results.) For the action of g ∈ G on the cosets one can then write g exp[ i ξ ( x ) · T ] = exp[ i ξ ′ ( x ) · T ] exp[ i u ( ξ ( x ) , g ) · J ] . (2.7)Also, let ψ denote any field transforming under a linear representation R ( h ), h ∈ H ,of the unbroken subgroup. Then the transformations g : ξ → ξ ′ ( ξ, g ) , ψ → ψ ′ = R ( e u ( ξ,g ) · J ) ψ (2.8)give a non-linear realization of G . If g ∈ H , the transformations (2.8) become linear,with ψ transforming under R and ξ transforming according to the representation R of H determined from hT h − = R ( h ) T , (2.9)i.e., in the present case, as a rank-2 symmetric tensor.So far we actually have treated G as an internal symmetry (all equations above holdat space-time point x ). To treat it as a space-time symmetry, it is very convenient,though not essential, to use the extension of the coset construction for space-timesymmetries introduced in [5]. One defines the non-linear realization of G as a spacetimesymmetry by replacing (2.7) by g exp[ ix µ P µ ] exp[ i ξ ( x ) · T ] = exp[ ix ′ µ P µ ] exp[ i ξ ′ ( x ′ ) · T ] exp[ i u ( ξ ( x ) , g ) · J ] . (2.10)This defines a non-linear realization of G = GL (4 , R ) on the coordinates and the fields ξ ( x ). Indeed, as it is easily verified by use of (2.4) - (2.5), for any g ∈ G , (2.10) implies x µ → x ′ µ = A µν ( g ) x ν , (2.11)with gP µ g − = A νµ ( g ) P ν , and g exp[ i ξ ( x ) · T ] = exp[ i ξ ′ ( x ′ ) · T ] exp[ i u ( ξ ( x ) , g ) · J ] . (2.12)4ere A ( g ) = S ( q )Λ( h ) denotes the vector representation of g = qh , h ∈ H , q ∈ G/H .Correspondingly, for fields ψ transforming under a linear representation R ( h ) of H , g : ψ ( x ) → ψ ′ ( x ′ ) = R ( e u ( ξ ( x ) ,g ) · J ) ψ ( x ) . (2.13)Again, for g = h ∈ H , these non-linear realizations reduce to linear transformationsunder the representations R ( h ) and R ( h ) for ψ and ξ , respectively. To construct the effective Lagrangian one needs covariant derivatives convenientlyobtained [4] from the Maurer-Cartan form γ − d γ , or, for space-time symmetries, itsmodification [5] corresponding to (2.10). So, lettingΓ = exp[ ix · P ] γ ( ξ ( x )) = exp[ ix · P ] exp[ i ξ ( x ) · T ] , (2.14)covariant derivatives are obtained by expanding Γ − dΓ in the group generators:Γ − dΓ = i ˆ ω α P α + i2 D αβ T αβ + i2 ω αβ J αβ . (2.15)Noting thatΓ − dΓ = exp[ − i2 ξ · T] (idx · P) exp[ i2 ξ · T] + exp[ − i2 ξ · T] d exp[ i2 ξ · T] , (2.16)a calculation (Appendix A) gives explicitly:ˆ ω α = dx µ e αµ (2.17) D αβ = − { e − , de } αβ (2.18) ω αβ = −
12 [ e − , de ] αβ . (2.19)Here the symmetric matrix e is defined by e αβ ≡ ( e − ξ ) αβ = (exp[ i ξ · T v ]) αβ , (2.20)where in the second equality the subscript v denotes generators in the vector represen-tation: ( J αβ v ) δγ = i (cid:0) δ αγ η βδ − η αδ δ βγ (cid:1) (2.21)( T αβ v ) δγ = i (cid:0) δ αγ η βδ + η αδ δ βγ (cid:1) . (2.22) It should be pointed out that (2.10) may actually be taken to define realizations of the Affinegroup, since, as one easily verifies, taking g = exp iǫ µ P µ induces translation shifts of the coordinateswith ξ ′ ( x ′ ) = ξ ( x ). In fact, local translations ǫ µ ( x ) may be accommodated, which implies that someform of general coordinate invariance must be lurking in the resulting effective theory. But we willnot follow this line of argument here. ω α and D αβ transform covariantly, i.e like the fields ψ , whereas ω αβ transform inhomogeneously (‘like a gauge field’): D αβ = ˆ ω γ D γαβ , ω αβ = ˆ ω γ ω γαβ . (2.23) D αβ gives the Goldstone field ‘covariant derivative’ D . Thus, one obtains D γαβ = − e − γκ { e − , ∂ κ e } αβ . (2.24)Similarly, ω γαβ gives the ‘spin connections’ and serves to define the covariant derivative Dψ of any field ψ transforming as in (2.13): D γ ψ ( x ) = e − γκ ∂ κ ψ + i ω γαβ J αβ ψ , (2.25)with the generators in the representation of ψ .As it is well known, however, this covariant derivative is not unique. (2.19), obtainedthrough (2.15), ensures the right transformation properties for Dψ . But adding tothe expression read off (2.19) and (2.23) appropriate covariently transforming terms,such as appropriate (linear) functions of D , , results into equally good definitions ofthe covariant derivative of ψ . Different such choices simply amount to reshuffling ofterms in the effective Lagrangian formed from D , ψ and Dψ , and are a matter ofconvenience; a particular choice may result into a more transparent structure of theeffective Lagrangian. For later reference we note, in particular, the expression ω γαβ = − e − γκ [ e − , ∂ κ e ] αβ − D αβγ + D βαγ = − e − γκ [ e − , ∂ κ e ] αβ + 12 e − ακ { e − , ∂ κ e } βγ − e − βκ { e − , ∂ κ e } αγ (2.26)which augments the expression following from (2.19) by the addition of the second andthird terms in (2.26). Note that (2.24) and (2.25), (2.26), are given, in accordance with(2.23), in the basis provided by the ˆ ω α ’s.Any Lagrangian that is invariant under H and is constructed from D , ψ and Dψ will now be invariant under the full group G . The most general effective Lagrangiandescribing the interactions of the Goldstone bosons and the particle fields at scalesbelow the symmetry breaking scale is then given by the most general H -invariantfunction of ψ , Dψ , D , and their higher covariant derivatives such as DDψ and D D .It is often the case that a simpler or more transparent form of the effective La-grangian is obtained by actually not using the above canonical parametrization for the An example is provided by the pion-nucleon chiral Lagrangian (in the parametrization choicehistorically first used for the pion field), where a straightforward way of getting a nucleon covariantderivative gives a form which obscures the fact that the axial coupling constant, as it occurs in theGoldberger -Treiman relation, is an independent coupling. To avoid this it is preferable to shift aterm from this covariant derivative to an interaction with an independent coupling. ξ , but instead introducing functions of ξ and ψ with simpler trans-formation properties under G . In particular, one may find appropriate functions thattransform linearly under the full group G rather than only the unbroken group H . Thisis discussed in detail in [4], where a characterization of such functions is given; here weapply the results in [4] in the present context.Take for ψ a covector under H = SO (3 ,
1) with components v α and let V µ = e αµ v α . (2.27)Then, as can be seen from (2.10) - (2.13), the quantities (2.27) transform linearly as acovariant vector under G = GL (4 , R ). Indeed: V ′ ( x ′ ) = ( e − ξ ′ ( x ′ ) ) v ′ ( x ′ ) = ( e − ξ ′ ( x ′ ) )Λ( u ( ξ ( x ) , g )) v ( x ) = A ( g )( e − ξ ( x ) ) v ( x ) = A ( g ) V ( x ) . Similarly, it is seen that, given a vector under H with components v α , the quantities V µ = e − µα v α transform linearly as the contravariant components of a vector under G .More generally, from any Lorentz tensor with components ψ α ··· α k β ··· β l , one may obtain an GL (4 , R ) tensor with components given by:Ψ µ ··· µ k ν ··· ν l = e β ν · · · e β l ν l e − µ α · · · e − µ k α k ψ α ··· α k β ··· β l . (2.28)For fields ψ in spinor representations of the Lorentz group, no corresponding quantitiestransforming linearly under GL (4 , R ) can be constructed, since there are no finite-dimensional spinor representations of GL (4 , R ).Of particular importance is the case when (2.28) is applied to the invariant tensor η αβ of the Lorentz group. Defining g µν ≡ e αµ e βν η αβ , (2.29)one obtains a GL (4 , R ) symmetric rank-2 tensor: g : g µν ( x ) → g ′ µν ( x ′ ) = ( e − ξ ′ ( x ′ ) ) αµ ( e − ξ ′ ( x ′ ) ) βν η αβ = ( e − ξ ′ ( x ′ ) ) αµ ( e − ξ ′ ( x ′ ) ) βν Λ γα ( u ( ξ, g ))Λ δβ ( u ( ξ, g )) η γδ = A κµ ( g ) A λν ( g ) g κλ ( x ) . (2.30)Its inverse g µν , defined by g µκ g κν = δ µν , then also transforms linearly, and is, equiva-lently, given by g µν = e − µα e − νβ η αβ . Again, the chiral baryon-meson Lagrangians provide a well-known example: the common textbookmeson (Goldstone) field parametrization by a unitary unimodular matrix transforming as a (¯3 , SU (3) × SU (3) is not the canonical one; but a linearly transforming one, albeit subjectto the unitary-unimodular constraints, that is much more convenient to work with, and somewhatakin to (2.29) below. .2 General Relativity as the effective theory We now note that the passage to the linearly transforming quantities (2.28) has theform of passage from the (non-holonomic) frame provided by the basis one-forms ˆ ω α and their dual basis vectors ˆ e α - defined through < ˆ ω α , ˆ e β > = δ αβ - to a coordinatebasis { d α , ∂ α } . The relation between these frames being fixed by (2.17), i.e.ˆ ω α = e ακ dx κ , ˆe α = e − κα ∂ κ , (2.31)the relation between tensor components is then indeed that given by (2.28). Takingthis geometric point of view, consider the covariant derivative ∇ acting on, say, a vector v = v α ˆ e α = V µ ∂ µ : ∇ ˆ e γ v = ∇ ˆ e γ v α ˆ e α = ˆ e γ ( v α ) ˆ e α + v α ω γαδ ˆ e δ = (cid:2) e − κγ v α,κ + ω γβα v β (cid:3) ˆ e α (2.32)with the standard definition of connection coefficients ∇ ˆ e γ ˆ e α = ω γαβ ˆ e β , (2.33)and commas denoting, as usual, partial derivatives w.r.t. x κ . Taking the generators J αβ in (2.25) in the vector representation (2.21) and comparing to (2.32) one then gets ∇ ˆ e γ v = ( D γ v α ) ˆ e α . (2.34)Specification of the connection coefficients ω γαβ defines a choice of a particular connec-tion structure. Here they are specified by (2.19), or any other equivalent choice in thesense described above - note that these satisfy ω γαβ = − ω γβα . On the other hand, interms of covariant derivative components in the coordinate frame one has ∇ ˆ e γ v = ∇ e − βγ ∂ β V µ ∂ µ = e − βγ V µ,β ∂ µ + V µ e − βγ Γ δβµ ∂ δ = e − βγ (cid:2) V µ,β + Γ µβκ V κ (cid:3) ∂ µ (2.35)= e − βγ ∇ β V µ ∂ µ (2.36)with the standard notations ∇ ∂ κ ∂ λ = Γ µκλ ∂ µ (2.37)for the coordinate frame connection components, and ∇ ∂ γ ≡ ∇ γ for the covariantderivative operator along a coordinate basis vector direction. From (2.34) and (2.36)we then get the relation between the covariant derivatives in the two frames: ∇ λ V µ = e γλ e − µα D γ v α . (2.38)8imilarly, for any type of tensor one obtains ∇ λ Ψ µ ··· µ k ν ··· ν l = e γλ e β ν · · · e β l ν l e − µ α · · · e − µ k α k D γ ψ α ··· α k β ··· β l , (2.39)and the obvious extension involving any number of derivatives. Note that (2.39) isconsistent with (2.28) as it should, since, by construction, covariant derivatives Dψ oftensors ψ transform as tensors. Furthermore, comparing (2.32) and (2.35) (or, equiv-alently, combining (2.33), (2.37) and (2.31)) gives the relation between the connectioncomponents in the two frames:Γ µλκ = ( e − e ,λ ) µκ − e γλ e − µα ω γαβ e βκ . (2.40)Applying (2.39) now to the tensor g µν defined in (2.29) gives ∇ λ g µν = e γλ e αµ e βν D γ η αβ = 0 (2.41)since, from (2.25), D γ η αβ = 0.Solving (2.41) for the Γ µκλ ’s in the familiar way gives thenΓ µλκ = 12 g µσ ( g σκ,λ + g σλ,κ − g κλ,σ ) , (2.42)i.e. the Christoffel symbols. This in turn determines a set of spin connection coefficientsthrough (2.40). Indeed, inserting (2.42) into (2.40) determines ω γαβ to be given by(2.26).Defining a curvature tensor in the usual way, i.e. r αβδγ v β ˆ e α = (cid:2)(cid:0) ∇ ˆ e δ ∇ ˆ e γ − ∇ ˆ e γ ∇ ˆ e δ (cid:1) − ∇ [ˆ e δ , ˆ e γ ] (cid:3) v , (2.43)a straightforward computation gives (cid:0) ∇ ˆ e δ ∇ ˆ e γ − ∇ ˆ e γ ∇ ˆ e δ (cid:1) v − ∇ [ˆ e δ , ˆ e γ ] v = e − ǫδ e − βγ h ( ∇ ǫ ∇ β − ∇ β ∇ ǫ ) V µ i ∂ µ . (2.44)Hence, with R µαβγ V α = ( ∇ β ∇ γ − ∇ γ ∇ β ) V µ (2.45)defining the curvature tensor components in the coordinate frame, one has the relation e − µα e ǫδ e ζγ e βκ r αβǫζ = R µκδγ , (2.46)as indeed required by (2.28). Note that the last term on the r.h.s. in (2.43) is crucialfor consistency in the non-holonomic frame, whereas, by (2.45), R µδγκ is given by thefamiliar Riemann tensor expression in terms of the Γ µλκ ’s.To recapitulate, we carried through the standard coset construction of the sponta-neously breaking of G = GL (4 , R ), as a space-time symmetry, to its Lorentz subgroup H = SO (3 ,
1) in terms of massless Goldstone fields and matter fields. These provide9 non-linear realization of G and a linear realization of H . We then proceeded to lookfor functions of these fields with linear transformation properties also under G . Havingobtained such quantities we found that their relation to the original fields, as givenby (2.28), (2.39), is that of the transition from a non-holonomic frame to a coordinateframe, the transformation being determined by the Goldstone fields. Furthermore, theexistence of the invariant tensor η αβ in SO (3 ,
1) translates into the existence of rank-2tensor transforming linearly under GL (4 , R ) whose covariant derivative vanishes. Thisin turn, selects a unique connection, which in the coordinate frame description is givenby the Christoffel symbols. We noted that this transition to (finite-dimensional) fieldslinearly transforming under GL (4 , R ) is possible only for tensor, but not spinor, rep-resentations, of SO (3 , g µν , defined in (2.29), serving as the metric tensor. To make thisexplicit we introduce some further notational conventions. All quantities introducedabove have been defined as functions of the SO (3 ,
1) tensor fields ψ and ξ with allindices raised and lowered by η αβ . In particular, GL (4 , R ) tensors have been definedas such functions. We now write E αµ = e αµ , E µα = e − µα , (2.47)and agree to raise and lower indices from the middle of the greek alphabet by g µν , andindices from the beginning of the greek alphabet by η αβ . This is easily seen to be aconsistent convention since, as it is easily verified, one has E µα E να = δ µν , E µα E µβ = δ αβ , E αµ E να = g µν , E µα E να = g µν , E αµ E µβ = η αβ , (2.48)and relations such as, for example, g µκ V κ = V µ , g µκ V κ = V µ , Ψ µνκ g κλ = Ψ µνλ , ψ αβγ η γδ = ψ αβδ = E αµ E νβ E δλ Ψ µνλ are equivalent in content to (2.27), (2.28).With these conventions then, E αµ serve as a symmetric tetrad (vierbein) connectinga local orthonormal frame to a ‘world’ coordinate system with metric g µν . The gen-eral effective Lagrangian in terms of these fields, rather than the original non-linearlytransforming ξ ’s, is now given by the sum over all possible G -invariant functions of themetric, Ψ and ∇ Ψ, the only possible leading term being the Hilbert-Einstein action.Spinor fields couple in the effective action through the vierbein in the usual manner.One may ask how the general coordinate invariance present in this effective ac-tion came about since such invariance was not input in the original setup; only thespontaneous breaking of G = GL (4 , R ) was postulated at the outset. The answer is implicit in the fact that G is treated as a space-time symmetry. After going overto variables giving a linear realization of G , this means that any tensor obtained by In this connection see also footnote 2. ∇ λ Ψ µ ··· µ k ν ··· ν l , transforms linearly(homogeneously) as a G -tensor also w.r.t. space-time indices, such as λ , introduced bythe differentiation. Any G -invariant monomial then remains invariant when the tensorsin it are subjected to transformations by matrices A ( g ), g ∈ G which have been madespace-time dependent, in particular A µν = ∂x ′ µ /∂x ν , for any differentiable x ′ ( x ). Inthis manner one effectively ends up with a non-linear realization of general coordinatetransformations over their linearly realized GL (4 , R ) subgroup, which is indeed howthey appear in the GR formalism.In summary, the argument is based solely on: (i) the assumption that there isa space-time GL (4 , R ) symmetry which is spontaneously broken to SO (3 , SO (3 ,
1) possesses an invariant constant tensor, i.e. η αβ . With no otherassumptions or inputs, application of the coset formalism of spontaneously brokensymmetries then leads, by straightforward derivation, to the conclusion above. The relation of the formalism of GR to spontaneous breaking of GL (4 , R ) was previ-ously considered in [6] and [7].The authors of [6] note the similarity of the formalism of GR to that of non-linear realizations of symmetries, and identify GL (4 , R ) as the natural relevant group.For somewhat obscure reasons, however, they choose not to follow through with thecomplete coset construction for GL (4 , R ), and as a result manage only to suggest ratherthan arrive at an exact correspondence.The authors of [7] take two groups, the conformal group and GL (4 , R ) (actually theaffine group) as their symmetry groups. They first carry out separate coset construc-tions for each group. They then impose what they call the ‘simultaneous realization’of the two groups by demanding that the GL (4 , R ) covariant derivative is expressedsolely in terms of the conformal group covariant derivative. They then argue that thisconstraint uniquely leads to GR. This procedure, however, is clearly problematic. Sucha constraint does not appear possible or even meaningful within the usual frameworkof Lagrangian field theory. In any event, as we saw, no such externally imposed con-straints are necessary to relate the effective theory of broken GL (4 , R ) to GR. Boththese papers, however, make the crucial observation of the existence of linearly trans-forming quantities such as (2.29), and their central role in establishing a correspondencewith GR. Our starting point was the assumption that there is a GL (4 , R ) space-time symmetrywhich is spontaneously broken down to SO (3 , GL (4 , R ) we found that, expressed in terms of such linearlytransforming fields, the effective theory assumes the form of gravitational theory inthe GR framework, the Hilbert action being the simplest term in the general effectiveaction. Needless to say, this form would be quite obscured in the original canonical orother non-linear parametrizations. In particular, the decoupling of Goldstone bosonsnot corresponding to the physical graviton degrees of freedom would generally not bemanifest; and arriving at the equivalence by direct computation of physical amplitudeswould be rather nontrivial.By design, the construction of the effective theory of broken symmetries [4] - [5]makes no assumptions about the nature of any underlying theory that may be responsi-ble for the assumed symmetry breaking pattern. The obvious question then is whetherthere is an underlying theory with good UV behavior, whose dynamics drives GL (4 , R )symmetry breaking leading to the effective theory description of its broken GL (4 , R )phase given above.Since here we are concerned with space-time symmetries, the most straightforwardand perhaps natural way to proceed is to consider field theories with GL (4 , R ), or SL (4 , R ) replacing SO (3 ,
1) as the global space-time symmetry. We will not exam-ine the ingredients necessary for constructing field theories with space-time symmetrygroup GL (4 , R ), or more properly its universal covering group GL (4 , R ) (we will notbother to always make this distinction for the purposes of this discussion), in any de-tail here. We will only point out a couple of apparently generic features of such fieldtheories.First, note that since there is no invariant constant tensor in GL (4 , R ), i.e. noanalog to the invariant η αβ of the SO (3 , GL (4 , R )-symmetric space-time is less ‘rigid’ than Minkowski space-time. A metric could be introduced only as one of the dynamical fields. Requiringgood UV behavior, together with this absence of an invariant metric, provides verystrong constraints. Suppose one looks for the analog of a gauge theory of fermionand vector fields. One may, in particular, look for the GL (4 , R ) generalization of theDirac equation. Taking the fermion field ψ ( x ) to transform according to an infinite-dimensional representation S ( ν ) (the representation label ν typically comprises four Another possibility involving alignment of internal and external groups via condensate formationwas suggested in [3]. The theory of representations of non-compact groups such as GL (4 , R ) or SL (4 , R ) is rathermore involved and richer, but also less developed, than the familiar representation theory of compact X µ (analogs of the γ µ matrices) definedby S ( ν ′ ) ( g − ) X µ S ( ν ) ( g ) = a ( g ) µν X ν , (3.1)where a ( g ) denotes the fundamental vector representation, g ∈ GL (4 , R ). A Diracoperator X µ ∂ µ can then be constructed. In addition, to obtain an invariant Lagrangian,one needs an intertwining operator β (analog of γ ) between the representations ν and ν ′ connected by the vertex operators. Given appropriate choice of representations,such operators can be constructed. One such construction was given in [10]. One maythen straightforwardly couple ψ to a vector field A µ transforming as a fundamentalrepresentation covector. Further terms, however, in particular kinetic energy terms,apparently cannot be written down in the absence of a metric. Such a gauge theorywould automatically be in the ‘super-strong coupling’ limit.Second, adjoining translations to form the general or special affine group, one mayapply Wigner’s argument to classify states of given momentum. Now, however, thelittle group is GL (3 , R ), or SL (3 , R ), so the states are classified according to unitaryirreducible representations of these groups. There is no notion of particle states inthe ordinary sense. The analog of single-particle states here are excitations of defi-nite momentum classified according to the infinite-dimensional unitary GL (3 , R ) rep-resentations. Amplitudes and correlation functions can be defined in a path integralquantization framework. There is, however, no physical interpretations in terms ofordinary particle physics. Such an interpretation emerges only after symmetry break-ing to the SO (3 ,
1) subgroup. SL (4 , R ) representations generally allow embedding ofan infinite sum of Lorentz spinors describing a tower of physical spins (see e.g. [11]).Note though that particle fields below the symmetry breaking scale may, in general, becomposites of the original fields. Also note in this connection that in the broken phasethe restrictions of [12] on the appearance of massless spin-2 particles are evaded as inGR; whereas in the unbroken GL (4 , R ) phase there is no invariant notion of helicity,and the argument in [12] cannot be applied. It remains to be seen, of course, whetherany consistent GL (4 , R )-symmetric theories with spontaneous breaking to the Lorentzgroup can be constructed.The author would like to thank P. Kraus for discussions. This work was partiallysupported by NSF-PHY-0852438. groups. Large classes of representations, including the so-called principal and discrete series, can beobtained by (one of the variants of) the method of induced representations [8]. There are no finite-dimensional spinor representations; and all unitary representations are of course infinite-dimensional.Certain representation classes have been studied in [9]. We note here that the so-called multiplicity-free representations (cf. first paper in [9]), i.e. those that upon reduction to the SO (3 ,
1) subgroupcontain representations of the latter only once, are not sufficient for building GL (4 , R ) or SL (4 , R )invariant field theories. In particular, to construct vector operators defined in (3.1) non-multiplicity-free representations are necessary. Appendix
Given some algebra of operators
A, B, . . . , define A { B } ≡ [ A, B ], so that A { B } =[ A, [ A, B ]] and so on. One then has the identities e iA Be − iA = ∞ X n =0 i n n ! A n { B } (A.1)and e − iA ( λ ) ddλ e iA ( λ ) = i ∞ X n =0 ( − i ) n ( n + 1)! A n { dAdλ } . (A.2)Then, from (A.1) and repeated use of (2.5) e − i ξ · T ( i d x · P ) e i ξ · T = i ∞ X n =0 i n n ! 12 n ( − ξ · T ) n { d x · P } = i ∞ X n =0 i n n ! i n ( ξ n ) αβ d x β P α = i (exp[ − ξ ]) αβ d x β P α . (A.3)From (A.2) and repeated use of (2.2) - (2.3) one gets e − i ξ · T d e i ξ · T = i ∞ X n =0 ( − i ) n ( n + 1)! ( 12 ξ · T ) n {
12 d ξ · T } = i ∞ X even n =0 ( − i ) n ( n + 1)! i n ξ n { d ξ } ) αβ T αβ + ∞ X odd n =1 ( − i ) n ( n + 1)! i n ξ n { d ξ } ) αβ J αβ (A.4)which, making use of (A.2) again, can be written as e − i ξ · T d e i ξ · T = i (cid:20) − { e ξ , d e − ξ } αβ T αβ −
12 [ e ξ , d e − ξ ] αβ J αβ (cid:21) (A.5)with { , } denoting the anticommutator as usual. From (2.16) and (A.3), (A.5) onethen has (2.15) and (2.17) - (2.19). References [1] J.D. Bjorken, Ann. Phys. (N.Y.) , 174 (1963); P.R. Phillips, Phys. Rev. ,966 (1966); H.C. Ohanian, Phys. Rev. , 1305 (1969).142] P. Kraus and E.T. Tomboulis, Phys. Rev. D 66 , 045015 (2002)[arXiv:hep-th/0203221]; S.M. Carroll, H. Tam and I.K. Wehus, Phys. Rev.
D 80 , 025020 (20009) [arXiv:0904.4680]; Z. Berezhiani and O.V. Kancheli,arXiv:0808.3181v1 [hep-th]; A. Kostelecky and R. Potting, Phys. Rev.
D 79 ,065018 (2009) [arXiv:0901.0662]; A. Jenkins, Int. J. Mod. Phys. D18, 2249 (2009)[arXiv:0904.0453].[3] E.T. Tomboulis, arXiv:1105.3720.[4] S.R. Coleman, J. Wess and B. Zumino, Phys. Rev. , 2239 (1969); C.G. Callan,S.R. Coleman, J. Wess and B. Zumino, Phys. Rev. , 2247 (1969).[5] D.V. Volkov, Sov. J. Particles Nucl. , 3 (1973).[6] C.J. Isham, A. Salam and J. Strathdee, Ann. Phys. (NY) , 98 (1971).[7] A.B. Borisov and V.I. Ogievetsky, Teor. Mat. Fiz. , 239 (1974) [Theor. Math.Phys. , 1179 (1975)][8] A.W. Knapp, Representation Theory of Semisimple Groups , Princeton UniversityPress (1986).[9] Dj. ˘Sija˘cki and Y. Ne’eman, J. Math. Phys. , 2457 (1985); A.B. Borisov, Rep.Math. Phys. , 141 (1978); V. I. Marchenko, JETP Lett. , 334 (1996).[10] J. Mickelsson, Comm. Math. Phys. , 551 (1983).[11] I. Kirsch and Dj. ˘Sija˘cki, Class. Quant. Grav. , 3157 (2002)[arXiv:gr-qc/0111088].[12] S. Weinberg and E. Witten, Phys. Lett.96B