Background-Independence from the Perspective of Gauge Theory
aa r X i v : . [ g r- q c ] D ec Background-Independence from the Perspectiveof Gauge Theory
Casey CartwrightAlex FlournoyAugust 6, 2018
Abstract
We consider two concepts often discussed as significant features of general relativity (particularly whencontrasted with the other forces of the Standard Model): background independence and diffeomorphisminvariance. We remind the reader of the role of backgrounds both as calculational tools and as part of theformulation of theories. Examining familiar gauge theory constructions, we are able to pinpoint whenin the formulation of these theories they become background independent. We then discuss extendingthe gauge formulation to gravity. In doing so we are able to identify what makes general relativitya background independent theory. We also discuss/dispel suggestions that ”active” diffeomorphisminvariance is a feature unique to general relativity and we go on to argue against the claim that thissymmetry is the origin of background independence of the theory.
We seek to clarify what is a surprisingly divisive is-sue on the role of diffeomorphism invariance in gen-eral relativity and in particular how it is related tothe background independence of the theory. A cur-sory survey of both published literature and popularinformation sources reveals a lack of consensus. Theviewpoint of this paper is that by constructing theo-ries based on gauging global symmetries, one can dis-tinguish between the steps of symmetrization and theremoval of background independence. This approachis obviously of relevance to the strong and electroweakforces and there is mounting evidence that it ap-plies equally to the gravitational interactions. Weapproach the discussion at several levels, hoping tomake the main points clear to physicists with varyingbackgrounds (no pun intended). As such, we begin byrefamiliarizing the reader with backgrounds used forcalculational convenience in the familiar example ofMaxwell’s electrodynamics using the field equationsand in its Lagrangian formulation. The latter allowsus to make certain useful definitions. We then turnto the formulation of electrodynamics as a gauge the-ory where we can most clearly see the separate stepsof symmetrization and removal of background inde-pendence, and also work in a paradigm immediatelyapplicable to the remaining fundamental forces. Ourattention then turns to the more subtle case of gravitywhere we give a brief review of the history and present status of attempts to formulate GR as a gauge theory.This allows us to address claims that diffeomorphisminvariance is the source of background independencein the theory.
Physicists are perhaps most readily familiar with thenotion of backgrounds from calculations in elemen-tary electrodynamics. In that setting a typical calcu-lation involves determining the behavior of a test par-ticle in response to a fixed set of sources. To be moreprecise, one often posits a (possibly time-dependent)charge distribution to act as a source. This source isused in Maxwell’s equations to solve for the relevantfield configurations. With the field configurations inhand, the motion of the test charge can be deducedusing the Lorentz force law in conjunction with New-ton’s laws of motion (for nonrelativistic test parti-cles). In such an analysis the backreaction of thesources due to the presence of the test charge is ig-nored (justified by its having a small mass comparedto the sources). In this scheme the source or its in-duced field configuration serves as a background forthe dynamics of the test particle. One can improvethe analysis by taking the backreaction into account.From an action viewpoint this analysis plays out1s follows: An action defined with fixed source termscoupled to the undetermined electric and magneticfields is extremized to obtain equations of motion(Maxwell’s) for the fields, which are then solved.These field configurations are then fed into an ac-tion for the test particle as fixed background fields,i.e. this action is not extremized with respect tothese fields. Extremizing this action with respect tothe test particle degrees of freedom then gives equa-tions of motion which (with additional boundary con-ditions) can be solved to determine the motion ofthe test particle. Clearly one could forgo the speci-fication of sources and instead specify a set of fixedfields to be used in the test particle action. In eithercase, the arbitrary assignment of some fixed terms inthe test particle action which influence its behaviorconstitute what can be referred to as a background.This leads us to a working definition of a background:Any degree of freedom appearing in an action withrespect to which the action is not extremized con-stitutes a background. Two comments are in order.First the notion of background here is relevant to theparticular action being used. Additionally, this use ofthe term ”background” is synonomous with the term”nondynamical” often encountered in the literature.In this context, taking into account backreaction canbe achieved by including all degrees of freedom as dy-namical components of a single action. Of course onewould still have to stipulate boundary conditions andas such these constitute a sort of background whichis only absent in cosmological scenarios [1].
In the preceding discussion, the split of the actioninto a background and test particle was a matter ofcalculational convenience. Our primary concern inthis paper is with the presence (and elimination) ofbackgrounds in the formulation of theories. To keepthe discussion more straightforward we will work inthe framework of field theory wherein all degrees offreedom are encoded in fields. This approach is alsomore aligned with our best understanding to date ofthe electroweak and strong interactions.While the gauge symmetry of electromagnetic po-tentials in relation to electric and magnetic fields is afamiliar result (often treated as not much more thana technical detail), its significance in the fundamen-tal structure of electromagnetic interactions is oftenunderstated. When one wants to understand elec-tromagnetic interactions and their similarity to thestrong and weak interactions of the Standard Model, gauge field theory is the indispensable common set-ting. Gauge field theory also has the aesthetic advan-tage of ascribing interactions to the presence of sym-metries. Qualitatively the construction of a gaugefield theory proceeds as follows: First one specifies anon-interacting field theory which possesses a global(space-time independent) symmetry. The global sym-metry is then made local (space-time dependent) typ-ically by modification of the derivative operator to acovariant form with the introduction of a compen-sating gauge field. The covariant derivative impliesa coupling between the original fields and the newgauge field, i.e. an interaction. However at this pointthe gauge field itself is nondynamical in the sense de-scribed above since it has no meaningful variationalrole in action. To render the gauge field dynamicalthe action must be augmented to include a gauge ki-netic term. With the inclusion of this term, variationof the action with respect to the gauge field becomesa meaningful operation.For a specific example, consider the Lagrangianfor a free complex scalar field L = 12 ∂ µ φ ∗ ∂ µ φ, (1)which enjoys a global phase invariance φ → e iqα φ, φ ∗ → e − iqα φ ∗ . (2)Here q is a constant which will eventually play therole of the interaction coupling (or charge) and α is the transformation parameter (like the angle in arotation). This transformation is only a symmetryof the Lagrangian if the parameter α is independentof spacetime, i.e. if the transformation is ”global”.Otherwise the derivative generates an additional termdue to the change in the parameter, e.g. ∂ µ φ → e iqα ∂ µ φ + iq∂ µ αe iqα φ. (3)We can promote this global symmetry to a local formwith parameters depending on spacetime position bysuitably modifying the derivative to a covariant form,i.e. ∇ µ φ ′ = e iqα ∇ µ φ (4)with the addition of a compensating gauge field ∂ µ → ∇ µ = ∂ µ + iqA µ . (5)Invariance under local transformations of φ is nowguaranteed so long as the gauge field suitably trans-forms, i.e. A µ → A ′ µ = A µ + iq∂ µ α. (6)It is important to take account of the theory to thispoint. We have a locally invariant Lagrangian whichnow includes an interaction between the scalar field2nd the gauge field due to the product of the scalarand gauge fields from the new term in the covariantderivative ∇ µ φ ∗ ∇ µ φ = ∂ µ φ ∗ ∂ µ φ + iq ( φ∂ µ φ ∗ − φ ∗ ∂ µ φ ) A µ + q A µ A µ φ ∗ φ. (7)There is however no kinetic (derivative) term for thegauge field, so varying the action with respect to thisdegree of freedom is meaningless. The gauge field atthis point constitutes a background as defined above.To use this action we would have to specify some(arbitrary) gauge field configuration. It is interest-ing to note that a subset of the choices for the gaugefield include those that are gauge equivalent to zero.Such choices may seem to indicate a nontrivial inter-action, but actually result from a poor choice of gaugesince they are physically indistinguishable from a freescalar field. However these particular choices are alsothe only ones consistent with the next step in the de-velopment of the theory. To proceed, we now providethe gauge field with a locally invariant kinetic term L = − F µν F µν , F µν = ∂ µ A ν − ∂ ν A µ . (8)It is now meaningful to vary the action with respectto the gauge field and hence it ceases to be a back-ground for the theory. Referring to the form of theLagrangian with vanishing kinetic term (7), we cannow see that this is a special case of the backgroundindependent version, i.e. those instances where thecompensating field is gauge equivalent to zero. Insome sense this picks out the subset of pure gaugebackgrounds as consistent with the fuller formulationof the theory. To this end, it makes sense to startwith free Lagrangians in the construction of gaugetheories in this manner. This discussion has beenpurely in terms of the action functional. This storycan be given a more geometric underpinning by start-ing with the bundle construction which identifies theobjects from which we build the action as pullbacksof principle gauge and associated vector bundles bylocal sections.An important takeaway from the preceding dis-cussion is the distinction between making a theorylocally invariant and rendering it background inde-pendent. From the gauge construction this is almostobvious, and indeed this discussion applies point forpoint to the strong and weak forces in the StandardModel. Nonetheless this has been a source of dis-agreement when applied to more complicated con-texts such as gravitation, to which we now turn. While we can strip away the background dependenceof the strong, electromagnetic and weak interactionsin the Standard Model, we should be careful to under-stand the degree to which the results are backgroundindependent. After the full gauge construction, thenewly introduced compensating fields are removed asbackgrounds by rendering them dynamical. Howeverthere still occur in the resulting actions degrees offreedom which are pre-selected but not determined bythe dynamics of the theory. These include the pre-viously mentioned boundary conditions (which canbe addressed by considering ”cosmological” theories),but also the geometry of the underlying spacetime aswell as its topology and dimension. Every physicistis familiar with the idea that the geometric sector ofthe remaining backgrounds will be addressed at somelevel by Einstein’s theory of general relativity (GR).But is there any sense in which this part of the storyplays out along the familiar lines of gauge theory?Surprisingly, despite a long history and an enormousamount of work, there still remains no complete con-sensus on how GR is realized as a gauge theory. Oneaspect of GR that complicates the discussion is thatthe symmetries expected to be part of the gauge pro-cedure are external, i.e. they act on spacetime, in-stead of acting on internal degrees of freedom. Whilethere exists reasonably well developed approaches interms of the action, there are some complications in-cluding the unavoidable presence of torsion and thelack of a geometric underpinning in terms of some-thing like a bundle construction. In a forthcomingpaper we will address gauge approaches to gravita-tion (beyond field theory) and what they have to sayabout the uniqueness of gravitation in contrast to theother Standard Model forces as well as making con-nections to theories of extended objects. For our ar-guments here it is sufficient to work at the level ofthe action in terms of fields.The history of gauge field theory approaches togravity actually began shortly after the birth of mod-ern gauge theory, circa the work of Yang-Mills [2].Utiyama first tried to obtain GR by gauging theLorentz group [3]. In his analysis he had to make sev-eral unjustified assumptions, but eventually arrivedat a theory akin to GR, however energy-momentumwas not conserved. Later Sciama gauged the Lorentzgroup in a theory already containing GR to isolatethe role of torsion as reflecting the geometric effectsof sources with intrinsic spin [4]. Kibble was the firstto fashion the more complete picture by starting inflat spacetime and gauging instead the full Poincaregroup [5] including not just the Lorentz transforma-3ions but spacetime translations as well. His formu-lation led to the presence of torsion, but also ac-counted for the more standard elements of GR. All ofthese approaches led not to pure GR, but rather toits generalization Einstein-Cartan theory. A decadeafter Kibble’s contribution, Cho fashioned a gaugetheory of the translational group R [6]. The re-sulting theory formulated by Cho is known as thelocal teleparallel equivalent of general relativity [7].The success of Kibble’s approach has been elaboratedupon and generalized in the exhaustive work of Hehlet al([8],[9]). It is interesting to note the durationof development and understanding of external gaugesymmetries. Unlike their internal counterparts themerits of external gauge symmetries have been dis-puted for nearly 50 years.At a glance, when one is faced with obtainingGR as a gauge theory, the obvious starting point isflat (non-gravitationally interacting) spacetime andthe relevant global symmetry transformations are thePoincare group (the semi-direct product of Lorentztransformations and translations SO (1 , ⋉ R ) act-ing on the spacetime coordinates as well as theLorentz group acting on the tensor indices of fields.If one tries to mimic too carefully the typical gaugeconstructions `a la Yang-Mills then one might focus onthose transformations that act at a point (linearly),i.e. the Lorentz group. This approach also lendsitself more readily to an underlying bundle struc-ture. This was the theme of Utiyama’s work. How-ever it was soon pointed out that since the sourceterm in the standard formulation of GR is the con-served energy-momentum tensor, it would be nec-essary to include the group of translations in thegauge construction. Indeed this argument also re-veals why spacetime torsion becomes a necessary in-gredient of the final theory anytime the Lorentz groupis gauged, since torsion is sourced by spin angularmomentum. Technical complications in these gaugeconstructions include accounting for the action onboth tensor indices as well as the coordinates uponwhich the field configurations depend. This requiresworking in the tetrad formalism with a local Lorentzframe defined at each point in spacetime. Addition-ally, since the action itself is integrated over space-time, its invariance under the local transformationsrequires not just a modification of the derivative op-erator, but the promotion of the Lagrangian to aLagrangian density. In any case, once invariance isachieved, the newly introduced compensating fieldsare rendered dynamical by adding to the Lagrangianappropriate gauge kinetic terms, e.g. the standardEinstein-Hilbert action. The final technical hurdleof giving the entire program a geometric underpin- ning in terms of some bundle-like structure is com-plicated by the translations. Work on this is still on-going [10],[11],[12],[13],[14],[15],[16],[17] with severalinteresting avenues to be discussed more criticallyin a forthcoming paper (Extended Objects and theBundle Structure of general relativity; manuscript inpreparation).What all of these formulations have in common isthe gauge field theory approach wherein some globalsymmetry in a non-interacting theory is gauged re-sulting in an interacting theory. They also exhibit thekey observation above that symmetrizing the theoryand making it background independent are two dis-tinct steps. In summary, one can conclude (as manyhave pointed out already) that the background inde-pendence of GR arises from the background geome-try being promoted from a fixed input to a true dy-namical component of the action. One should keepin mind as mentioned before that the resulting the-ory still has some residual background dependence inthe form of spacetime topology, dimension and anyboundary conditions imposed. Rovelli and others have claimed that what makes GRa background independent theory is its invariance un-der active diffeomorphisms. Here we pose two objec-tions to this conclusion. First, as has been pointedout by numerous authors, the distinction betweenactive and passive diffeomorphisms is ill-conceived.Once one accepts this conclusion, the formulation ofgravity as a gauge theory becomes more akin to thestandard case. It then follows, from the gauge theoryperspective discussed above, that invariance does notitself alleviate backgrounds, but rather making thecompensating degrees of freedom dynamical does.Defining diffeomorphisms as a differentiable mapsfrom one manifold to another (or itself) implies asmooth reassignment of the locations of points in themanifold. The notion of shuffling about the pointsin spacetime is what some authors call active diffeo-morphisms. This is to be (alledgedly) distinguishedfrom starting with one coordinate system and thensimply reassigning the coordinates. Rovelli has madea case for the distinction by imagining a sphere (likethe surface of Earth) with distinct points labeled bycities. He then imagines a wind map which movesthe air over the surface of the Earth to new locationsover a period of time. He posits that the wind map islike an active diffeomorphism. From day one to daytwo the wind could move the gloomy weather from4aris to Denver and the sunny weather from Denverto Paris. On the other hand he then considers co-ordinatizing the sphere, and in particular assigningParis and Denver coordinate values, e.g. ( P , P ) and( D , D ). At this point he may instead refer to theweather at ( P , P ) and ( D , D ) without referenceto the cities. Now he claims that by choosing newcoordinates that swap ( P , P ) ⇔ ( D , D ) he hasmade a change in the weather assignment at thesecoordinate values, but he clearly has not moved theweather from Paris to Denver and vice versa. Thisis is Rovelli’s basis for distinguishing between activeand passive diffeomorphisms.The problem with this distinction is the necessityof some underlying unchanging demarcation of points(cities) before the wind map or coordinates are con-sidered. Rovelli himself points out that these citiesconstitute a background which once removed renders the distinction between active and passive diffeomor-phisms needless. The rebuttal to this claim fromthe perspective of gauge theory is that backgroundsare not unchanging absolutes, but rather nondynam-ical. To be clear, if we have made an action locallysymmetric but not yet given the compensating fielda kinetic term, performing a gauge transformationhas a nontrivial effect on the compensating field (6).The field is not absolute, despite being nondynami-cal! To soften the blow, one may conclude that thereare certainly some theories wherein Rovelli’s distinc-tion is meaningful, but such scenarios are not part ofthe any standard gauge theory. To the degree thatit seems GR (and indeed the rest of the StandardModel) can be formulated in terms of gauge theory,the need to distinguish between active and passivediffemorphisms is absent.5 eferences [1] L. Smolin. The Case For Background Independence . 2005 arXiv Id 0507235v1.[2] C. N. Yang and R. Mills.
Conservation of isotopic spin and isotopic gauge invariance . Phys. Rev. , 96,1954.[3] R. Utiyama.
Invariant Theoretical Interpretation of Interaction . Phys.Rev , 101:1597–1607, 1955.[4] D.W. Sciama.
On the analogy between charge and spin in general relativity . 1962.[5] T. W. B. Kibble.
Lorentz Invariance and the Gravitational Field . Jour.Math.Phys. , 2:212–221, 1961.[6] Y. Cho.
Einstein Lagrangian as the translational Yang-Mills Lagrangian . Phys.Rev. D , 14, 1976.[7] J. Garecki.
Teleparallel equivalent of general relativity: a critical review . 2010 arXiv Id 1010.2654v3.[8] F. W. Hehl.
Gauge Theories Of Gravitation: A Reader with Commentaries . (Imperial College Press,2013).[9] F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman.
Metric-Affine Gauge Theory of Gravity:Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance . Phys.Rep. ,258, 1994.[10] D. Ivanenko and G. Sardanashvily.
The gauge treatment of gravity . Phys. Rep. , 94(I):1–45, 1983.[11] R. Tresguerres and E. W. Mielke.
Gravitational Goldstone fields from affine gauge theory . Phys.Rev.D ,62:9, 2000.[12] Romualdo Tresguerres.
Unified description of interactions in terms of composite fiber bundles . Phys.Rev.D , 66:1–15, 2002.[13] G. Sardanashvily.
Gauge gravitation from the geometric viewpoint . 2005 arXiv Id 0512115v2.[14] A. Mikovic and M. Vojinovic.
Poincare 2-group and quantum gravity . 2013 arXiv Id 1110.4694v3.[15] J. Baez and D. Wise.
Teleparallel Gravity as a Higher Gauge Theory . 2014 arXiv Id 1204.4339v3.[16] R. Tresguerres.
A proposal of foundation of spacetime geometry . 2014 arXiv Id 1407.8085v2.[17] P. Ho.