aa r X i v : . [ h e p - t h ] F e b The Fr¨ohlich-Morchio-Strocchi mechanism andquantum gravity
Axel MaasInstitute of Physics, NAWI Graz, University of Graz,Universit¨atsplatz 5, A-8010 Graz, AustriaFebruary 4, 2020
Abstract
Taking manifest invariance under both gauge symmetry and dif-feomorphisms as a guiding principle physical objects are constructedfor Yang-Mills-Higgs theory coupled to quantum gravity. These ob-jects are entirely classified by quantum numbers defined in the tangentspace. Applying the Fr¨ohlich-Morchio-Strocchi mechanism to theseobjects reveals that they coincide with ordinary correlation functionsin quantum-field theory, if quantum fluctuations of gravity and cur-vature become small. Taking these descriptions literally exhibits howquantum gravity fields need to dress quantum fields to create physicalobjects, i. e. giving a graviton component to ordinary observed parti-cles. The same mechanism provides access to the physical spectrum ofpure gravitational degrees of freedom.
In non-gravitational quantum field theories, global and local symmetriesplay fundamentally different roles [1–4]. Local symmetries localize theories,and are essentially auxiliary. This can probably be best seen from the factthat they can be removed by a choice of suitable variables, leaving theorieshaving only (almost) global symmetries [3–8]. This shows that in principlephysics should not depend on the treatment of local symmetries, especiallynot on any gauge choices. Though in practice this is not a very useful insight,especially due to the Gribov-Singer ambiguity [3, 9, 10], it is important con-ceptually. Taken to the extreme, this implies that effects like confinement,in the sense of the absence of colored states from the spectrum, are nothingbut a manifestation that they are gauge-dependent, and hence unphysical[3]. Global symmetries, on the other hand, have important observable conse-quences [4, 5]. While global charges are, strictly speaking, also not directly1bservable, differences in global charges are. Especially, this leads to physicaleffects like degeneracies, super selection sectors, and allowed and forbiddendecays. In this sense, global symmetries are physical.These insights can be used to formulate observables, e. g. in the standardmodel [1, 2, 4], in a manifest gauge-invariant way. Interestingly, the gauge-dependent degrees of freedom encode still information on the gauge-invariantphysics, which can be used to reconstruct observable physics. While this ingeneral requires non-perturbative methods, in many cases quantum fluctu-ations are small compared to classical physics, allowing a systematic treat-ment. This essentially reduces to expanding around a classical solution.E. g., in QED [3] this is done around the quantum-mechanical exact so-lution of the hydrogen atom. In electroweak physics, this is possible dueto the Fr¨ohlich-Morchio-Strocchi (FMS) mechanism [1, 2], which expandsgauge-invariant states around vacuum expectation values. It is this latterapproach, which will also be useful here. A review of the FMS mechanismin particle physics can be found in [4].Of course, an immediate question is, what happens in a situation withgravity. In that case coordinate transformations become local themselves,and thus the line between global and local symmetries seems to blur. Thisseems to be especially true for quantum fields on a fixed singular backgroundmetric, e. g. a Schwarzschild or Kerr metric. However, a full treatmentrequires quantum gravity, to put all entities on the same footing.The aim of the present work is to construct a manifestly invariant de-scription of objects in quantum gravity, which yield the usual particles ofquantum field theory if the quantum gravitational fluctuations become neg-ligible. This will also help in defining the role both of global and localsymmetries in the quantum gravity setup. Employing the FMS mechanismwill also allow to obtain an approximate calculation scheme to obtain theirproperties.Of course, this depends on the setup for quantum gravity. Here, canon-ical quantum gravity will be used, under the assumption that it becomeswell-defined in a path-integral approach, e. g. due to asymptotic safety [11–13] or some other non-perturbative mechanism [14, 15]. An important rolewill be played by the tangent space, which will be needed to give globalsymmetries a well-defined meaning, which leads to a somewhat unusual per-spective. However, it also allows to make connections to other, more general,approaches to gravity, like Kibble-Sciama [16–18] or spin basis [19, 20] ones.The details of the setup will be discussed in sections 2 and 3. This allows forboth well-defined spin quantum numbers and will act as a guiding principleto construct global symmetries of particles in quantum gravity.In section 4 first the emergence of conventional particles in this setupwill be studied using the FMS mechanism. To consider both global and localsymmetries the simplest example is Yang-Mills-Higgs theory, in the versionwhich forms the electroweak sector of the standard model. Adding fermions2o the mix would substantially complicate the issue of spin [18, 20], and willnot be considered here. Indeed, it will be seen how the physical spectrumreemerges when the particles are studied in situations where quantum grav-ity effects are small. At the same time, a constructive way will be obtainedhow to add both curvature effects as well as quantum fluctuations of gravity.A natural consequence of requiring manifest invariance requires to dis-cuss how a graviton emerges. In preparation for this, first the simpler caseof scalar geons [21, 22] will be discussed in section 4. To investigate thegraviton requires spin, as otherwise it is not possible to distinguish geonsand gravitons, and also particle physics excitations, as will be discussed insection 5.Taking the presented results at face value leads to interesting speculationfor their consequences for phenomenology, which will be done in section6. This poses interesting questions, which will require non-perturbativetechniques to fully answer. Simulations in the spirit of [23–25] are most likelysuited to directly answer these questions as well as functional methods [11–13, 26, 27], but this is also possible using other approaches [14, 15]. Thesefuture directions will be discussed in the summary in section 7.
The basic setup in the following will be to consider (four-dimensional) space-time as a collection of events, which can be enumerated, and have definiteneighboring relations . To every event is associated a flat, Minkowski tan-gent space, and a vierbein field e aµ [28], which connects the tangent spacesto the manifold of coordinates in the usual way, i. e. by mapping the corre-sponding unit vectors E into each other, E µ = e aµ E a , where Greek indicescount in the manifold and Latin indices in the tangent space. The vierbeinis required to be locally invertible, making this a bidirectional connection.Moreover, a metric on the manifold can then be constructed as g µν = e aµ e bν η ab (1)where η ab is the flat Minkowski metric in the tangent space. From the metricthe usual Christoffel symbols Γ νµρ and Christoffel symbols with mixed indicescan be constructed by using the vierbein. Especially, this defines the spinconnection Γ µab = e ρa e σb Γ µρσ . (2)The covariant derivative then takes the form D µ = e aµ D a = ∂ µ + Γ abµ f ba (3) When thinking of the underlying R which is used to define a manifold [28], this R could be used to enumerate the events and define neighboring relations. f ba are the usual generators of the Lorentz group in the correspondingrepresentation of the object in question.The metric is assumed compatible, i. e. D µ g ρσ = 0, what ensures thatthe causal connection between neighboring events is the same, no matterwhat is the starting event. Finally, this metric is taken to be torsion free,and thus Γ ρµν = 12 g ρσ ( ∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν ) (4)as usual.Under a general coordinate transformation in the manifold x µ → x µ + ξ µ ( x µ ) = J νµ x ν the vierbein transforms homogeneously e → J e and so doesthe spin connection Γ → J Γ. Such a transformation acts on the manifoldindices µ . It should be noted that rotations and Lorentz boosts are alsorepresented by such event-dependent translations [18, 29]. Especially, orbitalangular momentum is connected to these local translations.To also include spin, it is necessary to include a Lorentz group acting onthe tangent space, but not the manifold [18, 29]. Under this transformationthe vierbein transforms also homogeneously e → Λ e . Because of that, how-ever, this is actually immediately a local transformation, if the dynamicsdepend only on the metric, like in the Einstein-Hilbert action (8) below:Any local Lorentz transformation drops out immediately in (1). Especially,also the Christoffel symbols (4) are trivially invariant under it. This is nottrue for the spin connection (2), which transforms asΓ µ → ΛΓ µ Λ − + Λ D µ Λ − , (5)with the covariant derivative (3).It should be noted that both transformations do not commute [18]. Byintroducing the covariant derivative as (3), spin is defined in the Lorentz rep-resentations of the tangent space. This assignment does not mix spin withorbital angular momentum, which is defined in the manifold [18]. Thus, ten-sors with tangent indices transform like Lorentz tensors in particle physics,and can be associated to have a fixed spin, which is essentially their repre-sentation of the Lorentz group. The local part of the Lorentz symmetry actsthen merely as a trivial reparametrization symmetry. This will become morecomplicated once fermions are introduced, but this will not be investigatedhere.If one would want to make the Lorentz symmetry also dynamical inthe sense of a gauge symmetry, this would require to introduce additionalfields. One possibility would be to make the spin connection (2) such anindependent field [16–18], another one using dynamical Dirac matrices [19,20]. In general, this will allow for space-time torsion. These options willnot be considered here. In fact, in section 5 it will be seen that such localLorentz symmetry is actually potentially problematic at the quantum level,if one would like to keep spin as an observable.4rom these objects the Riemann tensor can be constructed as [18] R σµνρ = e aρ e σb η ac F cbµν (6) F abµν = 2 (cid:16) ∂ [ µ Γ abν ] + η cd Γ ca [ µ Γ dbν ] (cid:17) , (7)and the corresponding contractions create the Ricci tensor and curvaturescalar. A suitable classical action for this theory can be constructed as [18] S = 12 κ Z d x det( e ) (cid:16) e µa e νb F abµν + l (cid:17) (8)Here, κ and l are the usual combinations of Newton’s constant and thecosmological constant. The experience [11–15, 24, 25] strongly suggests thatthis action is insufficient at the quantum level, and that higher-order terms,e. g. of R and spin type [18], are necessary. As all calculations here willremain at lowest-order tree-level, this does not need to be specified yet,though the quantitative results below may, of course, change.In the following this theory will be coupled to Yang-Mills-Higgs theory, i.e. the weak/Higgs sector of the standard model. For the scalar the couplingto the spin connection vanishes automatically in the Lagrangian, as for thetrivial representation f ab = 0. It is assumed that there is also no couplingto the gauge field, as otherwise gravity already classically breaks the gaugesymmetry [18, 28]. Especially, in such a case this corresponds to a gaugeanomaly, i. e. the quantum theory would depend on the choice of gaugein the classical theory . Conversely, the angular momentum current wouldchange under a gauge transformation, which would have already at weakgravity consequences.Hence, the weak-gauge covariant derivative ∆ remains covariant onlywith respect to the gauge field. This yields the matter action as S m = Z d x det( e ) (cid:16) g µν (∆ pqµ φ qu ) † (∆ prν φ ru ) + V ( φ † φ ) + g µρ g νσ W iρσ W iµν (cid:17) (9)∆ pqµ = ∂ µ δ pq − igW iµ T ipq (10) W iµν = ∂ µ W iν − ∂ ν W iµ − gf ijk W jµ W kν , (11)where letters i, ... enumerate the adjoint representation of the gauge algebra, p, ... the fundamental representation of the gauge algebra, and u, ... the fun-damental representation of the custodial symmetry, i. e. the flavor symmetryof the Higgs degrees of freedom [4], both assumed to be an SU(2) group, inaccordance with the standard model.This theory features covariantly conserved angular momentum, whichis entirely made from orbital angular momentum, as the spin of the gauge It may be that this would actually be possible, see e. g. [30], but this will not beconsidered here. J uµ = ℑ tr (cid:16) T u X † ∆ µ X (cid:17) (12) X = φ − φ † φ φ † ! , (13)with T u the generators of the custodial symmetry, is covariantly conserved D µ J uµ = ∂ µ J uµ = 0 (14)as none of the building blocks couple to the spin connection, the couplingconstants f in (3) are all zero [18], and thus the conservation reduces to theordinary one. This is quite similar to the electromagnetic case [18], wherethe covariant derivative in the current conservation can also be replaced byan ordinary one. It is precisely due to the absence of a coupling of the gaugefields to the spin connection and the fact that the charge carriers are scalarsthat this is possible.To summarize, the theory has four different symmetries. One is the dif-feomorphism on the manifold, which takes the form of a local transformationof the vierbein or metric as the elementary degree of freedom. There is theLorentz transformation, which acts at every event only on the vierbein, buttrivially on the action. Nonetheless, the diffeomorphisms and the Lorentztransformation do not commute when applied to the vierbein. There is aglobal custodial transformation C , which acts only on the Higgs field, inthe same way at every event. Finally, there is a gauge transformation G .It transforms both the W field and the Higgs field in the internal space atevery event in a local way. In this context, the gauge-field acts like theconnection. Thus, under transformations J , C , G , and Λ the independentfields behave as e aµ → J νµ ( x )Λ ab e bν (15) X pu → G qp ( x ) C vu X qv (16) W pqµ → J νµ ( x ) G pr ( x ) G − qs ( x ) W rsν + J νµ ( x ) G pr (cid:0) ∂ ν G − ) (cid:1) qr (17)where W µpq = W µi T ipq and the change of evaluation coordinates x of theevents has been suppressed for brevity.6 The quantum theory
To obtain a quantum version of this theory requires at the moment variousassumptions. It will here be assumed that a path integral formulation Z = Z D e D φ D W e i ( S + S m ) (18)works. Note that in a sense of a geometrical definition the fields are functionsof events, i. e. every field configuration gives a field amplitude at every event.Only to actually calculate a quantity like the action coordinates need to beintroduced.Herein are made three central assumptions. One is that the vierbeinis the suitable dynamical variable. However, exchanging it for the metricwould lead to essentially no change in the remainder. The motivation tochose the vierbein instead of the metric is to have a dynamical variablecarrying the information about spin. The second is that the measure is asuitable Haar measure without adding further terms to the action to avoidpossible obstructions [31]. And finally that it is necessary to integrate overthe full, non-compact GL(4, R ) group, i. e. without restrictions of the possiblemanifolds. Especially, it is assumed that diffeomorphism orbits have all thesame (infinite) size.Note that the measure is also invariant under the local Lorentz transfor-mations and diffeomorphisms. However, local Lorentz symmetry enters in atrivial way. Especially, the partition sum is well defined without fixing thissymmetry, and its volume could be absorbed in the normalization. This isin contrast to the diffeomorphism symmetry, which would need treatmentbefore perturbative expressions could be calculated.That these assumptions are probably insufficient beyond tree-level isshown by the apparent necessity of counter terms and probable need forextended gravity actions [14, 15, 24, 25]. Also, at loop-order a dynamicaleffect like, e. g., asymptotic safety [11–13, 26] will be needed to make thetheory well-defined. All of this will be necessary when pushing the presentinvestigations beyond the tree-level ones to follow. However, as it will beseen, this may be only a relatively small quantitative effect except at themost extreme of situations.There is one important consequence of (18) which is true for it and forany of its extension which does not introduce absolute frames: Just as withquantum field theories [1, 2, 4, 32], this implies that quantities not invariantunder local diffeomorphism transformations, or gauge transformations, havenecessarily zero expectation value , as long as no coordinate system is fixed.The reason is that the path integral (18) sums over all possible values of thevierbein, and thus metric, with equal weight, as the action and measure are Note [4] that an individual measurement will very much yield a non-zero value - pro-vided the measurement process remains in quantum gravity as it is in quantum mechanics. R ) is anon-Abelian group, this may be involved due to the Gribov-Singer ambiguity[9, 10].Especially, this implies that h g µν i = 0. Likewise, any quantity carryingnon-contracted indices, no matter whether tangential ones, space-time ones,or belonging to the custodial or gauge symmetry, has necessarily vanishingvacuum expectation values. Just because every possible transformation ofit will be integrated over as well. Thus, the only non-vanishing vacuumexpectation values are those with fully contracted indices, which are alsoinvariant under local transformations. Thus, they can only be products ofoperators O a...r... ( x ), which are diffeomorphism and gauge invariant, and arecontracted as ω a a ...r r ... O a ...r ... ( x ) O ′ a ...r ... ( y ) ... (19)Herein is ω a constant tensor, built as a tensor product from arbitraryinvariant tensors of all involved groups of suitable rank. As a consequence,this requires to construct objects which transform in a suitable way underall symmetries to form physical observables.The simplest example are operators, which are completely scalar, andthus invariant. Two very interesting such scalar operators are O ( x ) = φ † ( x ) φ ( x ) , (20)which describes the physical Higgs particle [4], and O ( x ) = R ( x ) , (21)the local curvature. They will play an important role later one. It shouldbe noted that also operators like (20) and (21) depend on the event x , noton coordinates.In particular, an operator like (21) would also allow to characterize theaverage space-time structure, despite h g µν i = 0, by forming the expectationvalue (cid:10)R d x det( g ) R ( x ) (cid:11)(cid:10)R d x det( g ) (cid:11) . (22)If no event is special, the average space-time needs to be necessarily homo-geneous and isotropic, and thus this invariant average curvature is sufficientto characterize it. If it is suspected that the average space-time has a more Similar to particle physics [3, 5, 8], it could be possible to attain gauge invariance bya suitable dressing in the spirit of the Dirac string, see e. g. [33, 34]. The present approachhas a very similar relation to this as in the particle physics case [4]. h g µν ( x ) i again a meaning. Alternatives would be to determine distributionsof ds , which are also invariant quantities .The interesting question is now how to associate conventional particlephysics objects in a physical sense to operators like (20) and (21). Thesimplest physical object in particle physics is the particle itself. While thenotion of particle in itself is quite non-trivial [5], the fundamental quan-tity describing it is less ambiguous: The propagator. Any resemblance tophysical propagators will require a dependence on two events, e. g. D ( x, y ) = h O ( y ) O ( x ) i . (23)The points x and y are taken to denote the events, not the coordinates.Hence, at this point, the propagator is not a function of distance, but oftwo events. This dependence is coordinate-independent, and thus the wholeexpression is diffeomorphism-invariant. The dependence on the events canbe exchanged for any diffeomorphism-invariant characterization of the twoevents.In flat-space quantum field theory this is actually also true. But becausespace-time is static, the events are in a one-to-one correlation with coordi-nates, and thus it is not obvious. The characterization of the two eventsthen becomes just the ordinary distance, which is static as well, giving theusual notion of a propagator depending on the distance between two points.In the quantum gravity space-time is expected to be isotropic on averageas well. Hence, again only one quantity is needed for a diffeomorphism-invariant characterization of the relation between both events. However,this is more complicated now, as distance depends on the metric, and isthus dynamical as well. Thus, the distance between two events becomesitself an expectation value [23]. However, for every configuration there is aunique geodesic connecting the two events x and y [28]. Thus, a uniquelydefined expectation value for an invariant length r can be defined as r ( x, y ) = (cid:28) min z ( t ) Z yx dtg µν dz µ ( t ) dt dz ν ( t ) dt (cid:29) . (24)In this the minimization over the path z ( t ) connecting the events x and y should state to find the geodesic length. With this, the propagator (23)should be considered to be a function of the expectation value of this geodesicdistance, D ( r ( x, y )). This creates an invariant under all gauge symmetries,both local and space-time, and is hence a physical object. Of course, tocalculate (23) and (24) it is usually necessary to introduce again coordinates. The same statements should apply if the equations of motions of the quantum effectiveaction are solved. Especially, for a non-gauge fixed action without explicit coordinatesystem I would still expect h g µν i vanishes [1, 2]. That the expectation value of the vierbeins, and of the metric, vanishes isa consequence of diffeomorphism invariance and the path integral averagingover the whole group. Classically, of course, the Einstein equations haveonly solutions with g µν = 0.As the diffeomorphism invariance in (18) behaves like a gauge symmetry,it is possible to fix a gauge. This is essentially equivalent to fixing a coor-dinate system. As in gauge theories, this can be implemented by insertinga functional δ -function in such a way as to only pick up the contributionof a single representative of every diffeomorphism orbit. This will not alterthe value of any diffeomorphism-invariant quantity, though diffeomorphism-dependent quantities, like the metric, will change, and depend on the choice.Because of the non-Abelian structure, this procedure could [35–37] sufferfrom a Gribov-Singer ambiguity [9, 10]. However, the consequence of thiswill make the argument of the δ -function just (much) more involved. Butaside from this technical complication this will have no impact on the con-ceptual development here.It is possible to fix any coordinate system. Especially, it can be fixedsuch that the gauge-fixed vacuum expectation value of the metric no longervanishes. In this sense, it is very similar to what happens in Brout-Englert-Higgs physics, where the vacuum expectation value of the Higgs only appearsin some fixed gauges, but vanishes in other gauges or without gauge fixing[1, 2, 4, 38]. Furthermore, all gauge-invariant quantities remain untouched.E. g., the curvature, as an invariant quantity, has still the same value.The advantage is that there may exist gauges in which calculations be-come especially easy. E. g., just as the observed Fermi constant in a gaugewith non-vanishing Higgs vacuum expectation value can be given a simpleform and calculated at tree-level, so it is may be possible to construct gaugeswith a simple connection to one or more observables also in quantum gravity.Of course, the choice of such gauges will depend in general on the values ofthe parameters of the theory, i. e. Newton’s constant and the cosmologicalconstant.Consider, e. g., a situation where the observed curvature is that of a de10itter vacuum . Then it is possible to fix a gauge such that the vacuumexpectation value of the metric is the de Sitter metric, and just as in Brout-Englert-Higgs physics it is possible to split g µν = g cµν + γ µν , (25)where g cµν is the classical de Sitter solution, and γ µν are the quantum fluctu-ations satisfying h γ µν i = 0. Such a choice will be called curvature gauge inthe following. Note that only g µν and g cµν are separately necessarily genuinemetrics, but γ µν may not be. However, because γ µν will be mainly small inthe following, this will not be an issue at leading order.This choice is, of course, not necessary, only convenient. Another possi-ble split could be as well g µν = η µν + γ ′ µν , but then the curvature would beentirely created from the quantum fluctuations, and not from the splittedclassical part. Because the calculation of the curvature would then be hard,this would not be a good choice.The FMS idea [1, 2, 4] is to take an operator, and insert the split (25) intoits expression. This is merely an exact rewriting, which does not necessarilyhave any useful consequences. However, in BEH physics, the usefulnesscomes from the fact that physical quantities are dominated by the classicalpart, because the average amplitude of quantum fluctuations of the Higgsfield are small compared to the vacuum expectation value. Then, the FMSmechanism yields a way of determining physical non-trivial results [1, 2, 4].If now in the same sense the fluctuations around the classical metric aresmall, then this can also be used in quantum gravity in the same way. Asthe universe around us shows, just like for BEH physics, exactly such abehavior this seems to be not so a bad starting point. Of course, this is acoincidence due to the parameter values of gravity in our universe, just asis the case for BEH physics.Implementing this idea implies to expand any correlation function of anoperator O ( g µν ), in which the metric g µν appears n times, as h O ( g µν ) i = (cid:10) O ( g cµν ) (cid:11) + X i, Permutations of g c and γ (cid:10) O ( g cµν ; n − i, γ µν ; i ) (cid:11) (26)where the numbers in the second argument indicate how often the full,classical, and quantum metric in the observable appear. If the dependenceon g µν is non-linear, but analytic, the sum becomes a power series. This isespecially relevant if the inverse metric appears. If it is non-analytic, then This choice is for simplicity. Note that this is fundamentally different from a background-field approach. Themetric is split after gauge-fixing, and there is no separate symmetry transformations ofeither g cµν or γ µν . Only simultaneously transforming both in the same way is meaning-ful. A background metric in a background-field formalism would, instead, enjoy a fullindependent background diffeomorphism symmetry. g cµν = η µν yields for small γ µν flat-space-time quantum-field theory, and all the rightproperties for physics at, e. g., LHC. In this sense, experiment already tellsus that such a split must be possible in (canonical) quantum gravity.The avenues to this starts by noting that the expansion (26) permits anordering in the size of the contributions by counting powers of γ µν . Thisordering can now be used to calculate physical quantities. In particular, if O is a diffeomorphism-invariant quantity, then the right-hand sum must alsobe, even if the individual terms on the right-hand side are not. This showsin the present example immediately that the curvature is entirely given bythe first term, and all quantum corrections to it vanish or cancel in thecurvature gauge.Applying this to the propagators of section 3, it is useful to first look at12he argument, the invariant distance. Applying the expansion (26) yields r = min z ( t ) Z yx dtg cµν dz µ ( t ) dt dz ν ( t ) dt + (cid:28) min z ( t ) Z yx dtγ µν dz µ ( t ) dt dz ν ( t ) dt (cid:29) = r c + ρ. (27)Thus, the invariant distance r is the geodesic distance r c of the classicalmetric g c to which a quantum correction ρ is added. This immediately givesalso a test for the expansion. Only if | ρ/r c | ≪
1, for r c not light-like, itcan be expected to be a useful expansion. It should be noted that γ µν isdefinitely not small on individual configurations and can fluctuate locallywildly on individual configurations. The statement is essentially that allthese fluctuations compensate on average. The first important result isthat if g cµν = η µν this recovers that to leading order flat-space distancesare the arguments on which propagators depend, just as in ordinary flat-space quantum-field theory. Especially, at leading order no expectation valueneeds to be performed anymore, as can be explicitly seen in (27).Continue with g cµν = η µν . In this case the curvature also vanishes alreadyat leading order, recovering indeed flat space. As the operator O (20) doesnot depend on the metric, its propagator reads D ( r ) = D O ( y ) † O ( x ) E ( r ) = D O ( y ) † O ( x ) E ( r c + ρ ) (28)= D O ( y ) † O ( x ) E ( r c ) + X ∂ nr D ( r ) | r c ρ n + δ ( ρ ) , (29)where δ collects all non-analytic contributions in ρ . The first term in (29) isthe ordinary flat-space propagator. The second collects the quantum fluctu-ations of the metric on the distance between both events. Thus, as long as ρ is indeed small compared to r c , the remaining terms can be neglected, andthe propagator is the one in the sense of quantum field theory. In the short-distance limit, however, it can be expected that ρ at some point becomescomparable to r c , and then quantum gravity effects affect the interpretationof the two-point function as a propagator, in the sense of quantum fieldtheory.So far, however, only the argument has been evaluated using the FMSexpansion. The expectation value is still evaluated in a full quantum gravitysetting, and contains the full quantum fluctuations of the metric. It is nowthe second step of the FMS mechanism to apply a double expansion alsoto (cid:10) O ( y ) † O ( x ) (cid:11) ( r c ), once in terms of the metric, and once in terms of allother coupling constants.Because the operator O (20) does not explicitly depend on the metric,this amounts to evaluate the expectation value in a power series in γ . Thus D ( r ) = D O ( y ) † O ( x ) E g cµν ( r c ) + O ( γ µν ) (30)But then the first term is just the ordinary propagator in the fixed metric g cµν .For g cµν = η µν , this is the ordinary propagator of flat-space-time quantum13eld theory. In this sense, flat-space-time quantum field theory emerges asthe leading term in the FMS expansion of quantum gravity.In the standard model in flat space-time the propagator D can be ap-proximated by applying the FMS expansion a second time also to the Higgsfield, yielding [1, 2, 4] φ = v + Φ , (31)with h Φ i = 0 and v the Higgs vacuum expectation value. This yields finally D ( r c ) = v + v h Φ( x )Φ( y ) i + O ( v ) = v + v h Φ( x )Φ( y ) i tl + O ( v, g, λ ) (32)where the neglected terms only yield scattering thresholds. Especially, whenexpanding the term of order v to lowest order in the particle physics cou-pling, this implies that D is given by the tree-level Higgs propagator, andthus has exactly the same mass.The final result (32) is thus the following statement: For values of theNewton coupling and cosmological constants where the average fluctuationsof the full quantum metric around the classical metric is small, and overdistances where geodesics are approximately the flat-space-time geodesics,and the parameters of particles physics yield small fluctuations around therespective vacuum expectation values, the full gauge-invariant, diffeomor-phism invariant operator O (20) behaves like the observed Higgs particle.This gives a fully physical leading-order description of the Higgs in quan-tum gravity, which agrees well with experiments. In this sense, the FMSmechanism explains how systematically flat-space-time quantum field the-ory emerges as a diffeomorphism-invariant limit of quantum gravity. This isa remarkable result: Starting from a manifestly diffeomorphism and gauge-invariant composite operator in full quantum gravity, it was systematicallypossible to calculate what mass in the scalar channel would be measured atthe LHC: The one of the elementary Higgs.This result can be systematically improved, by adding higher ordersin the quantum corrections to the geodesics, the quantum fluctuations ofthe metric, and the particle physics fluctuations. However, this requires tosuitably deal with ultraviolet problems both of particle physics and quan-tum gravity. As all effects seem to be small enough [4, 39] over distancesrelevant for CERN experiments, this does not spoil the agreement with ex-periment. Of course, evaluating the quantum gravity and standard modelloop corrections requires an ultraviolet completion, which can, e. g., be dueto asymptotic safety, including the matter sector [40].There are many interesting directions how to augment the description ofthe Higgs with additional effects. One is clearly to go beyond the flat-space-time limit, both in the geodesic argument and in the classical expansionpoint. This yields quantum field theory in curved backgrounds [41, 42]. Theother would be to introduce quantum gravity fluctuations at leading non-trivial order, e. g. in the context of asymptotic safety [26, 27]. Of course,conventional particle physics effects can be included as well [4].14he next step is to consider what an operator like O (21) does, whichincludes explicitly the metric. This operator is, in fact, made up entirelyof the metric and the spin connection. Thus, in the FMS expansion (26)the first term is the classical curvature scalar. If g cµν is de Sitter or in flatspace time, this is just a constant, and in fact in both cases related to thecosmological constant, as expected in the scalar channel. Thus, the higherorders describe fluctuations on this dark energy background. Applying theFMS expansion yields, to lowest order, the expansion of D as D = (6Λ) + 3Λ h ( g µνc γ µν ( x )) + x ↔ y i (33)+ h ( g µνc γ µν ) ( x ) ( g ρσc γ ρσ ) ( y ) i ( r c ) + O ( γ ) . (34)The second term vanishes, as there is no absolute space-time. The thirdterm is the trace of the quantum metric, and describes thus a quantumexcitation over the vacuum, which to leading order will depend again onlyon r c . Thus, essentially this dilaton field creates scalar fluctuations aroundthe dark energy background. This could be considered to be a gravity ball,or geon [21, 22].Such a scalar particle, if reasonably stable and massive, is actually adark matter candidate. Its properties can be calculated in various approx-imations [26, 27, 41–43], and will depend on the chosen classical metric.What happens under the assumption that it is indeed dark matter will bebe explored more in section 6. To construct a manifestly invariant version of the W and Z bosons posesa fundamental problem. Because both particles need to be replaced byobjects invariant under the weak gauge symmetry requires them to not bejust the original gauge fields. The simplest operator to do this is actually thecustodial current operator (12) [4]. However, this operator has no definitespin. Attempting to create an operator with a definite spin yields locally J ua = e µa J uµ . (35)Such an operator certainly transforms in any tangent space locally as desired.The corresponding correlator is then D uvab = D J u † a ( x ) J vb ( y ) E . (36)The argument is thus very similar to the one which requires the globalcustodial symmetry to create the observed (approximate) triplet of physicalversions of the W and Z bosons [4]. The consequence of this is, as only thevierbein carries a tangent index, that the particles carrying spin considered15ere are effectively bound states of quantum gravity excitations and particlephysics excitations. At least when considering the operator structure.This yields now an challenging problem. As long as Lorentz transforma-tions in the tangent space are local, it would be possible to perform themat both events independently. If averaging over such random transforma-tions, an expression like (36) will vanish, for the same reasons as discussedin section 3. If the local Lorentz-symmetry would be gauged, like e. g. inKibble-Sciama gravity [18], the only possibility would be to add a dress-ing to (35), such that itself is invariant. But then no spin would remain,and thus no physically observable spin multiplets or selection rules. Similarconsiderations could apply to further generalizations of spin [19, 20].The situation is, however, different. In canonical quantum gravity thelocal Lorentz transformations are only a reparametrization symmetry, andnot part of a gauge symmetry. The situation is therefore rather more similarto insist to evaluate quantities using two different coordinate systems inordinary quantum field theory. Though such a theory is invariant underthe choice of a coordinate system, it is still necessary to choose the samecoordinate system to make meaningful statements.A possibility is hence to choose the ’same’ coordinate system in the tan-gent space at both events x and y in (36). How to achieve this in practicemay be non-trivial. One possibility is to define the integration measure in(18) only modulo local Lorentz transformations, keeping global, i. e. event-independent, Lorentz transformations. This could also be obtained by eithera ’gauge-fixing’ term or by an actually dynamical term which breaks the lo-cal Lorentz symmetry explicitly to a global one. This would yield a preferredcoordinate system in the arbitrary split in (1). Finally, an object of struc-ture Σ ab ( x, y ), which transforms under local Lorentz transformations likeΛ − ( y ) Σ ( x, y )Λ( x ) and which reduces to η ab both in case x → y and in flatspace, could be inserted into (36). This would be a kinematical analogue tothe Wilson line in ordinary gauge theories. This last option is non-trivial,because it is ad hoc not clear whether such a purely kinematic object wouldnot upset the notion of spin as a global quantity with measurable conse-quences. Still, it would be fundamentally different from a Wilson line as itdoes not involve a gauge symmetry, but only a reparametrization symmetry.The ultimate resolutions of this still requires further scrutiny.However, for the purpose at hand, an FMS-style approach, any of theseoptions will yield the same result at leading order. This can be seen as It may be possible to consider spin in the same way as four-momentum or energy assomething only creating an observable quantity in the low-energy/small curvature limit.Then one could alternatively also introduce, e. g., the spin connection as an additionaldynamical gauge field [18], and perform a further FMS expansion on it, which will at lead-ing (flat) order create spin as an effective global quantity, very much like four momentum.This avenue is not chosen here, because of the bias of the author to maintain spin as aglobal observable. D uvab = v (cid:10)(cid:0) ( e c ) µa W uµ (cid:1) ( x ) (( e c ) νb W vν ) ( y ) (cid:11) ( r c ) (37)+ v (cid:10)(cid:0) ǫ µa W uµ (cid:1) ( x ) (( e c ) νb W vb ) ( y ) + x ↔ y (cid:11) (38)+ (cid:10)(cid:0) ǫ µa W uµ (cid:1) ( x ) ( ǫ νb W vν ) ( y ) (cid:11) + ... (39) ( e c ) µa = δ µa = v h W ua ( x ) W vb ( y ) i ( r c ) + ... (40)where the vierbein was split analogously as before into a classical part andthe quantum part, e aµ = ( e c ) aµ + ǫ aµ . This implies that the leading contributionis just the ordinary W / Z propagator in flat space-time. Higher orders allcontain already at least three fields, and are thus either scattering statesor complicated bound states. Hence, in this controlled way also the W / Z propagator emerge as the ordinary elementary particles in the intermediatedistance regime. If a quantity like Σ would have been introduced, it wouldbecome η ab in the last step, and thus yield the same result as using (36)directly.The same applies to operators build entirely from the metric. Just likethe W and Z gauge bosons cannot be the physical excitations in the standardmodel, so neither can the metric act like a physical object. It is necessary toconstruct a suitable fully diffeomorphism invariant quantity. The simplestsuitable object is O ab = e µa e νb R µν , (41)i. e. the Ricci tensor projected into the tangent space, to make it a spintwo particle in the same ways as the vector particle above. Of course, thesame caveats apply here. Considering again the curvature gauge, the lowestnon-vanishing order in the propagator is D abde = const. + ( e c ) µa ( e c ) νb ( e c ) ρd ( e c ) σe h γ µν γ ρσ i + O ( γ ) . (42)As before, at intermediate distances this becomes the propagator of thequantum fluctuations of the metric around the gauge-fixed metric. At tree-level, this is a massless propagator, showing that the physical gravitationaltensor particle is massless as well, consistent with the expectation. It should,however, be noted that this is a prediction about the very involved boundstate operator (41) using the FMS mechanism .Confirming this in any non-trivial calculation would validate the FMSexpansion in quantum gravity, and would justify determining higher-ordercorrections. This would be especially interesting when it comes to the scalarexcitation, as this would yield an interesting phenomenology, as the follow-ing, very speculative, section explores. Note that massless composite particles with spin are already created similarly inconventional Yang-Mills theories [44–46]. Speculative phenomenology
The previous setup incites a number of very interesting options. This sectionis entirely speculative, but indicates possible interesting directions where theideas presented here could have consequences.The most interesting option is the existence of a scalar particle-like fluc-tuation around the dark energy background, which could play the role ofdark matter, the geon (34). There are two important ingredients to make iteven a candidate.One is that it is massive. At tree-level, this occurs because a non-vanishing cosmological constant induces such a tree-level mass, of order thecosmological constant [41, 42]. This makes the geon very light, but as ascalar, there is no stacking limit. Note that close to black holes the internalstructure will be resolved, and usual arguments against such a very lightdark matter candidate do not necessarily apply.The other is that it is sufficiently stable. As it is very light, its decaychannel is entirely into massless particles, i. e. photons and gravitons. Sincethere is no direct coupling to photons, only the graviton decay channel isinteresting, which is mediated by the matrix element M = D O ab O ba O E ≈ (cid:10) γ µν γ νµ γ ρρ (cid:11) + O ( γ ) . (43)The second equality uses the FMS mechanism, to estimate the matrix ele-ment at intermediate distances. At tree-level, this matrix element is sup-pressed by the Newton coupling, and thus this decay is very weak. Hence,on relevant time scales, the geon could be stable.The next question would then be its production in the early universe,as well as many other astrophysical constraints. However, this will happenpotentially at strong quantum gravity fluctuations, and thus may require togo beyond the leading-order FMS mechanism.The other phenomenological application to speculate about is what hap-pens to black holes or other singularities. Obviously, this needs to be alsorecast into diffeomorphism-invariant operators. Considering a Schwarzschildblack hole, this would be a scalar operator, i. e. such a black hole could ac-tually even have overlap with (34). There are two options. Either there is anon-decomposable operator describing it, i. e. one which cannot be decom-posed into separately diffeomorphism-invariant operators. Or it has overlapwith a product of separately diffeomorphism-invariant operators. In thiscase, such a black hole would actually be akin to a neutron star, which issimilarly described, but with baryon operators.Thus, a black hole would then be rather a geon star, an object made upof many individual quantum particles. The properties of black holes, likethe horizon, would then emerge as in-medium properties. The fact that, e. This is an observable process, and thus needs to be fully gauge invariant.
Herein, the FMS mechanism in quantum gravity is laid out, and its conse-quences of requiring manifest gauge invariance, both with respect to diffeo-morphism invariance and quantum gauge symmetries in quantum gravity,are taken. This shows how, very like the situation in flat-space-time quan-tum field theory, quantum gravity states can be described in terms of theelementary excitations in suitable gauges. It also gives natural descriptionsof objects, which behave like ordinary particles in space-times with negligi-ble curvature. Finally, it addresses the necessities needed for spin to becomean observable in the sense of a global symmetry.While the FMS mechanism allows to estimate the behavior of quantumgravity at tree-level and in regions of weak curvature, this is not sufficient forcalculations at loop level or at strong curvature, where the usual problemsof quantum gravity will reappear. Here, the present setup needs to besupplemented by (weak) non-perturbative physics, e. g. asymptotic safety.Of course, the present results are working under the assumption of aconventional form of quantum gravity. Replacing gravity with a differentstructure, e. g. string theory, may, or may not [48], yield different results. Itis also not obvious what happens if supersymmetry is thrown into the mix.But using different dimensionalities, especially with additional compactifieddimensions, additional fields, or a different action, will not qualitativelychange anything of the presented structural results.It should be noted that the present approach is related to loop quantumgravity in the same sense as ordinary gauge theory can be related to its19ormulation in terms of Wilson loops or other gauge-invariant variables. Itwould be equivalent on the level of observables, provided the same quanti-zation would be performed, but utilizes the quasi-local formulation of gaugedegrees of freedom at intermediate stages.The options arising for dark matter and black holes as particle-like ex-citations of the gravitational fields, taking up the ideas of geons, is veryinteresting. Although, it would be somewhat depressing from the point ofview of direct and indirect detection of dark matter. Still, this may yieldpotentially observable consequences for black hole-dark matter dynamics tobe explored.
Acknowledgments
I am grateful for many discussions on this subject to Reinhard Alkofer,Holger Gies, Astrid Eichhorn, Jan Pawlowski, H`elios Sanchis-Alepuz, andAndreas Wipf and to Malcom Perry for his questions on the FMS mechanismand black holes, which initiated this research. I am grateful to ReinhardAlkofer and Ren´e Sondenheimer for a critical reading of the manuscript andvaluable feedback.
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