Emergent gauge symmetries: Yang-Mills theory
Carlos Barceló, Raúl Carballo-Rubio, Luis J. Garay, Gerardo García-Moreno
PPrepared for submission to JHEP
Emergent gauge symmetries: Yang-Mills theory
Carlos Barceló, a Raúl Carballo-Rubio, b Luis J. Garay c,d and Gerardo García-Moreno c,a a Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía, 18008 Granada,Spain b Florida Space Institute, University of Central Florida, 12354 Research Parkway, Partnership 1,32826 Orlando, FL, USA c Departamento de Física Teórica and IPARCOS, Universidad Complutense de Madrid, 28040Madrid, Spain d Instituto de Estructura de la Materia (IEM-CSIC), Serrano 121, 28006 Madrid, Spain
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
Gauge symmetries are typically interpreted as redundancies in our descriptionof a physical system, needed in order to make Lorentz invariance explicit when workingwith fields of spin 1 or higher. However, another perspective on gauge symmetries is thatthey represent an effective decoupling of some degrees of freedom of the theory. In thiswork we discuss the extension of a mechanism for the emergence of gauge symmetriesproposed in a previous article [1] in order to account for non-Abelian gauge symmetries.We begin by examining the linearized theory and then move on to discuss the possiblenon-linear extensions via a perturbative bootstrapping process. In particular, we showthat the bootstrapping procedure is essential in order to determine the physical principlesunder which the decoupling observed at the linear level (and therefore, the emergence ofgauge symmetries) extends to the non-linear scenario. These principles are the following:low-energy Lorentz invariance, emergence of massless vector fields describable by an actionquadratic in those fields and their derivatives, and self-coupling to a conserved current.This serves as a step forward in the emergent gravity program by extending the mechanismto non-linear theories. a r X i v : . [ h e p - t h ] F e b ontents The search for a theory of quantum gravity, i.e., a theory which combines the principles ofgeneral relativity and quantum mechanics, has been one of the key cornerstones in funda-mental physics of the last century. Until now, it has not been possible to find a completelysatisfactory theory, although there are many illuminating approaches [2]. Among these ap-proaches towards constructing a theory of quantum gravity, we could distinguish whetherthe geometrical and spacetime notions, characteristic of general relativity, are emergent ornot. On the one hand, we have the theories that consider the geometrical degrees of freedomand the dynamics of spacetime as fundamental. Such theories typically try to apply quan-tization schemes to these degrees of freedom seeking for a background independent theoryof quantum gravity. Canonical quantum gravity in its modern formulation in terms of loopquantization [3] is the most popular approach within this category. On the other hand,we have approaches in which the fundamental degrees of freedom are not taken to be thespacetime itself but such a concept emerges with all of the properties of general relativity insome regimes of the theory, typically in low-energy limits. Within this category, we wouldinclude string theory [4, 5], but also theories that start from condensed-matter-like systemsas the substratum for emergence [6]. One of the main challenges of this last approach is toexplain when and how a diffeomorphism gauge symmetry can emerge in physical systems– 1 –hat do not include it in their microscopic description [7]. A related problem is the possibleemergence of gauge symmetries in particle physics [8]. The emergent paradigm is, somehow,the opposite direction to the one that has been explored the most by the community, whichis enlarging the gauge symmetry group of the Standard Model at high energies, instead ofbreaking it [9]. Grand Unification Theories [10] or Technicolor [11] are archetypal examplesof this direction of work.In this work we pursue a program for understanding emergent gauge symmetries ingeneral, introduced by some of the authors in a previous work [1]. In that work we pre-sented a linear system closely related to electrodynamics and described how an effectivegauge symmetry naturally emerges, with the only prerequisite of having a Lorentz covari-ant description with massless fields. Before addressing the more convoluted problem ofdiffeomorphism emergence, in this work we shall analyze how the Abelian description in [1]generalizes to non-Abelian symmetries, and then, also interacting non-linear theories ofrelativistic fields. Such a theory offers new difficulties that are not present in the lineartheory, and would bridge the gap between the simple linear electromagnetic case and themore complicated gravitational case, that we leave for a future work.We begin by reviewing in abstract terms (section 2) the main ideas of the mechanismpresented in [1] for the emergence of gauge symmetries. We emphasize the differencesbetween physical and gauge symmetries. In [1] it was proved that these abstract ideas areclearly implemented in a linear model tightly related to electrodynamics. In section 3 wewill introduce an equivalent linear system but now with a collection of Lorentzian fields A aµ which in a second stage will be subject to interactions among them. At this stage weagain find an emergence of gauge symmetries. We approach the problem by studying themost general Lorentz-invariant quadratic action for a set of relativistic fields and see howthe gauge symmetry of a system of uncoupled Maxwell systems emerges.The more novel issues of the present work start when discussing how to deform thelinear theory into a non-linear theory (section 4). To generate a non-linear theory we applya bootstrapping mechanism. The bootstrapping of theories of Yang-Mills type was workedout already in the seminal work by Deser [12], with the starting point being a linear theoryof vector fields invariant under both Lorentz and gauge transformations. Here we dropthe assumption of gauge invariance for the linear theory, which is essential in order todiscuss the emergence of gauge symmetries following the ideas discussed in [1]. Hence, ourbootstrapping procedure represents an extension of Deser’s analysis. We will discuss howconsistency arguments lead to the construction of a family of theories with emergent gaugesymmetries, characterized by the choice of a specific Lie Algebra of the same dimension asthe number of relativistic fields involved in the construction. Our analysis will also illustratesome aspects that were not addressed explicitly in previous works, such as the role played byboundary terms in the iterative procedure and their interplay with the possible uniquenessof the latter. We will also discuss the bootstrapping procedure for the charged mattersector, and discuss related issues such as the physical interpretation in our formalism of theso-called Gribov copies.Weinberg-Witten theorems [13] and Marolf’s theorem [14] are often invoked as imped-iments towards having a successful framework in which gauge symmetries of Yang-Mills– 2 –nd gravitational character can emerge. In fact, these results are often used to state thatsuch a program is condemned to fail from the beginning. One of the aims of this work isto provide a concrete framework which illustrates how these impediments can be bypassed,while also allowing to describe the emergence of gauge symmetries. Notation and conventions.
We work in four dimensional spacetime and we use thesignature ( − , + , + , +) . The symbol ∇ represents the covariant derivative compatible withflat metric of Minkowski spacetime. We use greek ( µ, ν... ) indices for the spacetime indices,latin indices from the beginning of the alphabet ( a, b... ) for the internal indices on the spaceof gauge fields, and from the middle of the alphabet ( i, j... ) for the internal indices withinthe flavor space of matter fields. In this section, we first clarify the meaning of gauge symmetries, making special emphasis onthe fact that local symmetries, i.e., those whose generators are functions of the position inspacetime, are not necessarily gauge symmetries. We then review the mechanism presentedin [1] for the emergence of gauge symmetries in certain systems.
For finite-dimensional systems it is straightforward to discern whether a given symmetryis physical or gauge by looking at the group parameters; physical symmetries have a finiteset of parameters, while gauge symmetries are always parametrized by functions [15, 16].In the infinite-dimensional case of a field theory, this shortcut does no longer work becauseall symmetries are now parametrized by functions [15].To circumvent this problem, one can perform a canonical analysis in phase space fol-lowing the procedure introduced by Dirac [17], in which the existence of gauge symmetrieswill manifest in the appearance of first-class constraints [18]. Alternatively, one can studythe Noether currents associated with symmetries, given that gauge symmetries are charac-terized by having identically zero Noether charges, contrary to physical symmetries whosecharges are non-trivial and can be used as coordinates parametrizing the space of solutionsof the theory. Throughout this work we will follow the second procedure, although bothare equivalent.More specifically, the current associated with a general gauge symmetry can be writtenas J µ = W µ + S µ , (2.1)where W µ is zero on-shell ( W µ | S = 0 , being S the space of solutions of the theory) and S µ = ∇ ν N [ νµ ] is a superpotential, i.e., the divergence of an antisymmetric tensor N µν , thatis identically (i.e., not only on-shell) conserved. Once the charge is computed on-shell as anintegral of J , it is clear that the first term does not contribute. Furthermore, the secondterm always produces a boundary term evaluated at the spatial boundary of the spacetimein virtue of Gauss theorem. Suitable boundary conditions supplementing the equations– 3 –f motion typically guarantee fall-off conditions such that this contribution also vanishes,rendering a trivial Noether charge [19]. The mechanism presented in [1] for the emergence of gauge symmetries is based on thischaracterization of gauge symmetries. In a system displaying no gauge symmetries a priori ,all Noether currents are non-trivial. However, if certain conditions are met that turn outto make some of these Noether currents trivial, the corresponding symmetries could beconsidered as emergent gauge symmetries.Let us consider a field theory depending on a collection of fields { φ a } a ∈ J . Let usrefer to them collectively as Φ . For simplicity we require that the system is free of gaugesymmetries . This means that all of the symmetries the theory might display will have non-vanishing Noether charges. These charges can be used to parameterize the space of solutionsto the equations of motion S . This follows from the fact that there always exists a completeset of symmetries whose Noether charges parameterize the space of initial conditions of thedynamical equations of the theory [20].These are in general complicated, contact and non-point symmetries [20] that cannotbe generally found explicitly, although their existence is always guaranteed as long as theinitial value problem is well posed. Otherwise, we would have that every system can beexplicitly solved in terms of these charges, which is not the case. Integrable systems arethose for which these symmetries can be explicitly found out.Let us now introduce a set of constraints defined as Ψ = 0 . This set of constraints,that might be satisfied only approximately, can be understood as the decoupling of some ofthe degrees of freedom. The situations we are interested in are those in which the subspace
U ⊂ S which we define as the subspace of S for which Ψ = 0 is non-trivial, in the sensethat there are non-trivial solutions Φ (cid:54) = 0 for which Ψ = 0 . Requiring the subspace U to benon-trivial is a necessary condition that determines whether a choice of constraints Ψ = 0 is suitable for our aim.In terms of Noether charges, we can define Q as the complete set of charges thatparametrize S . One can always find a parametrization such that the condition Ψ = 0 amounts to the selection of a subset Q ⊥ Ψ ⊂ Q with the requirement Q ⊥ Ψ = 0 . There existother sets of charges Q (cid:54)⊥ Ψ which parametrize the different solutions within the set U , i.e.,there are different systems of Noether-charge coordinates that one can use to distinguishsolutions inside U . In principle, one can decide to use as internal Noether coordinates in U ,those associated with symmetries that can be defined within U , i.e., symmetries which leavethe condition Ψ = 0 untouched. Let us denote by Q (cid:107) Ψ those coordinates. Then, dependingof the specific system and the condition Ψ , we can have two different scenarios. Non emergence of gauge symmetry:
One can find that the set of Noether chargesassociated with symmetries Q (cid:107) Ψ that preserve the subspace U (by assumption there willalways be some of them) is a proper system of Noether coordinates in U . This is the standard It is equally possible to consider the emergence of additional gauge symmetries in a system in whichsome of the symmetries are already gauge; however, this makes the discussion more convoluted withoutproviding additional conceptual insights. – 4 –ituation one can find. These charges essentially parametrize U without redundancies.Then, the projection onto U solely removes the freedom associated with the value of Ψ , or,equivalently it only leaves the freedom parametrized by the Noether charges Q (cid:107) Ψ . Emergence of gauge symmetry:
This happens when some of the physical symme-tries preserving the subset U have trivial Noether charges when restricted to U , i.e. when Ψ = 0 . In this case, we cannot find a properly constructed internal Noether coordinatesystem in U . Not all the different physical solutions in U can be distinguished by usingonly operations within U . However, they are clearly distinguishable from the point of viewof the entire theory. If all the probes one has about the system were through these charges,one could conclude that there exist equivalent classes of solutions in U which, being in-distinguishable, can be understood as representing a single physical configuration. Theseequivalence classes correspond to the emergent gauge orbits that appear within this sub-space. Then, apart from the reduction of freedom intrinsic to the projection onto U , inpractice there is an additional reduction of freedom since some of the configurations within U are physically identified (they belong to the same equivalence class that we have intro-duced). All in all, this process can be interpreted as the entire elimination of dynamicaldegrees of freedom (meaning whole solutions; recall that each degree of freedom is definedby a pair of initial conditions) when looking only at the sector of the theory characterizedby U .This mechanism for the emergence of gauge symmetries strongly relies on the natu-ralness with which a specific system of effective equations and constraints Ψ = 0 mightappear in a low-energy regime of a possibly much more complicated theory. Although quiteabstract, as here formulated, this mechanism for the emergence of gauge symmetries hasbeen proved to work for an extension of electrodynamics in [1]. We will pursue here a gen-eralization of the mechanism for Yang-Mills theory, the main novelty being the non-linearnature of the latter theory. The existence of a suitable bootstrapping procedure connect-ing non-linear theories with their linear limit will prove crucial for the definition of thedecoupling conditions in the non-linear theory.
Let us begin with the emergence of the linearized Yang-Mills gauge symmetry. The startingpoint of our discussion will be the most general Lorentz invariant action quadratic in a col-lection of vector fields A aµ and their first-order derivatives, where the latin indices ( a, b, c... )run from to N . At this level, these indices are just labels without a deeper physicalmeaning. Modulo boundary terms which do not modify the equations of motion, the mostgeneral action that we can construct for such a theory is the following: S = (cid:90) d x √− η (cid:20) − P ab F aµν F bµν + 12 ξ ab ( ∇ µ A aµ )( ∇ ν A bν ) − M ab A aµ A bµ + A aµ j aµ (cid:21) , (3.1)where the tensor F aµν is defined as F aµν = 2 ∇ [ µ A aν ] ; (3.2)– 5 – ab , ξ ab , M ab are symmetric, constant matrices; and j aµ are conserved currents ∇ µ j aµ = 0 representing the matter field content. Furthermore, we require that the matrix P ab benon-degenerate and positive definite, as otherwise some of the equations of motion will notbe of second order and the system will not correspond to N local propagating degreesof freedom with the appropriate sign for the kinetic term. We can eliminate the matrix P ab at the expense of changing the matrices ξ ab and M ab . Since P ab is a real symmetricmatrix, we can always find an invertible matrix R ab that transforms it to the identity, i.e.that R ab R cd P ac = δ bd . Then the field transformation A aµ → R ab A bµ (which also changes thematrices ξ ab , M ab , and the current j aµ , although we will keep the same symbols to avoid amore cumbersome notation), provides the following general action: S = (cid:90) d x √− η (cid:20) − F aµν F bµν + 12 ξ ab ( ∇ µ A aµ )( ∇ ν A bν ) − M ab A aµ A bµ + A aµ j aµ (cid:21) . (3.3)The Euler-Lagrange equations derived from this action are ∇ µ F µνb − ξ ab ∇ ν ∇ µ A aµ − M ab A aν = j νb . (3.4)The case ξ ab = M ab = 0 corresponds to the linearization of a Yang-Mills theory whosegauge group is of dimension N . This linearization is equivalent to a system of decoupledMaxwell equations.Let us focus on the general case in which both matrices ξ ab and M ab are non-degenerate.Since the currents j aµ are conserved, there is a physical symmetry of the theory, given bythe following transformations: A aµ → A aµ + ∇ µ χ a , j aµ → j aµ , (3.5)where χ a are not arbitrary functions as in the linearized Yang-Mills case, but they need toobey ( ξ ab (cid:3) + M ab ) χ a = 0 . (3.6)Clearly, we need to impose boundary conditions such that χ a vanish at infinity, ensuringthat there are no zero modes. We can compute the conserved quantities associated withsuch symmetries by considering the previous transformation to be infinitesimal and ap-plying Noether’s theorem. If we compute the current J µχ associated with the infinitesimaltransformations δA aµ = (cid:15) ∇ µ χ a we obtain J µχ = − F µνa ∇ ν χ a + ξ ab ∇ ν A aν ∇ µ χ b + j aµ χ a + M ab A aµ χ b = −∇ ν ( F µνa χ a ) + ξ ab ∇ ν A aν ∇ µ χ b + ∇ ν F µνa χ a + j aµ χ a + M ab A aµ χ b , (3.7)which has the form of a divergence of a superpotential (first term) plus additional terms.Once we evaluate this current on-shell we find J µχ | on-shell = −∇ ν ( F aµν χ a ) + ξ ab (cid:16) ∇ ν A aν ∇ µ χ b − χ b ∇ µ ∇ ν A aν (cid:17) . (3.8)As in the electrodynamics case [1], these symmetries are physical symmetries and not gauge.Actually, they are equivalent to the local symmetries discussed in the electrodynamics case– 6 –or a collection of real fields ϕ a = ∇ µ A aµ . This can be made explicit by noticing that, underthe hypothesis of a conserved matter current, the divergence of the equations of motion ( ξ ab (cid:3) + M ab ) ∇ µ A aµ = 0 , (3.9)are always source-free Klein-Gordon equations for the scalar fields ϕ a . Thus, the localsymmetries we have introduced carry non-trivial Noether charges. Indeed, they can be seento correspond to the Fourier components of the free field expansion [1].If we restrict ourselves to the subspace U given by the set of fields obeying ϕ a = ∇ µ A aµ = 0 , (3.10)which is quite natural since there are no Lorentz invariant sources that might produceexcitations on this scalar sector of the theory, the local transformations (3.5) and (3.6)become gauge symmetries as long as the matrix M ab identically vanishes, M ab = 0 . Thatis, the Noether charges in (3.8) all become zero. This is completely analogous to theelectrodynamics case, where we required the mass to vanish for the gauge symmetry toemerge. This is because the gauge transformations (3.5) do not leave the subspace U invariant unless the mass matrix M ab is equal zero, since they have the form (see 3.5 and3.6) ϕ a → ϕ a − (cid:0) ξ − (cid:1) ab M bc χ c / ∈ U . (3.11)It is just for M ab = 0 that we have ϕ a → ϕ a , (3.12)under the transformations (3.5). Moreover, as we have mentioned, once we restrict ourselvesto that subspace, the Noether charges associated with these transformations (3.8) becometrivial as it should happen: Q χ = (cid:90) Σ √ h d x ∇ µ ( F aνµ m ν χ a ) = (cid:90) ∂ Σ ∞ √ γ d x n µ m ν F aνµ χ a = 0 , (3.13)where Σ is a generic spacelike surface with normal (timelike) vector m ν , ∂ Σ ∞ is the bound-ary of Σ whose normal vector we call n µ , and h, γ are the corresponding induced metrics.Here we have used Gauss theorem and the last equality follows immediately from choosingappropriate boundary conditions to exclude unphysical solutions with nonvanishing fieldstrength at infinity. To summarize, we have a subset U of solutions selected by ϕ a = 0 that, when M ab = 0 , is invariant under symmetry transformations (3.5) and such that theNoether charges identically vanish in it, although not in the whole set of solutions. So,we have precisely all the conditions described in the previous section for the emergence ofgauge symmetries. Actually, we can identify these emergent gauge symmetries with thelinearization of Yang-Mills gauge symmetries, which constitute a gauge theory whose gaugegroup is U (1) ⊗ N . In fact, the analysis up to now is completely equivalent to that in [1],but now having several copies of vector fields.We have seen how linearized Yang-Mills theory has emerged from a theory without thisgauge symmetry. We will deal with the whole non-linear theory in the following section,– 7 –here we will apply a bootstrapping procedure. For the moment, let us discuss in thislinearized framework two observables that are fundamental in the theory: the energy-momentum tensor and the Yang-Mills current. We will omit the matter content and focuson the contribution from the A aµ fields exclusively in the rest of this section.Applying Hilbert’s prescription, we can find the symmetric energy-momentum tensor T µν which agrees with the one obtained following Belinfante’s prescription [21, 22] T µν = − F aµρ F ρaν + η µν (cid:18) − F aρσ F aρσ + 12 ξ ab ∇ ρ A aρ ∇ σ A bσ (cid:19) (3.14) + 2 ξ ab A a ( µ ∇ ν ) ∇ ρ A bρ − ξ ab η µν ∇ ρ (cid:16) A aρ ∇ σ A bσ (cid:17) . (3.15)The projection of T µν onto the subspace U is clearly built out of the tensor F aµν , beingby construction invariant under the transformations (3.5). Therefore, the stress-energytensor is unable to tell the difference between elements of the equivalent classes of solutionswithin U .On the other hand, we can build Yang-Mills currents in the system. These currents areassociated with rotations in the internal space represented by the latin index a . Startingfrom the action S given in (3.3), we can see that, for general ξ ab and M ab , there are nosuch symmetries in the complete theory. However, under certain restrictions we notice thatthe action is invariant under the rigid transformations A aµ → A aµ + f abc A µb ζ c , (3.16)where ζ c is an arbitrary set of constants and the constants f abc need to verify f abc = f [ ab ] c . (3.17)The necessary restrictions are the fields being massless, M ab = 0 , and ξ ab satisfying thefollowing condition (written for simplicity in terms of Ξ acd = ξ ab f bcd ): Ξ acd = Ξ [ ac ] d . (3.18)This condition guarantees that the term in the action proportional to ξ ab is invariant underthe transformations (3.16). This condition is satisfied for instance if ξ ab = λδ ab . (3.19)There may be more general situations in which Eq. (3.18) is satisfied. However, for sim-plicity we will restrict our analysis below to theories in which ξ ab satisfies Eq. (3.19).Associated with any symmetry of this sort we have a conserved current according toNoether’s theorem which reads J (1) aµ = f bca (cid:110) F µνb A cν − λ ( ∇ ν A bν ) A cµ (cid:111) . (3.20)It is instructive to check that this current is conserved upon imposing the equations ofmotion. Taking the divergence of Eq. (3.20) and suitably grouping the terms we find ∇ µ J (1) aµ = f bca (cid:16) ∇ µ F µνb − λ ∇ ν ∇ µ A bµ (cid:17) A νc + f bca (cid:0) F µνb ∇ µ A cν − λ ∇ ν A νb ∇ µ A µc (cid:1) . (3.21)– 8 –e have four terms in this divergence. The first two terms vanish on-shell, i.e., imposingthe equations of motion Eq. (3.4) with M ab = 0 and j aµ = 0 . The third term vanishesdue to the symmetry structure of the Lorentz and internal indices. Since the tensor F µνb isantisymmetric in its Lorentz indices F µνb = F [ µν ] b , its contraction with another tensor justpicks the antisymmetric part of such tensor. In our case, the antisymmetric part of ∇ µ A νc is proportional to F µνc . This means that the third term can be written as f bca F µνb F µνc = 12 f [ bc ] a F µν ( b | F µν | c ) = 0 , (3.22)where we have taken into account the antisymmetry of f bca in its two first indices, and thefact that the contraction of the Lorentz indices of the F -tensors is symmetric in the internalindices. Finally, the fourth term is manifestly the same kind of contraction of a symmetricobject with an antisymmetric one f bca ∇ ν A νb ∇ µ A µc = f [ bc ] a ∇ ν A ν ( b | ∇ µ A µ | c ) = 0 . (3.23)Thus, we have proved that the current given by Eq. (3.20) is conserved on-shell.Here we find the first difference with the respect to a single vector field. Having severalcopies of vector fields allows to prescribe interactions between them, which in turn allowsfor the A aµ fields to become charged themselves. As we will see, there are many differentpossibilities to prescribe interactions, essentially as many as different Lie Algebras of di-mension N . But the important observation at this stage is the following: even restricting tothe constraint surface ϕ a = ∇ µ A aµ = 0 , which eliminates the second term in (3.20), noneof the currents that we can build are invariant under the emergent gauge transformations.Notice however that the charges Q a obtained by integrating the zero component of thecurrents are indeed invariant under these transformations and non-trivial.Therefore, these charges are in principle observables that one could use to distinguishbetween different solutions within the equivalence classes associated with emergent gaugesymmetries. The presence of these charges can be interpreted as traces of the completetheory, recalling that it is not a gauge theory, ab initio . We will continue the discussion ofthis important issue in the discussion section. To finish the section, let us just mention thatthe currents (3.20) are the ones that we will use to couple the fields A aµ with themselves atfirst order, on the way towards building a proper Yang-Mills theory. Up to now we have seen that if in a complicated theory there is a low-energy regime with anemergent Lorentz symmetry for a collection of Lorentz-invariant vector fields, then one willimmediately deduce that the system develops for free, under the massless assumption, theappearance of emergent gauge symmetries. But once the linearization of Yang-Mills theoryhas emerged in our system, it is natural to analyze whether this emergence can be extended(perhaps in a unique way) to the non-linear regime. This question will be explored in thissection. – 9 – .1 General idea and bootstrap procedure
To answer this question, let us assume for simplicity that we are dealing with the theoryin vacuum (vanishing source currents, j aµ = 0 ). Including the matter content back in theequations of motion will be straightforward and we will comment on that in Sec. 4.3.Let us recall that the projection that we have made onto the subspace U for whichthe gauge invariance emerges is defined by the constraints (3.10), i.e., ∇ µ A aµ = 0 . Theseconstraints emerged when analyzing the equations of motion of the linear theory, and can betherefore considered on-shell from this perspective. Given that the bootstrapping procedureaims at deriving the action of a suitable non-linear completion of a given linear theory,the most straightforward procedure is not to include these constraints as a part of thebootstrapping procedure (otherwise, the problem would be equivalent to a gauge-fixedversion of the one considered by Deser in [12]). In other words, we will be analyzing thebootstrapping of Lorentz-invariant linear theories of vector fields with no gauge invariancea priori.In practice, the starting point for the bootstrapping procedure is the action S definedin Eq. (3.3) with ξ ab = λδ ab , M ab = 0 , and j aµ = 0 , namely S = (cid:90) d x √− η (cid:20) − F aµν F µνa + λ ∇ µ A µa )( ∇ ν A aν ) (cid:21) , (4.1)Let us recall that F aµν was defined in Eq. (3.2). The additional condition that we willhave in this perturbative reconstruction of the theory will be preserving the order of theequations of motion: We want the most general theory compatible with having second orderdifferential equations.The case λ = 0 corresponds to the usual description of the bootstrapping of Yang-Millstheory. In that case, for consistency reasons, any kind of self-interactions that we add to theequations of motion of the free theory need to be introduced via a conserved current [12, 23].The idea is to consider a small coupling constant g and introduce a conserved current presenton the free theory on the right hand side, that is any of the currents J (1) aµ in (3.20) withoutthe second term (as the latter vanishes when λ = 0 ): J (1) aµ = f bca F µνb A cν . Any currenton this set is bilinear in the A aµ fields, and will introduce the first non-linearities in thetheory. In other words, we will consider that J aµ = gJ (1) aµ to first order. While for λ (cid:54) = 0 one does not need to consider conserved currents, our discussions of the electrodynamicscase in [1] and linearized Yang-Mills in the previous section strongly suggest that this willbe a necessary ingredient for the emergence of gauge symmetries at the non-linear level. Inany case, it has been discussed in [23–25] that, even when considering conserved currents,there is no unique current that can be chosen at this stage of the procedure in a naturalway. A complementary approach towards making a consistent non-linear extension of thelinear spin-1 and spin-2 theories was put forward by Ogievetsky and Polubarinov [26–28](see also the brief review in [29]).This ambiguity in the choice of current comes from the possibility of adding boundaryterms to the action which translate into identically conserved additional pieces for thecurrent computed via Noether’s procedure. For the first non-trivial interaction J (1) aµ , suchpieces come from the possible boundary terms that we can add to the quadratic action S – 10 –hile keeping its linear, second order character and containing at most quadratic terms inthe time derivatives, S ,B = (cid:90) d x √− η Q abµνρσ ∇ µ (cid:16) A aν ∇ ρ A bσ (cid:17) , (4.2)where Q abµνσρ is constructed with the tensorial quantities available, namely δ ab and η µν : Q abµνρσ = δ ab B ( η µσ η νρ − η µν η ρσ ) , (4.3)with B an arbitrary constants. The contribution of this boundary term to J (1) aµ can bestraightforwardly computed, adding to the current in Eq. (3.20) terms of the form J (1 ,B ) aµ = (cid:16) f dca Q dbµνρσ + f dca Q bdρσµν + f dba Q cdµνρσ − f dca Q bdµσρν (cid:17) A cν ∇ ρ A bσ . (4.4)When using Eq. (4.3) this expression is simplified to J (1 ,B ) aµ = Bf bca ( η µσ η νρ − η µν η ρσ ) A cν ∇ ρ A bσ . (4.5)It is straightforward to check that the tensor f bcd ( η µσ η νρ − η µν η ρσ ) in the equation aboveis antisymmetric under the exchanges µ ↔ ρ , ν ↔ σ and b ↔ c independently, which inparticular implies that J (1 ,B ) aµ is identically conserved. These ambiguity is inherent to thebootstrapping procedure, as this procedure by itself does not prefer one choice of currentor another. The specific current that we use (namely, the specific value of B ) needs to begiven as an input.In summary, we will consider the conserved source given in Eq. (3.20) (with possiblecontributions from boundary terms added to it) as the source of the equations of motion atfirst order, even if such conservation is not required from the perspective of bootstrapping.Hence, this represents an additional assumption of our construction, that at this stage canbe motivated by the invariance under the rigid transformations (3.16), which singles outthis conserved current. We will provide additional motivation for this choice below, oncethe implications that it has for the bootstrapping procedure become clear. We will also keepin mind the inherent ambiguity in the choice of a current that is associated with boundaryterms, as discussed above, and eventually explain how to deal with it.Now, to be able to derive these currents from an action principle, we need to add aterm of order g to the action, S = S + gS , such that J (1) aµ = δS δA aµ . (4.6)Adding this new term to the action will modify the current obtained via Noether’s procedureby a term of order O (cid:0) g (cid:1) , in addition to possibly imposing some consistency conditions onthe original symmetry that we identified in the free theory. Thus, we will have J aµ = gJ (1) aµ + g J (2) aµ . We have again the presence of ambiguities in the choice of J (2) aµ . Thesecome now from the possibility of adding additional boundary terms to S of the same order– 11 –nd containing the same number of fields it contains. This new piece added to the actionwill require adding a term of order O (cid:0) g (cid:1) to the action S = S + gS + g S such that J (2) aµ = δS δA aµ . (4.7)This will iteratively generate an action of the form S = ∞ (cid:88) n =0 g n S n , (4.8)where at each step we may produce additional constraints and the functionals S n are builtin order to match the contribution to the current generated by the n − term. At eachstep, we produce ambiguities with the same nature as the ones we have already discussed,as at each order we can add additional boundary terms to the action that translate intoadditional pieces for the current.At the end of the day, we will have that we can break the action into a free and aninteracting part S = S + S I , such that the variation of the interacting part will give us thewhole current to which we couple the free term δS I δA aµ = J aµ . (4.9)Although when applied to gravity this procedure requires the sum of an infinite series [25],we will see that for the Yang-Mills theory this procedure stops at order O (cid:0) g (cid:1) and S n ≡ ,for n > . As an additional part of the bootstrapping procedure we will also discuss how thisaffects the charged matter sector. Then in the next section we will discuss what happenswhen applying the bootstrapping to the unconstrained theory. Let us begin the process with the action S given in Eqs. (4.1) and (4.2). We have alreadydiscussed that we will select the current in Eq. (3.20), plus possible contributions fromboundary terms (4.4), as the source J (1) aµ at first order. Before discussing the role ofboundary terms, let us focus on the term in this current that is proportional to λ , namely J (1) aµ ( λ ) = λf bca ∇ ν A νb A µc . (4.10)As the equation to be solved in order to obtain S , namely Eq. (4.6), is a linear equation,we can consider independently the piece of the action S λ ) that leads to the piece of thecurrent above under its variation. As it is discussed in App. A, there is no choice of S λ ) that can lead to this current. This implies that the bootstrapping procedure can becompleted for the current associated with the rigid symmetries (3.16) if and only if J (1) aµ ( λ ) vanishes, which generically leads to the same condition that we identified at the linear levelwhen discussing the emergence of gauge symmetries, namely ∇ µ A aµ = 0 .This condition now appears as a requirement that must be satisfied in order to be ableto find an action (and therefore, to proceed with the bootstrapping) for the choice of current– 12 –t first order. This constraint appears then as a structural requirement of the bootstrappingprocedure. As in the analysis of the linear theory described in Sec. 2, this constraint ensuresthat the scalar degree of freedom encoded in A µ decouples. Let us stress that the ambiguityassociated with boundary terms cannot change this conclusion, as shown in App. A. Inphysical terms, this implies that this decoupling is a robust condition that must be satisfiedfor every conserved current associated with the symmetry under the transformations (3.16).Hence, in the following we will assume that the fields are divergenceless, which we willimplement in the action through a Lagrange multiplier to be added to S , which we canwrite without loss of generality (more details are provided in App. A) as S = (cid:90) d x √− η P bca µνρσ ∇ µ A bν A cρ A aσ , (4.11)where P abc µνρσ = η µν η ρσ α (cid:16) f abc − f cab (cid:17) + η µρ η νσ (cid:16) β f abc + β f cab + β f bca (cid:17) − η µσ η νρ (cid:16) β f abc + β f cab + β f bca (cid:17) . (4.12)The condition that this action leads to J (1) aµ when variations with respect to A aµ areconsidered implies the following algebraic relation: P bca σνρµ − P abc σµνρ = f bca ( η µσ η νρ − η µν η ρσ ) + Bf bca ( η µν η ρσ − η µρ η νσ ) . (4.13)This translates into an incomptible set of equations for the parameters ( α, { β i } i =1 , B ) .Thus, the system of equations that follows from (4.13) has no solution as long as we do notimpose further constraints on the components of the tensor f abc . This system of equationshas no solution as long as we do not impose further constraints on the components ofthe tensor f abc that reduce its number of independent components, which may result ina compatible system. A natural condition to impose is full antisymmetry of f abc . Thenaturalness of this choice stems from the fact that, once we consider the action (4.11),requiring the constants f abc to obey the Jacobi identity f ade f bcd + f bde f cad + f cde f abd = 0 (4.14)ensures that the transformations (3.16) remain symmetries of the theory with action S + gS . Eq. (4.14) is a consistency condition on the original symmetry that we advanced thatmight appear. Otherwise we would break the bootstrapping procedure. In fact, withoutthis consistency condition the iterative procedure cannot be continued. So let us assumeit in all what follows. Taking into account that the tensor f abc obeys the Jacobi identityand we have an Euclidean metric δ ab in that space, we conclude that f abc needs to becompletely antisymmetric. Hence, f abc can be understood as the structure constants of acompact semi-simple Lie Algebra [30]. Imposing this ansatz with the full antisymmetry of f abc , the tensor P bca σνρµ reduces to P bca σνρµ = βf abc ( η µσ η νρ − βη µν η ρσ ) . (4.15)– 13 –lugging this ansatz in the algebraic relation (4.13), we obtain the a compatible system ofequations for the parameters ( β, B ) whose unique solution is β = B = − ." y quitaría lasecuacionesWe have thus obtained S = − (cid:90) d x √− η (cid:104) f abc F aµν A µb A νc + ϑ a ∇ µ A aµ (cid:105) , (4.16)where we have added explicitly the Lagrange multipliers ϑ a that enforce the required con-straints on the fields A aµ for the self-consistency of the bootstrapping. As already noticedby Deser [12], the direct current J (1) aµ in Eq. (3.20) does not lead to the action in Eq.(4.16). The ambiguity in the definition of the current due to boundary terms in S mustbe taken into account in order to provide additional contributions to the current necessaryfor the bootstrapping procedure to work. We notice that these general ambiguities werenot emphasized enough in the past, as it was considered that the bootstrapping proceduredid not require to take these ambiguities into account. The reason for this was that thefist-order formalism used by Deser in [12] leads to Yang-Mills for a trivial choice of theseboundary terms, while this is no longer true for the second-order formalism defined usingthe vector fields A aµ . However, this does not imply that the second-order formalism cannotbe used for the bootstrapping procedure, as we have seen explicitly that the self-consistencyof the iterative procedure is enough to select the necessary boundary terms so that thereis a unique solution, up to a choice of a semi-simple Lie-Algebra of the same dimension asthe number of fields involved in the construction. The important role played by boundaryterms in the gravitational case was discussed in [24 ? , 25]. It would be interesting to have aclear understanding of the similarities and differences between Yang-Mills and gravitationaltheories from this perspective.In the next step, the previous S term produces a contribution to the current given by J (2) µa = f bcd f bea A µd A σc A eσ . (4.17)Notice that we have not added any boundary terms to S to build J (2) µa . This choiceis precisely the choice that fullfills our criteria of providing a result which implements adeformation of the original gauge symmetry and preserves the number of degrees of freedom.The action that leads to this current is S = 14 (cid:90) d x √− ηf bcd f bea A µd A σc A eσ A aµ . (4.18)The iterative procedure happens to stop here. The reason is that the S term does notcontribute to the current J aµ : It contains no derivatives of the A aµ fields and it is strictlyinvariant under the rigid transformations (3.16), thus making its contribution to the currentcomputed via Noether’s procedure identically zero.The final action S = S + gS + g S can then be written, after conveniently reorganizingthe terms, as S = (cid:90) d x √− η (cid:20) − F aµν F µνa + λ ∇ µ A µa )( ∇ ν A aν ) + gϑ a ∇ µ A aµ (cid:21) , (4.19)– 14 –here ϑ a are Lagrange multipliers and the non-linear field strength tensor F aµν has theYang-Mills field form F aµν = 2 ∇ [ µ A aν ] + gf bca A bµ A cν . (4.20)In this way we have found the free Lagrangian of a non-Abelian Yang-Mills theory in theLorenz gauge. The conditions (3.17) and (4.14) on the coefficients f abc that determinethe symmetry transformations prescribe the structure of a particular Lie Algebra (they arethe structure constant of the corresponding algebra). In the process of making the theorynon-linear we are selecting a particular form of Lie Algebra among those of dimension N .Then, we can write A µ = A aµ T a , with T a representing the generators of a semi-simple andcompact Lie Algebra which satisfy the algebraic relations [ T a , T b ] = if abc T c .The equations of motion resulting from this action are invariant under the deformationof the linear transformations (3.5). These transformations are given by the exponentiationof the transformations A aµ → A aµ + ∇ µ χ a + gf abc A bµ χ c , (4.21)with χ a obeying the following constraints that guarantee that the transformations abovedo not make the fields leave the subspace U defined by the Lorenz condition: (cid:3) χ a + gf abc A µb ∇ µ χ c = 0 . (4.22) The coupling to the matter content can also be obtained perturbatively via another boot-strapping process, as we have advanced in the previous section. Let us make explicit howthis is done, for instance, for a set of fermionic fields as if we aimed at constructing an emer-gent QCD theory. Let us represent the label corresponding to diferent flavors of fermions ψ i with latin indices i, j, k and assume that those indices run from to M . In the logic ofour iterative construction we first use as source of the A aµ fields a free fermionic currentwhich has the form j aµ = ¯ ψ i γ µ ˜ T aij ψ j , (4.23)where the matrices γ µ are Dirac gamma matrices and ˜ T aij are certain unspecified matrices.The free equation for the fermionic fields guarantees that this current is conserved no matterwhat ˜ T a we use. To see this, we just need to take the divergence of this current and use theequations of motion at zero order. Under these conditions, the decoupling of longitudinaldegrees of freedom is ensured and we have the emergent gauge symmetries that we alreadyhad in the electrodynamics example.We want to obtain the equations for A aµ from a Lagrangian in such a way that thesource of the field includes both the first order non-linearities of the fields A aµ and thisfermionic source current. In a first iteration, we therefore prescribe an action of the form S + S + S f (see (4.16)) with S f = (cid:90) d x √− η (cid:104) ¯ ψ j γ µ (cid:16) iδ ij ∇ µ − q a ˜ T aij A aµ (cid:17) ψ i (cid:105) . (4.24)Here the constants q a represent the charges associated with each of the U (1) copies of thesystem that we found in the linearized theory. From a physical point of view, it implies– 15 –hat the fermionic source not only affects the fields A aµ but that in return they also affectthe fermionic fields. Then, the resulting new equation for the fermions no longer need tofulfill an exact conservation condition, only conservation up to O ( q a ) . This is reasonable,as potentially only a sum of fermionic plus Yang-Mills currents should be conserved, andmoreover, a proper conservation will only appear when closing the bootstrapping procedure.As an aside, let us mention that by choosing the matrices ˜ T a to be commuting matrices,or equivalently, multiples of the identity, one would be able to maintain the conservationof the fermionic current at this first order, but at the cost of killing any possibility forthe action to be invariant under rigid rotations of Yang-Mills type. This structure for thefermionic current would be consistent with a theory without the S term, that is a lineartheory in which a set of non-interacting Maxwell field are coupled to a set of non-interactingfermionic fields. But this is not what we are seeking for here.The previous action for the fermions has another problem equivalent to that appearingwith the action S in equation (4.16) for the field A aµ : As it stands it is not directlyinvariant under rigid rotations (3.16). On the one hand, it is clear that this action can onlyhave the chance to be invariant under rigid rotations if the fermionic fields simultaneouslytransform as ψ i → ψ i − iχ a ˜ T aij ψ j . (4.25)On the other hand, not all sets of couplings q a and matrices ˜ T aij allow for a rigid symmetry,which one needs to continue the bootstrapping procedure. In the case of the S action,requiring the existence of a rigid symmetry restricted the form of the coefficients f [ ab ] c tothose closing a Lie Algebra (Jacobi property). In the bootstrapping process one has toactively select a specific Lie Algebra. It is now clear that the bootstrapping procedure doesnot select or point towards a concrete one. Now, requiring the presence of a rigid symmetryin the fermionic sector implies setting all the q a to a single g and requiring the matrices ˜ T aij to be precisely fermionic-space representation of the same Lie Algebra selected for the A aµ sector: [ ˜ T a , ˜ T b ] = if abc ˜ T c . (4.26)We will denote these specific matrices T aij (without a tilde). Again, unless one forces thetheory to follow this specific rule one does not obtain a consistent theory.Once we make this choice, there do not appear any more restrictions in the fermionicsector at the next order O (cid:0) g (cid:1) and, as such, the bootstrapping process is identical in itsnext step to the one described in previous sections. The result is that we need to add thematter term S f = (cid:90) d x (cid:2) ¯ ψ j γ µ (cid:0) iδ ij ∇ µ − gT aij A aµ (cid:1) ψ i (cid:3) , (4.27)to the action (4.19).Once the iterative process has finished, the final fermionic current plus the Yang-Millscurrent is the one that is conserved; neither of them is divergenceless separately. Equiv-alently, we can rephrase this assertion saying that the fermionic current is not conservedbut covariantly conserved. For instance, the covariant conservation of the matter current isthe necessary condition in the non-linear theory for the decoupling of degrees of freedom.– 16 –oreover, this action is invariant with respect to the infinitesimal gauge transformations(4.21), if we additionally perform an infinitesimal local transformation of the form ψ i → ψ i − iχ a ( x ) ˜ T aij ψ j . (4.28)in the fermionic sector.Thus, we arrive to the conclusion that the result of the bootstrapping process is thatin order to have a consistent theory we need all the coupling constants q a to be the same g .Furthermore, we need the matrices T aij (that determine the interactions between fermionsand A aµ fields) to be representations of the Lie Algebra defined by the selected constants f abc . Similar comments apply straightforwardly to other kind of matter coupled to thefields A aµ . The standard approach to construct a gauge theory assumes that there are redundancies inour description from the start. Configurations related by gauge transformations, i.e. thosewith vanishing Noether charges, represent the same physical state. Gauge symmetries defineclasses of equivalence within the configuration space of fields. In such situations, the gaugefixing conditions, like the Lorenz gauge ∇ µ A aµ = 0 . (4.29)are introduced in order to choose a representative of each class of equivalence. However,such conditions do their job well if there is a unique representative of those classes thatrespects the condition. The problem is that for non-linear gauge theories, the Lorenz gaugedoes not cross each gauge orbit once, as was shown by Gribov [31, 32]. In general thereexist more than one configuration of the fields A aµ related by gauge transformations, all ofthem obeying the condition (4.29). They are typically refered to as Gribov copies.This is, for instance, problematic from the point of view of defining the quantumtheory via path integral techniques. The gauge fixing conditions are needed in order tomake sense of the theory by summing over physically inequivalent configuration and theLorenz gauge condition is typically used because it is explicitly Lorentz invariant. Atthe perturbative level, this ambiguities are not relevant because we are exploring smalldeviations in field space from the background solution (which is typically the A aµ = 0 configuration, although it could be any other stationary point of the action where we canbase our Gaussian perturbative expansion of the theory [33]). The Lorenz gauge condition isgood enough to ensure that there are no Gribov copies around these saddle points but, whenexploring the non-perturbative regime of the theory, nothing forbids them to appear. Thus,one enters in conflict with defining the theory non-perturbatively. Gribov actually arguedthat these ambiguities could have a huge impact in the structure of the quantum theory.He even provided arguments supporting how an appropriate treatment of these features(for example, reducing the path integral to a region absent of Gribov copies, often called By gauge orbit of a certain configuration we mean the subspace within field configurations that can beobtained via a gauge transformation, i.e., the subspace of gauge-equivalent configurations to a given one. – 17 – fundamental modular region) could be related to the color confinement characteristic ofgauge theories. This is because the restriction of the integration to that region has theeffect of generating a linear increase of the interactions between color charges in the deepinfrarred. Thus, it is a possible mechanism for the explanation of confinement, although aconclusive analysis is not yet available: The theory becomes strongly coupled in that regimeand the typical perturbative computations are not reliable [31, 32].The procedure we have followed to construct a gauge theory is quite different fromthe standard approach. In our formalism, we began with a theory that had no physicalsymmetries and the gauge symmetries emerged after a suitable projection onto a naturalsubspace of the theory described, precisely, by the Lorenz condition (4.29). The emergentgauge symmetries we refer to are the transformations given by the exponentiation of (4.21).This means that these symmetries are the set of transformations acting on the A µ = A aµ T a fields as A µ → Ω( x ) A µ Ω − ( x ) + i Ω( x ) ∇ µ Ω − ( x ) , (4.30)with Ω( x ) = exp ( iχ a ( x ) T a ) , where the functions χ a ( x ) have to vanish asymptotically.Our construction points out that the configurations related by these emergent gaugetransformations are really different physical solutions, it is only that it is difficult to op-erationally differentiate them. Therefore, being faithful with our construction we shouldnot eliminate these redundancies from the path integral. This offers the first instance of adistinction between the standard Yang-Mills theory and our emergent Yang-Mills theory.Any such difference would appear in the non-perturbative regime. The emergent Yang-Mills construction that we have developed leads to several interestingobservations that go beyond the electrodynamics case discussed in [1]. When studyingstandard Yang-Mills theory it is usual to read that the theory does not possess meaningfullocal currents (see section 2.6 of [34] for a discussion of this point). We have seen thatthis is already a characteristic of the linear theory: there are no gauge-invariant Yang-Millscurrents. In fact, this is just an instance of the Weinberg-Witten theorem [13]. However,this cannot be taken as evidence that Weinberg-Witten implies that no Yang-Mills the-ory can emerge from a condensed-matter-like system, the very reason behind this beingthe emergent nature of gauge symmetries in such a framework. If gauge symmetries areemergent, field configurations that are equivalent at low energies are not equivalent fromthe perspective of the high-energy theory. Alternatively, internal observers that experienceonly the low-energy physics cannot distinguish operationally between certain configurations,while external observers that are aware that the description used by the internal observeris limited to low energies can certainly distinguish between them (see e.g. [35] for anotherexample in which the distinctions between internal and external observers are discussedexplicitly). Hence, it is not needed to demand the existence of certain observables (e.g. acurrent) that are gauge invariant as a self-consistency condition necessary for emergence.Conversely, that such a current does not exist in the low-energy description cannot be takenas an indication of the impossibility of embedding this description in an emergent frame-– 18 –ork in which the difference between configurations that are equivalent at low energies hasa definite operational meaning. We hope that this clarifies that claims in the literaturethat the Weinberg-Witten theorem forbids the emergence of certain theories are using atoo narrow notion of emergence. At most, Weinberg-Witten theorem could be taken as anindication that Lorentz and gauge symmetries must emerge simultaneously, which is indeedcompatible with our discussion.If we were only talking about interpretative choices, then the internal observer positioncould be argued to be superior. However, it is well known that the importance of havingalternative interpretations is that they suggest different extensions when the time comes.Here we have already identified one potential physical difference between both interpreta-tions: the need to eliminate or not Gribov copies from the path integral. This might alreadygo beyond an interpretative issue.
In this work we have presented how an emergent Yang-Mills theory with an emergent gaugeinvariance can represent part of the dynamics of a system with more degrees of freedom andno gauge invariance a priori. Our main result is the existence of a natural coupling schemethat leads to the emergence of gauge symmetries. The logic underneath the construction isthe following:1. One can start from a very complex theory, for instance from a condensed-matter-likesystem, with a very large but finite number of degrees of freedom.2. We restrict our attention to theories in which there is a low-energy regime which canbe effectively described in terms of a set of weakly coupled relativistic fields A aµ andfermionic matter fields ψ i (examples of how this can happen can be found in [35]and references therein). The important point here is that some collective excitationsacquire a relativistic behaviour which, in condensed-matter-like systems, is typicallyassociated with the presence of Fermi points [36–38].3. In addition, it is typically not difficult to find situations in which the effective fieldsby themselves are massless (representing sound-like excitations). Later, there mightbe a Higgs-like mechanism giving mass to some of the fields.4. Then, one can write down the most generic quadratic Lagrangian compatible withthese considerations and show the existence of a symmetry under rigid transformationsthat provides with a conserved current. The existence of this symmetry allows us todefine a non-linear completion in which the relativistic vector fields couple to thiscurrent. For the non-linear completion to be derivable from an action principle, thevector fields must satisfy a constraint under which the non-linear theory developsemergent gauge symmetries.In summary, the emergence of gauge symmetries relies only on a few simple principles:low-energy Lorentz invariance, emergence of massless vector fields describable by an action– 19 –uadratic in these fields and their derivatives, and self-coupling to a conserved currentassociated with specific rigid symmetries. Self-consistency between these principles leadsto the emergence of gauge symmetries described by semi-simple Lie-Algebras of the samedimension as the number of massless vector fields. Any theory satisfying these principlesat low energies must therefore be describable by a Yang-Mills theory.The emergence process can be understood as depending on a particular parameter,something like the temperature in a condensed matter system. Using this language, wewould say that below a certain temperature this effective field theory provides a convenientdescription of the system; above this temperature the description could be very different andnot easily related to the former. Therefore, we need not imagine the emergence procedureas something that can be described in terms of an effective field theory (with more degreesof freedom than the one presented here), and such that the Lorenz condition is non-zeroin one regime of the theory but becomes zero dynamically in another regime. There areindications that dynamical mechanisms which completely suppress degrees of freedom inan effective field theory are accompanied by different pathologies.We finish this work by recalling the structural similarity between Yang-Mills theoryand general relativity. We will devote a future work to investigate whether we can extendthe scheme of this paper to gravity. In fact, the motivation to analyze the electromagneticand Yang-Mills cases comes from the gravitational case. It is indeed reasonable to assumethat, if gravity is emergent, being this force special in the sense that describes the causalstructure of the spacetime itself, then most surely the rest of interactions would also beemergent. In any case, we expect that the present analysis will pave the way towards themore complicated gravitational case. Acknowledgments
GGM thanks Julio Arrechea for useful comments. Financial support was provided by theSpanish Government through the projects FIS2017- 86497-C2-1-P, FIS2017-86497-C2-2-P(with FEDER contribution), FIS2016-78859-P (AEI/ FEDER,UE), and by the Junta deAndalucía through the project FQM219. CB and GGM acknowledges financial supportfrom the State Agency for Research of the Spanish MCIU through the “Center of Excel-lence Severo Ochoa” award to the Instituto de Astrofísica de Andalucía (SEV-2017-0709).GGM acknowledges financial support from IPARCOS (Instituto de Física de Partículas yel Cosmos) through “Ayudas para cursar estudios de Doctorado del Instituto de Física dePartículas y del Cosmos”.
A Some algebraic considerations
In this appendix we show that there is no action that, after variation with respect to A aµ ,leads to the contribution proportional to λ , namely Eq. (4.10), to the current J (1) aµ . Letus consider a generic Lagrangian at first order L (1)( λ ) = P bca µνρσ ∇ µ A bν A cρ A aσ . (A.1)– 20 –ts variational derivative is given by δL (1)( λ ) δA µa = P bca σνρµ ∇ σ A bν A cρ + P bac ρνµσ ∇ ρ A bν A cσ − P acb νµρσ ∇ ν (cid:16) A cρ A bσ (cid:17) . (A.2)This variational derivative can be rewritten as δL (1)( λ ) δA µa = (cid:16) P bca σνρµ + P bac σνµρ − P abc σµνρ − P acb σµρν (cid:17) ∇ σ A bν A cρ . (A.3)Hence, if we use the equation above for the piece of the current in Eq. (4.10), we obtainthe equation P bca σνρµ + P bac σνµρ − P abc σµνρ − P acb σµρν ∝ γ bca η µρ η νσ . (A.4)Let us note that we have replaced f bca with a generic (that is, not satisfying specificantisymmetry requirements) tensor γ bca .The equation above can be further simplified using the fact that P bca σνρµ is symmetricunder the simultaneous exchange of c ↔ a and ρ ↔ µ . As a result, we have the simplifiedequation P bca σνρµ − P abc σµνρ ∝ γ bca η µρ η νσ . (A.5)Let us make the ansatz for P bca σνρµ consisting in the most generic tensor linear in γ abc and quadratic in η µν . The symmetries of P bca σνρµ allow us to write the following ansatz: P bca σνρµ = η µρ η νσ (cid:16) α γ abc + α γ acb + α γ bac + α γ bca + α γ cab + α γ cba (cid:17) + η µσ η νρ (cid:16) β γ abc + β γ acb + β γ bac + β γ bca + β γ cab + β γ cba (cid:17) + η µν η ρσ (cid:16) β γ abc + β γ acb + β γ bac + β γ bca + β γ cab + β γ cba (cid:17) . (A.6)Plugging this ansatz in Eq. (A.5), we obtain a system of algebraic equations for the coef-ficients { α i } i =1 and { β i , } i =1 , which turns out to be an incompatible system. Given thatNoether currents are not unique as discussed above, it is necessary to check that the intro-duction of boundary terms given in Eq. (4.2) does not allow for a solution to exist. In thepresence of boundary terms, Eq. (A.5) is modified to P bca σνρµ − P abc σµνρ = γ bca η µρ η νσ + Bγ bca ( η µν η ρσ − η µρ η νσ ) . (A.7)Let us recall that, for the purposes of the discussion in this appendix, we are replacing f abc with a more general γ abc . The analysis of the corresponding system of equations forthe coefficients { α i } i =1 and { β i , } i =1 shows also that there is no solution.For completeness, let us also consider the case in which γ abc = f abc is totally anti-symmetric, which as we have discussed in Sec. 4 appears as a necessary condition for thebootstrapping procedure to work. In this case, the most general ansatz we can make for P abcµνρσ is given by Eq. (4.15). Plugging this ansatz in Eq. (A.7) leads to an incompatiblesystem for the parameters ( β, B ) . Thus, we conclude that it is not possible to derive thepiece of the current in Eq. (4.10) (plus identically conserved pieces coming from boundaryterms) from a Lagrangian containing only the vector fields A aµ .– 21 – eferences [1] C. Barceló, R. Carballo-Rubio, F. Di Filippo, and L. J. Garay. From physical symmetries toemergent gauge symmetries. JHEP , 10:084, 2016.[2] S. Carlip. Quantum gravity: A Progress report.
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