# Toward a Mechanism for the Emergence of Gravity
## 3 Two Examples: Yang-Mills Theories and Linearized Gravity
### Yang-Mills Theories
From this point onwards, there are two possible situations which we discuss below. **Massive case \(M_{ab}\
eq 0\). ** In this case no gauge symmetries emerge, since the putative emergent gauge symmetries from Equations (5) and (6) do not leave \(\mathcal{U}\) invariant. Actually, they translate into the following map for \(\varphi^{a}\)  
\[\varphi^{a}\to\varphi^{a}-\left(\xi^{-1}\right)^{ab}M_{bc}\chi^{c}, \tag{9}\]  
which takes \(\varphi^{a}=0\) to a non-vanishing value for \(\varphi^{a}\
eq 0\). **Massless case \(M_{ab}=0\). ** In this case gauge symmetries emerge naturally, since the transformations from Equations (5) and (6) become gauge symmetries by leaving the scalar fields \(\varphi^{a}\) invariant (note that the last term of (9) now vanishes). It is better to pause at this point to summarize what we have obtained so far and what are we pursuing for the non-linear theory. We began with the most general Lorentz-invariant vector field theory for a set of \(N\) vector fields \(A_{\mu}^{a}\). If these vector fields represent massless excitations, we have shown that the coupling to a conserved current leads to a set of sourceless Klein-Gordon equations for the scalar sector of the theory. Since no excitations can be produced within that subspace, we have removed the scalar sector by projecting onto the subspace \(\varphi^{a}=0\). The resulting effective field theory that we have found displays emergent gauge symmetries given by Equation (5), with \(\chi^{a}\) satisfying \(\square\chi^{a}=0\). Notice that these transformations are just the residual gauge symmetries of linearized Yang-Mills theory after the Lorenz gauge is chosen. The next natural step would be to study non-linear vector-field theories. Among the possible non-linear vector-field theories, Yang-Mills theories are special since their gauge symmetry avoids the propagation of ghosts when one linearizes them. Furthermore, they are such that they can be derived from the linear theory through a bootstrapping procedure in which one analyzes the self-coupling of the theory [19]. However, such procedure does not lead to a unique theory by itself due to the ambiguities inherent to the definition of a Noether current [20; 21], at least not with further specifications [22]. The combination of the mechanism introduced here for the emergence of gauge symmetries with the bootstrapping procedure allowed us to make an incursion on the non-linear vector-field theories that are free from instabilities. We present here the main idea and refer the reader to [15] for further technical details. The idea of the bootstrapping procedure is the following. Consider a linear field theory displaying a symmetry and hence having a conserved current which we call \(J_{1}^{a\mu}\). We assume that our theory couples perturbatively to such current in such a way that we recover a smooth limit for the free theory when we take the coupling constant to be zero. We call such a coupling constant \(g\). Thus, schematically we will have equations of motion of the form  
\[\frac{\delta S_{2}}{\delta A_{a\mu}}=gJ_{1}^{a\mu}+\mathcal{O}\Big{(}g^{2} \Big{)}. \tag{10}\]  
To derive the right-hand side from an action principle, we would need to add a suitable interacting term \(S_{3}\) to the action such that  
\[\frac{\delta S_{3}}{\delta A_{a\mu}}=J_{1}^{a\mu}. \tag{11}\]  
However, notice that \(S_{3}\), which needs to be also symmetric under the transformations that we introduce for the bootstrapping procedure, would contribute to the conserved current due to [PERSON]'s theorem. Hence, we would obtain a contribution to the current of the form \(gJ_{2}^{a\mu}\) which we would want to derive from an action principle, from \(S_{4}\). In this way, we would obtain an infinite set of constraints for the successive actions \(S_{n}\) and \(S_{n+1}\). More concretely, we would have  
\[\frac{\delta S_{n+1}}{\delta A_{a\mu}}=J_{n-1}^{a\mu}, \tag{12}\]  
where \(J_{n-1}^{a\mu}\) is obtained from \(S_{n}\) through [PERSON]'s theorem.