# Toward a Mechanism for the Emergence of Gravity
## 2 A Mechanism for the Emergence of Gauge Symmetries  
The first step in the logic of our construction is to consider a non-relativistic condensed-matter-like theory that in some regime can be accurately described in terms of a set of Lorentz-invariant and weakly coupled fields over a flat background. The emergence of Lorentz invariance in this kind of systems is typically associated with the presence of Fermi-like points in the theory [4; 5; 7]. Here, we take for granted this first step and analyze under which conditions this effective field theoretic description develop emergent gauge symmetries. We will work in configuration space to avoid introducing [PERSON]'s machinery for the classification of constrained systems [9]. Thus, we need a characterization of gauge symmetries within this formulation. Such characterization is offered by [PERSON]'s theorem [16]. Generally, given a symmetry we can find a conserved current associated with it. If the symmetry is a gauge symmetry, the Noether charges associated with the corresponding current are trivial. Hence, for a gauge symmetry the current can always be written as  
\[J^{\mu}=W^{\mu}+S^{\mu}, \tag{2}\]  
with \(W^{\mu}\) being a term that is zero on shell and \(S^{\mu}=\
abla_{\
u}N^{\
u\mu}\) a superpotential, i.e., an identically conserved term. Once the theory is provided with suitable fall-off conditions, the contribution of the superpotential term to the charges identically vanishes, as it can be seen by applying the Gauss theorem [17]. This characterization of gauge symmetries is the starting point for our discussion of emergent gauge symmetries. Let us consider a field theory characterized by a set of fields, which we call \(\Phi\), with a well-posed initial value problem. For the sake of simplicity, we assume that such system is free of gauge symmetries from the beginning. There always exists a set of symmetries whose Noether charges parametrize the space of initial conditions [18], and hence the whole space of solutions which we call \(\mathcal{S}\). These symmetries are in general complicated symmetries that cannot be found explicitly; they can only be found for integrable systems. Knowing that they exist is enough for our purposes. Now let us consider a set of constraints defined by the condition \(\Psi=0\). Here, \(\Psi\) represents a set of equations that combine the fields \(\Phi\) and their derivatives. Let us represent the subspace defined by this constraint as \(\mathcal{U}\subset\mathcal{S}\). We demand such a space to be non-trivial in the sense of having more than one element. This means that there exist solutions for the constraint \(\Psi=0\), apart from the trivial one in which all the fields vanish, \(\Phi=0\). Let us denote \(\mathcal{Q}\) the complete set of charges that parametrizes the space of solutions \(\mathcal{S}\). It is convenient to choose a parametrization of the space of charges, i.e., a set of coordinates in charge space, such that the subspace \(\mathcal{U}\) is characterized by the condition that some of the charges vanish. In other words, we choose a subset of charges \(\mathcal{Q}^{\perp}_{\Psi}\subset\mathcal{Q}\) such that \(\Psi=0\) if and only if \(\mathcal{Q}^{\perp}_{\Psi}=0\). Furthermore, it is useful for our purposes to choose the remaining coordinates, which we denote as \(\mathcal{Q}^{\parallel}_{\Psi}\), in such a way that they leave the subspace \(\mathcal{U}\) untouched. This means that the symmetries associated with the charges \(\mathcal{Q}^{\parallel}_{\Psi}\) can be defined within \(\mathcal{U}\) since they preserve the condition \(\Psi=0\). Now we have two possibilities depending on the values of the charges \(\mathcal{Q}^{\parallel}_{\Psi}\) within the subspace \(\mathcal{U}\). **Non-emergence of gauge symmetry:** This corresponds to the standard situation in which the charges \(\mathcal{Q}^{\parallel}_{\Psi}\) parametrize the space \(\mathcal{U}\) without redundancies.