Galilean field theories and conformal structure

Journal of High Energy Physics, Apr 2018

Abstract We perform a detailed analysis of Galilean field theories, starting with free theories and then interacting theories. We consider non-relativistic versions of massless scalar and Dirac field theories before we go on to review our previous construction of Galilean Electrodynamics and Galilean Yang-Mills theory. We show that in all these cases, the field theories exhibit non-relativistic conformal structure (in appropriate dimensions). The surprising aspect of the analysis is that the non-relativistic conformal structure exhibited by these theories, unlike relativistic conformal invariance, becomes infinite dimensional even in spacetime dimensions greater than two. We then couple matter with Galilean gauge theories and show that there is a myriad of different sectors that arise in the non-relativistic limit from the parent relativistic theories. In every case, if the parent relativistic theory exhibited conformal invariance, we find an infinitely enhanced Galilean conformal invariance in the non-relativistic case. This leads us to suggest that infinite enhancement of symmetries in the non-relativistic limit is a generic feature of conformal field theories in any dimension.

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Galilean field theories and conformal structure

JHE eld theories and conformal structure Arjun Bagchi 0 1 Joydeep Chakrabortty 0 1 Aditya Mehra 0 1 0 Kalyanpur , Kanpur 208016 , India 1 Indian Institute of Technology Kanpur We perform a detailed analysis of Galilean eld theories, starting with free theories and then interacting theories. We consider non-relativistic versions of massless scalar and Dirac eld theories before we go on to review our previous construction of Galilean Electrodynamics and Galilean Yang-Mills theory. We show that in all these cases, the eld theories exhibit non-relativistic conformal structure (in appropriate dimensions). The surprising aspect of the analysis is that the non-relativistic conformal structure exhibited by these theories, unlike relativistic conformal invariance, becomes in nite dimensional even in spacetime dimensions greater than two. We then couple matter with Galilean gauge theories and show that there is a myriad of di erent sectors that arise in the non-relativistic limit from the parent relativistic theories. In every case, if the parent relativistic theory exhibited conformal invariance, we nd an in nitely enhanced Galilean conformal invariance in the non-relativistic case. This leads us to suggest that in nite enhancement of symmetries in the non-relativistic limit is a generic feature of conformal eld theories in any dimension. Conformal and W Symmetry; Conformal Field Theory; Space-Time Symme- - Galilean Algebraic aspects Representation theory Scalar elds Fermionic elds Vector elds 4 Interacting Galilean eld theories I Yukawa theory Electrodynamics with scalar elds Electrodynamics with fermions 5 Interacting Galilean eld theories II Pure Yang-Mills theory Yang-Mills with fermions Galilean SU(2) Yang-Mills with fermions 6 Discussions A Conditions of scaling of scalar elds B Conditions of scaling of fermionic elds B.1 Free theory B.2 Galilean Yukawa theory B.3 Galilean electrodynamics with fermions 2.1 2.2 3.1 3.2 3.3 4.1 4.2 4.3 5.1 5.2 5.3 1 Introduction 2 Galilean conformal eld theories: general considerations 3 Free Galilean eld theories C Scaling of spinors in SU(2) Yang-Mills coupled with fermions 1 Introduction The textbook formulation of quantum eld theories, especially gauge theories, is intimately linked to Lorentz invariance. Most day-to-day phenomena are, however, governed by nonrelativistic physics. It is thus very useful to understand Galilean invariance and its role in quantum eld theories in a deeper way than has been attempted so far in literature. tries when one considers non-relativistic sectors of relativistic (conformal) eld theories. This points to a lot of undiscovered physics and related mathematical structures in the Galilean world. Conformal invariance and conformal eld theories (CFTs) play a central role in the understanding of relativistic quantum eld theories (QFTs) today. In the parameter space of all Lorentz invariant QFTs, the relativistic CFTs arise as xed points of renormalisation group ows. QFTs thus can be understood as RG ows away from xed points governed by conformal symmetry. The very ambitious programme of classifying (and understanding) all relativistic QFTs could thus be rephrased in terms of a classi cation of all relativistic the non-relativistic version of conformal invariance.1 So what is non-relativistic conformal invariance? There are several versions of non-relativistic conformal invariance and the jury is still out on which one of them is the \true" non-relativistic conformal symmetry algebra. Below we go on to describe non-relativistic eld theories and some of their conformal extensions. 1) spatial directions xi which are spatially isotropic and homogeneous are invariant under translations of space and time and spatial rotations: H : Pi : Jij : t ! t0 = t + a; xi ! x0i = xi + ai; xi ! x0i = J ij xj : Bi : xi ! x0i = xi + vit [Pi; Bj ] = ij M: (1.1) (1.2) (1.3) (1.4) (1.5) These symmetries can be augmented by Galilean boosts: The group fH; Pi; Jij ; Big is called the Galilean group and can be understood as a contraction of the Poincare group. One can add a central extension to this Galilean group Here M is the mass of the system under consideration. The Galilean group with the central term included is called the extended Galilean group or the Bargmann group. 1For recent progress in this direction in two dimensional BMS or GCA invariant eld theories, see [2, 3]. Condensed matter systems found in nature are sometimes characterised at their critical point by non-relativistic conformal eld theories which exhibit a di erent scaling between space and time. D : t ! t0 = zt; xi ! x0i = x i where z is called the dynamical exponent (and is a parameter of the transformation). The group fH; Pi; Jij ; Dg is called the Lifshitz group and the eld theories with the Lifshitz group as their underlying symmetry are called Lifshitz eld theories [ 5 ]. The extended Galilean group together with the Galilean boosts and the dilatation HJEP04(218) operator (1.6) make up what is called the Schrodinger group. For z = 2, there is a further enhancement of this group to include a single non-relativistic special conformal generator C. C : t ! t0 = t 1 + t ; xi ! x0i = x i 1 + t The Schrodinger group derives its name from the fact that it turns out to be the maximal kinematic symmetry of the free Schrodinger equations [6{8]. Together with the mass, the total number of generators in this group are 13 in D = 4. This is to be contrasted with the relativistic conformal group, which contains 15 generators in D = 4. The process of group contraction, which e.g. gets us systematically from the Poincare group to the Galilean group as we take the speed of light to in nity, preserves the number of generators. It is clear that the Schrodinger group would not appear as a contraction of the relativistic conformal group, as the number of generators of the two groups are not the same. The contraction of the relativistic conformal algebra will lead us to what is called the nite Galilean Conformal Algebra (GCA) [9]. This symmetry, governed by the GCA is thus in a sense the \true" non-relativistic limit of the relativistic conformal symmetry. We will be concerned with this version of non-relativistic conformal symmetry in our work. A detailed review of the GCA would appear in section 2. Motivations of this paper. In this paper, we will investigate Galilean eld theories and their emergent conformal structure. We will consider relativistic eld theories that enjoy conformal invariance and follow a systematic non-relativistic limit in order to dene the Galilean eld theories. We will then investigate whether the conformal structure carries over to the non-relativistic case. The symmetry algebra of interest in these nonrelativistic theories would be the GCA. This paper is a continuation and elaboration of the earlier works [10, 11], where we have addressed the Galilean Electrodynamics and Galilean Yang-Mills theories. Speci cally, we would rst attempt to understand the non-relativistic versions of scalar elds and fermionic elds. These matter elds would then be added to the gauge elds as we build towards a complete understanding of interacting eld theories in the Galilean regime. We would be studying of symmetries of the equations of motion (EOM) of these individual systems, as opposed to looking for the symmetries of the respective actions. This is because de ning actions are often problematic in the non-relativistic case. One of the crucial reasons behind this is the fact that the metric degenerates in the non-relativistic limit { 3 { and Riemannian structures associated with the spacetime geometry need to be replaced by Newton-Cartan structures. We will not have much to say about actions in this paper. In order to escape the complications posed for de ning actions, we shall restrict ourselves to investigating the limits on relativistic EOM and see how to reduce to the Galilean EOM. Using methods outlined in earlier work [10, 11], we shall then investigate the symmetries of these equations. An equivalent treatment of our investigations in this paper should be possible in an action formulation. We leave this to future work. The nite GCA, as we will go on to review in the next section, admits in nite dimensional extensions in all spacetime dimension (which we will at times refer to as the in nite GCA). In dimensions D > 2, this is an enhancement of symmetries that is found in the non-relativistic limit and is not the result of a contraction of the relativistic conformal algebra, which is SO(D; 2). All the non-relativistic systems that we consider in this paper are descendants of parent relativistic conformal eld theories. We nd that all of these Galilean eld theories are not only invariant under the nite GCA, which is the contraction of symmetries of the parent system, they are invariant under the full in nite dimensional extension of the GCA.2 The above is a tantalising hint that there is a generic enhancement of symmetries in the non-relativistic limit of any conformal eld theory and that this enhancement is in nite dimensional. In nite symmetries are a pointer to the integrability [13] and ultimately the complete solvability of a system. So it seems that the non-relativistic limit of any conformal eld theory would lead to a Galilean sub-sector that is integrable. This may have profound consequences and can perhaps be useful to understand better the theory of strong interactions [14, 15]. Outline of this paper. With the above motivations in mind, we would go on to address the construction of Galilean eld theories in this paper. The next section contains a comprehensive review of Galilean conformal symmetry and its representation theory. In section 3, we address free elds. We start out with the simplest of examples of the scalar eld. Fermions are then addressed. All of this is new material. We then go on to review our earlier construction of Galilean Electrodynamics [10], which built on seminal work in 1970's by Le Bellac and Levy-Leblond [12]. Work on non-relativistic eld theories, similar in spirit are [16{20]. In section 4, we address how to couple di erent elds together and build interacting Galilean theories. In this section, we consider scalar and then fermionic elds coupled to Galilean Electrodynamics and study the symmetries of the EOM in each case. In section 5, we continue to build on interacting theories now by considering Galilean Yang-Mills theory. After a brief recap of the construction of [11], we focus on the Galilean SU(2) theory with fermions. Section 6 contains a summary of our paper along with further remarks. We devote three appendices to a detailed discussion of generalised limits in the Galilean theory. 2Whether or not there exist non-relativistic conformal eld theories which do not have a relativistic Galilean conformal eld theories: general considerations In this section we will provide a quick summary of the GCA along with a description of its representation theory. The details of the representation theory constructed here will be used in the coming sections to nd the symmetries of the EOM of several non-relativistic process of going to the Galilean framework which entails taking the speed of light to in nity. We will parametrise this by: This process e.g. on the Lorentz boost yields: In order to extract the Galilean boost, we will de ne xi ! xi; t ! t; ! 0: !0 making the generator nite in the limit. The algebra thus obtained in the limit from the relativistic conformal algebra is called the nite GCA. The generators of this algebra can be written as: L(n) = Jij = A more familiar identi cation is L( 1;0;1) = H; D; K and M i( 1;0;1) = Pi; Bi; Ki where H; D and K are respectively the Galilean Hamiltonian, dilatation and temporal special conformal transformation. On the other hand, Pi; Bi and Ki represent momentum, Galilean boost and spatial special conformal transformation. Jij generates homogeneous SO(D 1) rotations. The full algebra can be written as [L(n); L(m)] = (n m)L(n+m); [Mi(n); Mj(m)] = 0; [L(n); Jij ] = 0; [L(n); M (m)] = (n i m)Mi(n+m); [Jij ; Mk(n)] = M[(jn) i]k (2.3) with n; m = 0; 1. The algebra admits the usual Virasoro central charge 1c2 (n3 n) n+m;0 in the [L; L] commutator. One of the most interesting observations [9] about this algebra is that even if we let the index n of (2.2) run over all integers, the algebra (2.3) closes. This is thus an in nite extension of the algebra which was previously obtained by a contraction from the nite { 5 { (2.1) (2.2a) (2.2b) relativistic conformal algebra.3 We will refer to this in nite dimensional algebra as the GCA from now on. The GCA is thus in nite dimensional for all spacetime dimensions, unlike the relativistic conformal algebra which is in nite only in D = 2. One of the novel claims of [9] was that there would be an in nite enhancement of symmetries in the nonrelativistic limit of all relativistic conformal eld theories. In the later sections of this paper, we provide evidence to this conjecture. 2.2 Representation theory We will construct the representations of GCA in a method similar to that of the relativistic conformal algebra. The states in the theory are labelling by their weights under the HJEP04(218) dilatation operator L(0) [22, 23]: L(0) j i = j i: i The operators L(n); M (n) lower the weights for positive n while they raise the weights for negative n: L(0)L(n) j i = ( n)L(n) j i; L(0)M (n) i j i = ( n)Mi(n)j i: If we wish to have a bounded spectrum of , we cannot lower the weights inde nitely. Hence we de ne the notion of GCA primary states as (2.4) (2.5) (2.6) (2.7) (2.8) i L(n) j ip = M (n) i j ip = 0 8n > 0: We now have two di erent options in labelling these states further. Since { 6 { [L(0); Mi(0)] = 0; [L(0); Jij ] = 0; [Mk(0); Jij ] 6= 0: we see, the states can be labelled either by Mi0 or Jij but not with both of them. In this paper, we would label our states with Jij as we would be interested in operators with spin. These GCA primaries would be called scale-spin primaries. The following discussion closely follows earlier work [10, 11]. For representations labeled by the boost generator, the reader is referred to [22] (and to [23] for the D = 2 case). Now returning to the above described scale-spin representations, the action of Jij on the primaries in a particular representation is given as Jij j ip = ij j ip: The representations of the GCA are built by acting with the raising operators (L(n); M (n) for negative n) on these primary states. We postulate the existence of a state-operator 3It is worth pointing out here that there is an even larger extension [9] where the rotation generator is given an in nite lift according to However, we will not be interested in this extension as the systems we go on to examine do not exhibit this symmetry. It is unclear to us presently why this symmetry is absent. correspondence in the GCA in close analogy with relativistic CFTs:4 j ip = We will move interchangeably between states and operators in the discussion that follows. The action of the nite part of GCA on the above described primaries is given as: L Jij ; (0; 0) = ij (0; 0); M ( 1) i ; (t; x) = L(0); (0; 0) = (0; 0); points (t; x) as follows. According to our conventions, where U = etL( 1) xiMi( 1) (t; x) = U (0; 0)U 1 ; . For a general GCA element O, we therefore use: [O; (t; x)] = U U 1OU; (0; 0) U 1 and then exploit the Baker-Campbell-Hausdor formula and the commutation relations of the GCA to evaluate U 1 O U . This is straightforward for J ij and L(0): [Jij ; (t; x)] = x[i@j] (t; x) + ij (t; x); ) (t; x): An important point is the role of boost on the operators. For that, we look at the relevant Jacobi identity: [Jij ; [Bk; (0; 0)]] = [Bk; ij (0; 0)] + k[i Bj]; (0; 0) : In order to solve the above equation for elds having di erent spins we use: [Bk; (0; 0)] = a' + b k + ~b k + sAk + rAt ki + : : : ; where = f'; ; ; At; Aig and ' is a scalar eld, ( ; ) are 2-component fermions and Ai; At are spatial and temporal components of a vector eld. The dots indicate higher spin elds. For example, if we take (0; 0) = At, then to satisfy the Jacobi identity we have to take [Bk; (0; 0)] = sAk. The action of boost on other elds can be shown in a similar manner. We are yet to determine the constants appearing in (2.14). The determination of these constants demands inputs from dynamics. We will clarify how these constants get xed for particular physical systems. 4This is not a strict requirement, but it makes life substantially simpler if we can move between states and operators in our analysis. The state-operator map can be motivated as arising from the limit from the parent relativistic CFT. { 7 { (2.9) (2.10a) (2.10b) (2.10c) (2.11) (2.12) (2.13) (2.14) HJEP04(218) The action of boost on general operator (x; t) at nite spacetime points gives: : (2.15) The action of Ln and Min on a general operator (t; x) of weight at arbitrary spacetime points is thus given by (t; x) t n 1n(n+1)xkU [Mk(0); (0; 0)]U 1 ; [Ml(n); (t; x)] = tn+1@l (t; x) + (n + 1)tnU [Ml(0); (0; 0)]U 1 : (2.16) HJEP04(218) The action of L(n) and M (n) on i particular theory. 3 3.1 Free Galilean eld theories Scalar elds (t; x) will be used to nd the invariance of EOM of a As we just reviewed, the non-relativistic limit of conformal algebra can be systematically constructed by using the scaling on spacetime (2.1) and it has been claimed that there is an in nite enhancement of symmetries in any dimensions in this Galilean limit. It has been shown in recent work that some systems, viz. Galilean Electrodynamics [10] and Galilean Yang-Mills theories [11], exhibit the aforementioned in nite dimensional symmetry. In this paper, we aim to build upon this and show that any relativistic conformal eld theory exhibits this phenomenon. To this end, we rst look at free Galilean scalars and fermions. We will then use this construction to build interacting theories in the Galilean regime, and especially to couple matter to Galilean gauge theories that have been earlier discussed in [10, 11]. We will only consider massless theories as the presence of mass breaks conformal invariance. For the purposes of this paper, we shall con ne our attention to D = 4 unless otherwise stated. Relativistic scalar theory. We start with a brief description on relativistic conformal transformations and the conformal invariance of massless scalar eld theory. In order to set notation, we recall the conformal algebra in D = 4 has the following generators: Conformal generators: D~ = = The conformal algebra in D dimensions is isomorphic to so(D; 2). To highlight the di erences with the GCA (2.3), few important commutation relations are indicated below: [P~i; B~j ] = ij H~ ; [B~i; B~j ] = J~ij ; [K~i; B~j ] = ij K~ ; [K~i; P~j ] = 2J~ij + 2 ij D~ : The right hand side of all these commutators are zero in the nite GCA, while all other commutators stay the same. (3.1) (3.2) (3.3) { 8 { where ~ is the scaling dimension of the eld . In D dimensions, the scaling dimension of a scalar eld and a vector eld are ~ = D2 2 , whereas for a fermionic eld ~ = D2 1 . The transformation under special conformal transformation is given as: 2x ) (x): Consider now a real massless scalar eld '(x) in D-dimensional spacetime. The Lagrangian is given as The corresponding equation of motion is: 1 2 L = We now describe the conformal transformations of elds and review the formalism of checking for conformal symmetry from the point of view of the EOM following [21]. Poincare transformations of a multi-component eld : (x); L ) (x): Transformations under scaling takes the following form: The above Lagrangian and equation of motion are manifestly invariant under Poincare transformations. We wish to examine the action of the dilatation and special conformal transformation on the equation of motion: (3.9) ~ D 2 2 2 We have already stated that ~ = D 2 for scalar elds. This indicates that massless scalar eld theory is scale invariant as well as conformal invariant in all dimensions D. Galilean scalars. Now we work out the details of the non-relativistic limit of the free massless scalar eld theory we discussed above. This will provide the prototype for all the sections to follow, where we will examine the symmetries of EOM arising from di erent eld theories. The non-relativistic EOM for the free scalar theory can be found by using the scaling of the co-ordinates (2.1) on the relativistic equation of motion (3.8). This simply gives rise to: (3.10) Eq. (3.10) is obviously a rather uninteresting example as all the time derivatives have disappeared from the equation. But this would be instructive as an example for what is to follow. For checking the invariance of (3.10), we will consider the transformation of the eld variable (t; x) under an in nitesimal transformation generated by Q. We will check { 9 { (3.4) (3.5) (3.6) (3.7) (3.8) whether the transformed eld variable satis es the EOM. The in nitesimal change in the eld variable is given by: where " is the parameter corresponding to particular transformation under consideration. If Q generates a symmetry of the EOM, we would have: " (t; x) = ["Q; (t; x)] ; (t; x) = (x; t) = 0 is the schematic form for EOM. For the present case, (t; x) is a scalar eld '(t; x). We will check for the symmetries of the EOM. For checking the invariance under K; D and Kl, we use (2.16): As expected, the non-relativistic equation of motion of the massless scalar theory is invariant under the nite GCA. Note here that the di erential operator form of the dilatation operator D (2.2) does not change from the relativistic one D~ (3.2). Hence the dilatation eigenvalue ~ does not change as we go from the relativistic theory to the non-relativistic sector, where we have labeled it as . From now on, we shall not distinguish between the two, i.e. we set ~ = . The non-trivial part of the analysis of symmetries for the equation (3.10) is the proof that it exhibits an in nite dimensional symmetry under the extended GCA. By using the details of the representation theory discussed earlier, speci cally (2.16), we check the transformation of (3.10) under the generators of the in nite algebra L(n) and Mi(n): '] The EOM are hence invariant under in nite GCA in all dimensions. 3.2 Fermionic elds Next we consider massless fermions. The massless relativistic Dirac Lagrangian is L = i (x) (x) = 0; (3.11) (3.12) (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) 2 . where (x) is a 4-component Dirac fermion. The equation of motion is: where are the gamma matrices which follow the Cli ord algebra: f ; g = The theory is, by construction, Poincare invariant (3.4). We will focus on dimensions D = 4 in this analysis as we are going to use particular representations of the gamma matrices. The action of scale (3.5) and special conformal transformations (3.6) on (3.19) are: (3.20) 3 2 where we have used = ( steps. Eq. (3.19) is thus conformally invariant. + ) and = 4 in the intermediate Galilean fermions. We now look into the non-relativistic limit of massless Dirac eld theory and study its symmetry structure. Some previous work in this direction include [24, 25]. We will decompose the fermionic eld (t; x) into two component spinors: (3.21) (3.22) (3.23) (3.24) as our (3.25) (3.26) To get the non-relativistic equation, we will scale the spinors as along with scaling of spacetime (2.1). We would have also taken ! ; ! choice of scaling of the spinors. But (3.23) has a Z2 symmetry which means that the exchange of two spinor will have no e ect on the EOM. The action of Galilean boosts of fermions can be obtained by taking the non-relativistic limit on the Lorentz boosts in (3.4): B 0i = B 0i = : i 2 Comparing with (2.14), the arbitrary constants for this theory that label the scale-spin representations are xed to b = 0; ~b = 12 . This, along with the assumption that the dilatation eigenvalue is inherited unchanged from the relativistic theory as argued before, are the required input from dynamics in the general Galilean conformal representation theory elaborated in section 2. In the non-relativistic limit, massless Dirac equation reduces to = 0: 5We consider fermions in the Weyl representation in appendix B and show that the non-relativistic limit there leads to the same results. and then take the non-relativistic limit. We will perform our analysis in Pauli-Dirac representations:5 where i are the Pauli matrices. The relativistic EOM for the spinors ( ; ) are given by = ! i = 0 = 1 0 ! 0 1 ; 0 ! ! ; The relativistic EOM (3.19) were invariant under the relativistic conformal group in D = 4. A scaling limit of these equations leads to (3.26) and the same limit on the conformal group leads to the nite GCA. It is thus expected that the non-relativistic equations will be invariant under the nite GCA. We will now check this from an intrinsic Galilean perspective with required input from dynamics. At an operational level, this means that we check the symmetries of the EOM using (3.12) along with (2.16). The EOM under scale transformation remains trivially invariant. Checking for the invariance under spatial and temporal special conformal transformations gives (3.27) 2ti(1 ~b i i) (3.28a) Now fb; ~bg = f0; 21 g from above, and = 32 . Substituting this in (3.27){(3.28b), we get: 1 i 2 i = 2i 3 2 = 0: Therefore, the equations (3.26) are invariant under the nite GCA, as was expected. Now we extend the analysis to the in nite modes of the GCA. Under Ml(n), we have The EOM are invariant under all Ml(n). Similarly, checking for the invariance under L(n): = 0: (3.33) 3 2 Hence, the EOM are also invariant under all Ln. 3.3 Vector elds The study of Galilean electrodynamics stemmed from the work of Le Bellac and LevyLeblond in [12]. Our analysis in [10] was motivated by the search of symmetry in this nonrelativistic version of electrodynamics. Maxwell equations in relativistic electrodynamics are classically conformally invariant in D = 4. It is thus natural to expect that the nonrelativistic version of Maxwell's theory would exhibit non-relativistic conformal invariance in D = 4. The emergence of the nite GCA as symmetries is almost guaranteed if one performs the limit properly. The non-triviality is, as always, the existence of in nitely extended symmetries in the non-relativistic limit. In [10], our discussion of Galilean electrodynamics was based on the potential formulation. In addition to the scaling of spacetime, here we needed to prescribe how the scalar (3.29) (3.30) (3.31) (3.32) and vector potential scaled in the Galilean regime. In keeping with the earlier analysis of Le Ballac and Levi-Leblond, we found that we could scale the 4-vector potential A in two di erent ways: The rst limit is the Electric limit, where the electric e ects are much stronger than the magnetic e ects (jEj situation is reversed ( E j j jBj). The second limit is known as Magnetic limit where the jBj). From the point of view of the underlying spacetime, the origin of the two di erent limit can be traced to the degeneration of the spacetime metric. In the Galilean world, contravariant and covariant vectors behave very di erently since there is no spacetime metric to take one description to the other. The above scalings appear because we treat the original relativistic four potential once as a contravariant vector and then as a covariant vector before performing the limit. The EOM of Galilean electrodynamics (in absence of sources) in these two sectors are: Electric limit: Magnetic limit: (3.35a) (3.35b) For the invariance of the EOM under the whole GCA, we took the required inputs from the representation theory (2.16) and used the schematic form (3.12). We found that this theory has the underlying in nite dimensional symmetry of the GCA in D = 4. For further details, we leave it to the readers to look at [10]. For other recent works in this direction, the reader is pointed to [17{19]. The conclusions section contains a discussion of how the analysis of [18] di ers from our earlier work [10]. 4 4.1 Interacting Galilean eld theories I Yukawa theory We have so far dealt exclusively with free theories. Sceptic may argue that the in nite symmetry of the non-relativistic massless free scalars and fermions theories as well as Galilean electrodynamics arose precisely because of the absence of interactions. To alleviate such concerns, we begin our construction of interacting Galilean theories. In this section, we will construct non-relativistic versions of Yukawa theory, scalar electrodynamics and electrodynamics with fermions. We will begin our discussion with the action of massless relativistic Yukawa theory S = Z d4x 1 2 g ' : Here, the underlying symmetry is U(1) global and ' and are the singlet scalar, and non-singlet Dirac elds respectively with coupling constant g. The EOM are = 0; = 0: (4.1) (4.2) The action of dilatation (3.5) on EOM is given as g ) = ( 1 2 g' ) = 2 Thus, in D = 4, the theory is scale invariant only if we take the scaling dimensions for the scalar eld and 2 = 32 for the Dirac eld. This can be seen both from the EOM and the action. The action of special conformal transformation (3.6) on EOM is g ) = (2 1 2( 1 = 0; (4.4a) 1)gx ' = 0: (4.4b) The EOM have Galilean conformal invariance in D = 4. (4.3a) (4.3b) With the input of the scaling dimensions of the elds, we see that the EOM (4.2) are also invariant under special conformal transformation. Galilean Yukawa theory. In constructing the non-relativistic limit of Yukawa theory, we will follow the analysis in subsection 3.2, where we decomposed the Dirac spinor into two component spinors and used Pauli-Dirac representation of gamma matrices. The relativistic EOM (4.2) can be written down in terms of ( ; ): g' = 0: (4.5) To nd the non-relativistic limit of Yukawa theory, we will use the scaling of the coordinates (2.1) and the elds as Using this limit on (4.5), we get g ' = 0: The scale-spin representation of GCA is determined by input from dynamics as was the case for the free theories. Here we are dealing with spinors ( ; ). For the Galilean Yukawa theory, the limit from the relativistic theory xes the values of the constants to: f ( 1 ; 2); (a; b; ~b)g = 1; 3 2 ; 0; 0; : 1 2 We are now in a position to check for the invariance under GCA using (3.12) and (2.16). The EOM are trivially invariant under Ml(n). Checking for the invariance under L(n), g '] = in(n + 1)tn 1 2 + (1 1)(n + 1)tng '; (4.9) 3 2 Now we will add matter to the free Maxwell theory and will try to understand the nonrelativistic limit and its symmetries. The dynamics of scalar electrodynamics is given by the Lagrangian density: L = 1 4 F F (D ') (D '); (4.11) @ A is the electromagnetic eld strength and D ieA is the covariant derivative with e being the electric charge. The EOM are given as ieA (D ') = 0: (4.12) It is obvious that the above equations are invariant under Poincare transformations (3.4). Checking the invariance of EOM under scale transformation (3.5), we nd: ie' (D ') + ie(D ') '] = 2ie( 2 1)[' (D ') (D ') '] 2e2A ' ']; ieA (D ')] = ie( 1 ieA A ']: (4.13a) (4.13b) (4.14) (4.15) (4.16) (4.17) By usual scaling arguments, the theory is seen to be invariant under dilatations in D = 4. Under special conformal transformations (3.6): ie' (D ')+i(D ') '] = (2 1 + 6 2D)F + (2 1 2x e2A ' '] ie(4 2 4)x [' (D ') ieA (D ')] = 1)[A ' + x @ (A ') + x A (D ') iex A A '] + (4 + 4 2 2D)D ': The EOM are invariant under special conformal transformations in D = 4, when we put in the values of 1 ; 2 . it into two real elds Galilean scalar electrodynamics. We will now move to the non-relativistic limit of Scalar Electrodynamics. To take the limit on the complex scalar eld ', we will decompose ' = p ( 1 + i 2): This decomposition and the subsequent scalings would lead to a non-trivial with non-zero interaction pieces. After decomposition, the relativistic equations (4.12) become 2 2 1 that the gauge eld can be scaled in two ways (3.34), even here we will have two limits of the non-relativistic limits for Galilean scalar electrodynamics. 1 2 Electric sector. In this limit, we will scale the scalar and vector potentials identically as done in the electric limit of Galiean Electrodynamics. The scaling of the elds are given as Ai ! Ai; At ! At; 1 ! 1 ; 2 ! To make things more readable, we will denote the left hand side of the rst EOM as A, second as B and fourth as C. Thus, the equations in these notations become Before going into the process of nding the invariance of the EOM, let us look into the remnants of gauge invariance in this limit. The gauge transformations for relativistic scalar electrodynamics are given by '(x) ! eie (x)'(x); A (x) ! A (x) + @ (x); where (x) is an arbitrary function of spacetime. In terms of real scalar elds, the gauge transformation reads: The non-relativistic version of gauge transformation in this limit reads: (x) ! 2 (x): 1(x) ! 1(x) e 2(x); At(x) ! At(x); The EOM (4.19) are invariant under the new gauge transformations. Returning to the check of the invariance of EOM under scale and special conformal transformations, we nd that the values of the constants which x the details of the representation theory are required. These, as before, are taken to be the ones originating from the free parts. For Electric sector, the values of the constants are The in nitesimal transformations are 1(x) = e 2(x); 2(x) = e 1(x): Applying the scaling (4.18) to (4.21) and checking the scaling for , we nd that the scaling needs to be f ; 2); (a; r; s)g = f(1; 1); (0; 1; 0)g: 1 2 : (4.18) (4.19a) (4.19b) (4.19c) The invariance under dilatation is given as: The invariance under special conformal transformations is Similary for K, we have D + + 2e( 2 eAi 22)]: [D; B] = ) eAi 22): ) HJEP04(218) (4.28a) (4.28b) (4.29) (4.30a) (4.30b) (4.31) (4.32) (4.33) (4.34) (4.35a) (4.35b) (4.35c) (4.35d) (4.36) We will denote the left hand side of the rst EOM as C, the second as D and the third as E . Thus, the EOM in this notation become Hence, we see that the EOM are invariant in D = 4. Let us now check the invariance under in nite dimesional GCA. The EOM are trivially invariant under Ml(n). Under L(n), D + 1)n(n + 1)tn 1 1) [L(n); C] = (D 1 e2Ai 22 ) 2 2)n(n + 1)tn 1eAt 2 eAi 22)]; The EOM are invariant under M (n) and L(n). l Magnetic sector. In Magnetic sector, the scaling of the elds are given by The EOM are Ai ! Ai; At ! At; 1 ! 1 ; 2 ! 1 2 : eAt 22] = 0; We will now look into the gauge transformations in this limit. The gauge transformation for magnetic sector can be found by taking the scaling of the elds (4.34) and as (x) ! (x): Using the limit on (4.21), the transformation becomes 1(x) e 2(x); 2(x): The above transformation leave the EOM invariant. We will now nd the invariance of EOM under scale and special conformal transformations. The values of the constants are HJEP04(218) taken as The invariance under scale transformation is given by Checking the invariance under Kl: In a similar fashion, under K: f ; 2); (a; r; s)g = f(1; 1); (0; 0; 1)g: [D; C] = ( 1 [D; D] = ( 1 e2Ai 22 ) + e(2 2 eAt 22]; eAi 22]; [Kl; C] = 0; @i@i[Kl; 2] = 0; [Kl; D] = 0; [Kl; E ] = 0: [K; C] = 2( 1 [K; D] = 2t( 1 e2Ai 22 ) eAt 22]; eAi 22]; The EOM are invariant under nite GCA in D = 4. We will now check the invariance under in nite extention of GCA. It can be seen that the EOM are trivially invariant under Ml(n). Under L(n), we have [L(n); C] = (n + 1)( 1 ) [L(n); D] = (n + 1)tn( 1 ) + e(n + 1)tn(2 2 eAt 22]; eAi 22]; [L(n); E ] = ( 1 (4.37) (4.38) (4.39) (4.40) (4.41a) (4.41b) (4.41c) (4.42) (4.43a) (4.43b) (4.43c) (4.44) (4.45) (4.46) We now consider the non relativistic limit of U(1) gauge elds coupled to fermions. The Lagrangian density for the relativistic case is L = 1 4 F F + i D ; @ A is the electromagnetic eld strength and D ieA is the covariant derivative with e being the electric charge. The corresponding EOM are + e = 0; i D = 0: We can check that the Lagrangian density and EOM are invariant under the nite local gauge transformations (x) ! 0(x) = eie (x) ; (x) ! 0(x) = (x)e ie (x); where (x) is the real and nite local parameter of transformation. Next, we will check for invariance of (4.48) under scale (3.5) and special conformal transformations (3.6). Under scale transformation, (4.47) (4.48) (4.49) (4.50) (4.51) (4.52) (4.53) HJEP04(218) The EOM are invariant, since 1 = 1 and Galilean electrodynamics with fermions. To take the non-relativistic limit, we rst decompose the Dirac spinor into two components spinors as in subsection 3.2. By doing so, the relativistic equations (4.48) become e( y + y ) = 0; + y j ) = 0; (4.54) (i i@i + e iAi) + (i@t + eAt) = 0: (4.55) We know that the gauge eld can be scaled in two ways (3.34). Similarly, we can also scale the spinors ( ; ) in two ways in accordance to the limit of gauge eld under consideration. + e + e A 1)e A 3)e ; The equation is invariant for 1 = 1 and 2 = 32 . Under special conformal transformations, 6)x e 1)x e A 2 = 32 . : + F ; + e A ] = 2( 1 + i(2 2 3) : Again input from dynamics xes the values of the coe cients as f(r1;s1);(r2;s2);(r3;s3);b1;b2;~b1;~b2g = ( 1;0);( 1;0);(0; 1);0;0; 12; 12 : All the EOM are trivially invariant under Ml(n). The invariance under L(n) is given by [Ln;@i@iA(1;2)] = 0; t The EOMs are invariant under Ln. EMM case. The elds in EMM limit scales as The EOM for this limit are given as Ai ! Ai1; At1 ! At1; Ai2;3 ! Ai2;3; At2;3 ! At2;3; 1;2 ! 1;2; 1;2 ! 1;2: (5.33) 1 EEM case. The elds in EEM limit scales as The EOM in this limit are given as @i@iA(1;2) = 0; i i@i 1;2 = 0: t gAi2A1) gAi2@iAt1 = 0; t { 25 { (5.27a) (5.27b) (5.27c) (5.27d) (5.27e) (5.28) (5.29) (5.30) (5.31) (5.32) (5.34a) (5.34b) (5.34c) (5.34d) (5.34e) (5.34f) The values of the coe cients xed by dynamical input are f(r1; s1); (r2; s2); (r3; s3); b1; b2; ~b1; ~b2g = ( 1; 0); (0; 1); (0; 1); 0; 0; ; : All the EOM are trivially invariant under Ml(n). The invariance under L(n) is given by [Ln; (5:34c)] = (n + 1)( 1 [Ln; (5:34d)] = (n + 1)( 1 [Ln; (5:34e)] = in(n + 1)tn 1 [Ln; (5:34f )] = in(n + 1)tn 1 2 2 3 2 3 2 1 + ( 1 2 + ( 1 1)tn(n + 1)(gT112At1 2); 1)tn(n + 1)(gT211At1 1); We see that the EOM are invariant under Ln. MMM case. The gauge elds and spinors scales as A i ! Aa; A a i a t ! Aa; 1;2 ! t 1;2; 1;2 ! 1;2: EOM in this limit are given by As seen above, the EOMs are invariant under Ln. Next, we shall look at the invariance of the equations under L(n)'s. The values of the coe cients, as xed by dynamics, are fra; sa; b1; b2; ~b1; ~b2g = 0; 1; 0; 0; ; All the EOM are trivially invariant under Ml(n). The invariance under L(n) is given by n(n + 1)tn 1 ( 1 1;2: (5.35) (5.36) (5.37) (5.38) (5.39) (5.40) (5.41) (5.42a) (5.42b) (5.43) (5.44) (5.45) Discussions In this paper, we have performed a detailed analysis of Galilean eld theories and examined their symmetries. In particular, we have focussed on theories which are obtained by a systematic non-relativistic limit from a parent relativistic conformal eld theory. We started out by considering free theories like the massless scalars and fermions and revisited our previous constructions of Galilean gauge theories, viz. Galilean electrodynamics and Yang-Mills theory. We then went on to consider matter (both scalars and fermions) added to these Galilean gauge theories. A conjecture. We saw that in all our eld theory examples, if the parent theory exhibited relativistic conformal invariance, the theory obtained in the limit was invariant under the in nite-dimensional Galilean conformal algebra. We thus conjecture the following: Any relativistic conformal eld theory, in any spacetime dimension, contains a nonrelativistic subsector, the symmetries of which are dictated by the in nite dimensional Galilean conformal algebra. So, we are claiming that there is a generic in nite enhancement of symmetries in the non-relativistic limit of any CFT, even for spacetime dimensions D > 2. We have shown that this is the case for a very wide variety of examples. Of course, a proof of this is still lacking and there are several unanswered questions. Let us list some of these and some related puzzles below. The SU(2) puzzle. Our rst puzzle already manifests itself in our present work. As is explained in detail in the appendix C, if we take arbitrary scaling of fermions in the four sectors of the NR SU(2) theory, there are around O(1500) di erent limits that one could construct. As we have already mentioned in the previous section, we consider restrictions on our limits of the following form: The limit should give back the free NR Dirac equations (3.26) when the gauge elds are turned o . The EOM in the limit should reduce to appropriate Galilean SU(2) eqs. (5.6){(5.9) when matter elds are turned o . Amazingly, these consistency requirements immediately bring down the O(1500) possibilities to 46 di erent limits in the SU(2) theory. Again, rather surprisingly, these 46 sectors lead to only 19 di erent sets of EOM, after accounting for various exchange symmetries. If we wish to discard \uninteresting" sectors where there are no fermion-gauge eld interaction terms, we get a further reduction to a set of 15. As we have stated before, all these di erent sets of EOM exhibit invariance under the in nite GCA. This enormous reduction in the number of possible distinct sectors in the NR theory seems to indicate that there is something deeper at play which we are missing. It is also possible that by some more consistency requirements, we would be able to cut down further on the possible non-relativistic sectors of the SU(2) theory. It is, of course, extremely important to understand the general structure of the limit in the SU(2) theory before embarking on an analysis of the generic SU(N ) theory. Further remarks. Recently, [18] re-examined the symmetry structure of non-relativistic electrodynamics and found that the EOM in both the electric and magnetic limit exhibited in nite dimensional symmetries. The symmetries discovered were even larger than we discussed in our earlier work [10], but contained the GCA as an in nite sub-algebra. It appears that the reason why the authors of [18] discovered more symmetries than our earlier work [10] was that they did not consider that the values of obtained from the relativistic Maxwell theory, and chose one which led to the enhanced symmetries. They also found that the in nite symmetries existed for all dimensions, which again boiled down to this freedom in the choice of . The curiosity of this observation is that usual Maxwell theory is only relativistic conformally invariant in D = 4 and not in higher D. So it seems that by tinkering around with the non-relativistic conformal weights, one can perhaps eradicate anomalies in relativistic theories in their non-relativistic limit. Also, the analysis of [18] would point to the fact that non-relativistic conformal systems perhaps exist even without the existence of any parent relativistic conformal theory, thus making the parameter space of non-relativistic conformal systems even richer that the relativistic ones. We would like to revisit our analysis of matter elds coupled to Galilean gauge theories in the framework of [18] to see whether more symmetries emerge by relaxing conditions on the Galilean conformal weights. As we just mentioned, the existence of extended in nite dimensional symmetries in the Galilean limit of Electrodynamics even for D > 4 in [18] raises interesting possibilities about the anomaly structure in Galilean theories. In particular, it seems to suggest that L1 and M1, which are the descendants of the relativistic special conformal generator, curiously become symmetries although this was clearly not the case in the relativistic Maxwell theory. If there are no subtle loopholes in the arguments of [18], there seems to be a very real possibility that the existence of in nite symmetries may wash away anomalies arising from parent relativistic theories. Some work on non-relativistic anomalies can be found in [31{33]. We wish to re-examine Galilean anomalies from our limiting perspective. Explicitly, we wish to address the possible restoration of Galilean conformal symmetries in the non-relativistic version of Quantum Electrodynamics. A very natural avenue to which we would like to extend our work is Supersymmetric gauge theories. In particular, we would like to examine N = 4 Super Yang-MIlls (SYM) theory. This is one of the rare relativistic eld theories that is conformally invariant in the quantum regime. It is expected that non-relativistic sectors of N = 4 SYM would exhibit the in nitely extended (appropriately supersymmetrised version of) GCA as its underlying symmetry algebra.6 This would be the rst step to a whole host of questions, some of which we alluded to in the introduction. The emergence of in nite symmetries are pointers to integrability. If indeed the Galilean version of SYM exhibit in nite symmetries, this could indicate that these are sectors which exhibit quantum integrability and this would thus be a new integrable sector in SYM over and beyond the usual planar sector. Examining the planar sector in the limit should also be an interesting exercise as existing relativistic integrable sectors would perhaps get augmented by newer structures appearing from the symmetry enhancement in the limit. 6See e.g. [28{30] for attempts at supersymmetrising the GCA in D > 2. N = 4 SYM is of course dual to Type IIB string theory on AdS5 S5. So we should be able to explore the dual theory to the non-relativistic sectors of N = 4 SYM. Here we should be able to put on rm footing the idea of a Newton-Cartan like AdS2 R3 emerging from the Galilean limit of AdS5 as was put forward in [9]. There are numerous possible directions of future work on the bulk side that would stem from a better understanding of this picture. currently lack.7 On a similar note, Newton-Cartan structures would also emerge on the eld theory side. It would be useful to formulate the eld theoretic considerations in this and earlier works in a more geometric formalism that would be intimately linked to Newton-Cartan geometry. This should also facilitate an action formulation of these theories which we eld theories. Finally, we would like to comment on a related eld where our present considerations would be very useful. Instead of taking the speed of light to in nity, one can consider rather peculiar theories where the speed of light is taken to zero. Field theories with this feature are called Carrollian eld theories and are closely related to their Galilean cousins. Carrollian eld theories have recently emerged as theories of interest as their conformal extensions, Carrollian CFTs have been found to be putative dual theories to Minkowskian spacetimes. The Carrollian Conformal Algebra (CCA) is isomorphic to the Bondi-Metzner-Sachs (BMS) algebra in one higher dimension. The BMS algebra is the asymptotic symmetry algebra on the null boundary of Minkowski spacetimes [34, 35] and it has been known since 1970's that these algebras in three and four dimensional Minkowski spacetimes are in nite dimensional. Recently, there has been a resurgence of activities in the investigation of infra-red physics related to the BMS group and there have been developments linking the BMS group to long known soft theorems and memory e ects in an infrared triangle of relations. For a review of these topics, the reader is referred to [36] and the references within it. In three bulk spacetime dimensions, the BMS algebra takes the form [37] [Ln; Lm] = (n m)Ln+m + Here Ln are the so-called super-rotations, that form the Di (S1) of the circle at null in nity and the Mn are the super-translations, which are angle dependent translations of the null direction. The central terms cL and cM arise in the algebra and for Einstein gravity [37], they are cL = 0; cM = 1=4G, where G is the Newton's constant. The attentive reader would have noticed already that this above algebra (6.1) is isomorphic to the GCA in 2d which follows from (2.3). This connection was noticed rst in [38] and later exploited in a variety of works (e.g. [39]{[44]) in an attempt to formulate the holographic correspondence for 3d at spacetimes. The reader is referred to [45, 46] for a more comprehensive review 7See however [18, 19] for some action formulations that necessitate the incorporation of auxiliary elds. of the eld. The use of the 2d GCA in this case is justi ed as the contractions c ! 1 and c ! 0 yields the same algebras starting from two copies of the Virasoro algebra. In other words, the GCA and CCA are isomorphic in 2d. Interestingly, this algebra has also appeared as symmetries of the worldsheet of tensionless strings [47, 48] and in the context of the ambi-twistor string [49]. In higher dimensions, the GCA and CCA di er as the number of contracted directions for the GCA and the CCA don't remain the same. The GCA is a non-relativistic limit and hence, as we have discussed through out the present paper, the contraction to use is xi ! xi; t ! t. The CCA, on the other hand, requires us to take the speed of light to zero and hence the contraction required is xi ! xi; t ! t. In [45] , gauge theories in D = 4, viz. Electrodynamics (see also [17]) and Yang-Mills theories, in the Carrollian limit were constructed and it was found that in nite dimensional enhancements similar in spirit (but di erent in the actual algebraic details) to the ones we have discussed in this paper are also present there. An on-going investigation is whether matter added to these Carrollian eld theories would also generate the in nite conformal structures, now in the ultra-relativistic regime. These considerations would be very useful for understanding the at limit of the parent Maldacena correspondence that relates AdS5 S5 to N = 4 SYM. The comments about integrability and anomalies that we mentioned earlier would also be very pertinent in the ultra-relativistic context. Acknowledgments It is a pleasure to thank Rudranil Basu, Poulami Nandi, Kostas Skenderis, Wei Song and Marika Taylor for helpful discussions and Sunando Patra for help with Mathematica for the detailed calculations of the appendices. We would like to acknowledge the hospitality of the following institutes/universities during various stages of this work: Albert Einstein Institute (AM), University of Southampton (AB), Universite Libre de Bruxelles (AB, AM), Vienna University of Technology (AB, AM), Tsinghua University, Beijing (AB). The work is partially supported by DST Inspire faculty fellowships (AB and JC), a Max Planck mobility award (AB), and an SERB Early Career Research Award (JC). A Conditions of scaling of scalar elds Here, to start with, we induce most general scaling of the scalar elds ( and ) along with the scaling of gauge elds. Then we note down the inequalities between consistent with EOM of free theory. With the allowed values of and and which are we compute the non-relativistic equation of motions in the limit ! 0. It is worthy to mention that only a particular choice of and leaves some relics of the interacting theory. The relativistic EOM for scalar electrodynamics are given as 2 eA ( 12 + 22)] = 0; 2 1 e2A A e2A A Below we use the electric and magnetic limits of gauge elds to nd out the interplay between and Electric sector. The gauge and the scalar elds are scaled in the electric limit as ; and the EOM are read as We nd the following relations eAt( 2 +2 12 + 2 +2 22)] = 0; (A.5) HJEP04(218) + 1 > 0; (A.7) that are compatible with EOM of free theory. The allowed region in in gure 1a. The interaction terms disappear for all other choice except plane is shown = 1; The non-relativistic equations for this particular set of values are given in (4.19). Magnetic sector. The scaling of gauge and scalar elds are given in magnetic limit as At ! ; and the EOM are read as eAt( 2 +2 12 + 2 +2 22)] = 0; (A.9) eAj ( 2 +2 12 + 2 +2 22)] = 0: (A.10) We derive the inequalities that satisfy the EOM of the free theory as + The allowed domain is depicted in gure 1b. The interactions are non vanishing only with = 0; 1, and the derived non-relativistic equations are given in (4.35). B Conditions of scaling of fermionic elds We look at the arbitrary scaling of fermionic elds in the free and interacting theories. We then try to nd the conditions on the scaling by plugging them into the relativistic EOM. If the conditions are satis ed, we get back the free or interacting non-relativistic EOM. We also focus on how to incorporate the non-relativistic scaling of the fermions in the Weyl representation. By doing so, we want to show that the non-relativistic equations we got are representation independent. (a) Electric case (b) Magnetic case We start with the massless Dirac equation Using Pauli-Dirac representation (3.21) and (3.22), the EOM become Now we consider the following scaling of the fermion elds i ! which are plugged into (B.2), and we nd We note the constraint for these scalings are given as The non-relativistic equations of massless fermion theory = 1 or = 1: are achieved for The relation between the Weyl representation and Pauli Dirac representation is ; W = = S D = S ! ; u v ! (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) scaling of elds, i.e., , we nd 2 where `W' and `D' stand for Weyl and Pauli-Dirac representations respectively with S = p1 (1 + 5 0). The gamma matrices are in the Pauli-Dirac representation and using the u = p ( 1 2 1 2 ); v = p ( + ): Feeding (B.8) into the Weyl equations (i The EOM in the non-relativistic limit are thus representation independent. In passing we note that we can choose = 1; = 0 and this leads to the same equations as (B.6) with B.2 Galilean Yukawa theory We consider a Yukawa theory with an underlying U(1) global symmetry. Here, ' and are the singlet scalar, and non-singlet Dirac elds respectively with coupling constant g. The relativistic EOM are = 0: (B.10) Consider the scaling of the scalar (') and fermion ( ; ) elds as In this limit, the EOM become ! ! : 2 +2 y ) = 0; leading to the following relation among the scaling of fermions + 1 0; + 1 0; 1 = 0 or 1 = 0: The non-relativistic equations of massless Yukawa theory considered in the main text are regained with = 0; = 1. B.3 Galilean electrodynamics with fermions The relativistic EOMs for spinor Electrodynamics are given as + y j ) = 0; (B.16) (i i@i + e iAi) + (i@t + eAt) = 0; (B.17) Here, we are interested in the scaling of spinor elds that are consistent with the EOM of free theory and also leaves some interaction terms between the fermion and gauge elds. (B.8) (B.9) HJEP04(218) (B.11) (B.12) (B.13) (B.14) (B.15) (a) Electric case (b) Magnetic case The scaling of the gauge elds and the scalar elds are given as and the EOMs are read as Our demands are ful lled with At ! At; Ai ! Ai; e[ 2 +1 y + 2 +1 y ] = 0; 2 + 1 > 0; + 2 > 0; 2 + 1 > 0; and the EOM are At ! At; Ai ! Ai; e[ 2 +2 y + 2 +2 y ] = 0; + y j )] = 0: The following relations are achieved to satisfy our criteria: + along with (B.5). The equations (4.57) are achieved with = 1; = 0. If we take the values as = 1 and plug it into (B.19) and in the fermionic equations, we again get back the same interaction pieces given in (4.57b). We note that these constants possess two set of values which correspond to two sets of equations with same interaction pieces. Magnetic sector. In this limit, the elds go through the following scaling: (B.18) (B.19a) (B.19b) (B.20) (B.21) (B.22) (B.23) (B.24) along with (B.5). The non-relativistic equations (4.64) are consistent with Similarly, for magnetic case, we can also take the values of the constants as = and get back the same equations as (4.64) expect for one which is given as 2 ; = 12 (B.25) Due to this equation, the two di erent sets give rise to two di erent EOM. We have to keep in mind about these limits of U(1) case when we speak about the equations in SU(2) case. It can be seen clearly that if we switch o the self interactions and decouple the fermions, we get back the U(1) equations. HJEP04(218) C Scaling of spinors in SU(2) Yang-Mills coupled with fermions To start with, we adopt the most general scaling of fermionic elds within the SU(2) YangMills theory and nd out the consistent relations among these scalings. Then we make a speci c choice of and and take the limit ! 0 to get back the non-relativistic Here, we consider the di erent sectors of Galilean SU(2) Yang-Mills theory where fermions are in fundamental representation. Then we implement the electric and magnetic limits of gauge elds to nd out the interplay between and to match up our requirement. EEE sector. In EEE sector of Galilean SU(2) spinor Yang-Mills, the scaling of the elds are given as A ia ! Aia; A a a t ! At ; 1 ! 1 1; 2 ! 2 2; 1 ! 1 1; 2 ! 2 2: (C.1) Next, we write the conditions of the scaling of spinors which we can found by using the fact that the EOM at least contains the free parts. The conditions are as follows: i + j 1; i 2; 2; i 1; i 1 with i; j = 1; 2: (C.2) X i X i i In the table, we mention the cases where the structure of EOM remain invariant if we exchange fermion elds suitably. For example, we can see that if we take the values of the coe cients ( 1 = 1; 2 = 1; 1 = 0; 2 = 0), the corresponding EOM is denoted by Ae. Now, if we look at the case where ( 1 = 0; 2 = 0; 1 = 1; 2 = 1), we see that the EOM is also denoted by Ae but with the exchange symmetry given by ( 1 , 1 ; 2 , 2). To make our analysis bit simple, we will de ne certain quantities like Ae; Be; : : : that will be related to di erent sets of EOM. Since, we are in SU(2) case, we will take a = 1; 2; 3 and m = 1; 2. The EOM are given as follows: Ae case: @jAia) + gf abcAtb@jAtc + g( ym jTma n n + ym jTma n n) = 0; { 35 { 1 2 1 2 EOM 0 0 1 1 Ce 9. 0 1 1 0 with (C.2). 1;2 1) @t@jAta + @i(@iAja @jAia) + gfabcAb@jAtc + g( y1 jT1an n + yn jTna1 1) = 0; t HJEP04(218) Ee case: @j Aia) + gf abcAb@j Atc + g( y1 j T1a2 2 + y j T2a1 1) = 0; t 2 HJEP04(218) Gauge transformation. We now look at the gauge transformation in this limit. The relativistic Yang-Mills theory is invariant under gauge transformations of the form m = i a(T a )m; A a = a + f abcAb c; 1 g m = i aTma l l; m = i aTma l l: where f abc are the structure constant. The gauge transformation of ( ; ) elds are given as We will generalize our strategy that we have done in details in [11]. We apply the scaling (C.1) on equation (C.3) and check what scaling of a keeps the equation nite. We nd that the scaling needs to be The gauge transformation in this limit are a 2 a: Ata = 0; Aia = 1 1 = 2(i aT1a1 1) + 1 = 2(i aT1a1 1) + 2 1+2(i aT1a2 2); 2 1+2(i aT1a2 2); 2 = 2 = 1 2+2(i aT2a1 1) + 2(i aT2a2 2); 1 2+2(i aT2a1 1) + 2(i aT2a2 2): Taking one set of values of the coe cients ( 1; 2; 1; 2) from the table along with limit and plugging them into (C.7) gives us the required gauge transformation for the spinors in that sub case of the EEE sector. Together with (C.6), we can then nd the gauge invariance of the EOM associated with the given values of the coe cients. (C.3) (C.4) (C.5) (C.6) (C.7) ! 0 EOM Aeem Ceem Beem Feem 2 1 2 1 1 2 0 1 2 0 1 2 1 Aeem ( 1 , Ceem ( 1 , Beem ( 1 , Feem ( 1 , 1 1 1 ; 2 , ; 2 , ; 2 , 1 ; 2 , 2) 2) 2) 2) and 2 with (C.9). 1;2 The non-relativistic scaling of the elds in EEM sector are given as A1;2 1 ! ! At1;2; 2 ! 2 2 A 3 i ! A ; i 3 1 ! 1 1 t ! 2 ! 3 A ; t 2 2 : (C.8) The conditions of the scaling of spinors are X X i 1 2 ; i 1 2 i + j 1; 1; 1; with i; j = 1; 2: (C.9) We de ne the quantities like Aeem; Beem; : : : that will be related to di erent sets of EOM. The EOM in EEM sector are given as: Aeem case: g( y1T131 1 + y2T232 2) = 0; 3) i@t 1 + gT112At1 2 + gT122At2 2 + i i@i 1 = 0; 4) i@t 2 + gT211At1 1 + gT221At2 1 + i i@i 2 = 0: @j Ai(1;2)) + g( y1 j T1(21;2) 2 + y j T2(11;2) 2 1) = 0; g( y1T131 1 + { 38 { HJEP04(218) Feem (Free) case: 3) i@t 1 + gT112At1 2 + gT122At2 2 + i i@i 1 = 0; 4) i@t 2 + gT211At1 1 + gT221At2 1 + i i@i 2 = 0: HJEP04(218) Gauge transformation. To get the gauge transformation, we have to also take the scaling of the gauge parameter a as 1;2 2 1;2; 3 3 : Therefore, the gauge transformations of gauge elds reads Similarly, for ( ; ) elds, we have A1;2 = 0; A1;2 = t 2 = 2 = 1 = (i 3T131 1) + 2 1+2(i 1T112 2 + i 2T122 2); 1 2+2(i 1T211 1 + i 2T221 1) + (i 3T232 2); 1 = (i 3T131 1) + 2 1+2(i 1T112 2 + i 2T122 2); 1 2+2(i 1T211 1 + i 2T221 1) + (i 3T232 2): For a particular set of values of ( 1; 2; 1; 2) along with invariant under these transformations. ! 0 limit, the EOMs will be EMM sector. The scaling of the elds in EMM sector are given as follows: A 1 i ! 1 ! 1 A ; i 1 1; A 1 t1 t ! A ; 2 ! 2 2; A2;3 ! Ai2;3; 1 ! 1 1; A2;3 t ! 2 ! At2;3; 2 2 with the conditions of scaling of spinors as X X i 1 2 ; i 1 2 i + j 1; 1; 1; i with i; j = 1; 2: (C.14) We de ne some quantities like Aemm; Bemm; : : : and so on. The EOM in EMM sector are given as: (C.10) (C.11) (C.12) (C.13) 1 2 1 2 EOM 1. -12 -12 12 12 Aemm 0 1 1 Cemm 4. 0 1 Demm 1 1 5. 12 12 12 12 Aemm ( 1 , 1; 2 , 2) gAi3(@iAj2 gAi3Aj2) g( y1T122 2 + y2T221 1) = 0; g( y1T131 1 + y2T232 2) = 0; 5) i@t 1 + gT112At1 2 + i i@i 1 = 0; 6) i@t 2 + gT211At1 1 + i i@i 2 = 0: Bemm case: gAi3(@iAj2 @jAi2) + g( y1 jT112 2 + y2 jT211 1) = 0; g( y1T131 1 + y2T232 2) = 0; { 40 { Cemm case: Demm case: gAi3(@iAj2 gAi3Aj2) 3) @i(@iA2 @tAi2 + gAi3At1) + gAi3@iAt1 = 0; t 5) i@t 1 + gT112At1 2 + i i@i 1 = 0; 6) i@t 2 + gT211At1 1 + i i@i 2 = 0: gAi3(@iAj2 gAi3Aj2) 3) @i(@iA2 @tAi2 + gAi3At1) + gAi3@iAt1 = 0; t gAi2A1) gAi2@iAt1 = 0; t 5) i@t 1 + g iT122Ai2 2 + i i@i 1 = 0; 6) i@t 2 + g iT221Ai2 1 + i i@i 2 = 0: (C.15) (C.16) (C.17) We consider the scaling of a to understand the gauge transformation in this limit. It is given by 1 ! 2 1; 2;3 ! 2;3: The gauge transformations of the gauge elds becomes At1 = 0; Ai1 = g1@i 1 + Ai2 3 Ai3 2; Ai2;3 = At3 = g1@t 3 + At1 2: Similarly, for ( ; ) elds, we get 1 = (i 3T131 1) + 2 1+2(i 1T112 2) + 2 1+1(i 2T122 2); 2 = 1 2+2(i 1T211 1) + 1 2+1(i 2T221 1) + (i 3T232 2); 1 = (i 3T131 1) + 2 1+2(i 1T112 2) + 2 1+1(i 2T122 2); 2 = 1 2+2(i 1T211 1) + 1 2+1(i 2T221 1) + (i 3T232 2): Therefore, the EOMs come out to be invariant under these transformations. 10. 11. 1 F2m F3m Am Bm F2m ( 1 , 2; 1 , 2) F1m ( 1 , F2m ( 1 , F2m ( 1 , F3m ( 1 , Am ( 1 , Bm ( 1 , 1; 2 , 1; 2 , 2; 2 , 1; 2 , 1; 2 , 1; 2 , 2) 2) 1) 2) 2) 2) 1;2 MMM sector. The scaling of the elds in MMM sector are given as A a i i ! Aa; A t ! Aa; 1 ! t 1 1; 2 ! 2 2; 1 ! 1 1; 2 ! 2 2: (C.18) The conditions for the scaling of spinors in this limit are i + j 2; i 1; 1; i with i; j = 1; 2: (C.19) X i i 1 Following the above analysis, we de ne certain quantities like Am; Bm; : : : that are related to di erent sets of EOM. The various sets of EOM in MMM sector are g( y1T1a1 1 + y2T2a2 2) = 0; Am case: { 42 { Free MMM sector. F1m case: F2m case: F3m case: (C.20) (C.21) (C.22) Gauge transformation. In MMM sector, the gauge transformation can be found by taking the scaling of potentials and a as The gauge transformations of gauge elds in this sector are given as a Ata = 1 Aia = 1 g( ymTma n n) = 0; g( y1T1a1 1) = 0; 1 = (i aT1a1 1) + 1 = (i aT1a1 1) + 2 1+1(i aT1a2 2); 2 1+1(i aT1a2 2); 2 = 2 = 1 2+1(i aT2a1 1) + (i aT2a2 2); 1 2+1(i aT2a1 1) + (i aT2a2 2): The EOMs will come out to be invariant under (C.21) and (C.22), if we take a particular set of values of the coe cients given in the table. Open Access. 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Arjun Bagchi, Joydeep Chakrabortty, Aditya Mehra. Galilean field theories and conformal structure, Journal of High Energy Physics, 2018, 144, DOI: 10.1007/JHEP04(2018)144