New formulation of Galilean relativistic Maxwell theory

In this paper, we discuss Galilean relativistic Maxwell theory in detail. We first provide a set of mapping relations, derived systematically, that connect the covariant and contravariant vectors in the Lorentz relativistic and Galilean relativistic formulations. Exploiting this map, we construct the two limits of Galilean relativistic Maxwell theory from usual Maxwell's theory in the potential formalism for both contravariant and covariant vectors which are now distinct entities. Field equations are derived and their internal consistency is shown. The entire analysis is then performed in terms of electric and magnetic fields for both covariant and contravariant components. Duality transformations and their connection with boost symmetry are discussed which reveal a rich structure. The notion of twisted duality is introduced. Next we consider gauge symmetry, construct Noether currents and show their on-shell conservation. We also discuss shift symmetry under which the Lagrangian is invariant, where the corresponding currents are now on-shell conserved. At the end we analyse the theory by including sources for both contravariant and covariant sectors. We show that sources are now off-shell conserved


I. INTRODUCTION
The formulation of non-relativistic limit of classical field theories received considerable attention recently. It has found applications in holography [1], in studying non-relativistic diffeomorphisms (NRDI) [2][3][4][5][6], condensed matter systems [7][8][9], fluid dynamics [10,11], gravitation [12,13]. This formulation is tricky and markedly different from the relativistic case. Covariance in non-relativistic physics is subtle due to the absolute nature of time. The lack of a single non-degenerate metric in the non-relativistic limit poses some additional difficulties. Here we are interested in the non-relativistic limit of Maxwellean electrodynamics which is invariant under galilean transformations. The basic construction of Galilean electrodynamics was first given by Le Bellac and Levy-Leblond [14] back in 1970's. This was done in the field formulation. A similar field based analysis was done in [15] using embedding techniques. Further directions in this type of analysis were provided in [16]. Other references on different aspects of Galilean electrodynamics and gauge theories are [17][18][19][20][21][22][23].
In this paper we provide a detailed analysis of galilean relativistic Maxwell theory with and without sources. While earlier findings are reproduced we also find several new results with new interpretations. We know there are two distinct non-relativistic limits possible for electrodynamics known as electric and magnetic limit [14,16]. We derive these two limits from the Lorentz transformation of an arbitrary four vector. Our derivation of the non-relativistic scaling relations are consistent with [14,16]. All previous works so far treated only the contravariant components of any vector quantity but the novelty of our treatment is that we have considered both contravariant and covariant components separately (which are distinct quantities in the non-relativistic case) and hence help us to explore the rich symmetries involved in the theory. We then derive the Lagrangians for both electric and magnetic limits from which the equations of motion are obtained. We also show that Maxwell's equations under non-relativistic limit (electric and magnetic) yield same equations as those we get from the non-relativistic Lagrangians. This implies the internal consistency of the limiting process. We observe that at the equations of motion level if we replace the covariant vector components by the corresponding contravariant ones then the electric limit and magnetic limits get interchanged. We then define the galilean electric and magnetic fields for contravariant and covariant cases. Along the way, we discuss the dualities and point out some of the subtleties involved in the process. In this regard we observe one unique aspect of the duality symmetries called twisted duality which is valid only in the galilean limit. Especially we show that the transformations of the electric and magnetic fields under galilean boosts is connected with the familiar duality transformations. Next we move to discuss gauge symmetry. We have shown that we can choose different gauge parameters for contravariant and covariant four potentials as they represent different entities in the galilean limits. We then compute the galilean version of the Noether currents and explicitly show their on-shell conservation. We discuss shift symmetries which play an important role in the study of low-energy effective lagrangians in the context of Goldstone's theorem. Recently people have explored shift symmetries from different aspects [24,25]. We compute a DAE Raja Ramanna fellow † rabin@bose.res.in; soumya557@bose.res.in corresponding currents and their conservations in this limit. In the end we introduce sources and write down the Lagrangians for appropriate galilean limits (electric and magnetic) and the equations of motion just like the sourceless case. The paper is organised as follows: in section II we derive mapping relations between relativistic and non-relativistic vectors for electric and magnetic limit for both contravariant and covariant vectors. In section III we derive the non-relativistic lagrangian for both limits and write down the equations of motion. We discuss Maxwell's equations in terms of fields and explore the duality relations in IV. Gauge symmetry, Noether currents and their conservations are discussed in V. In section VI we discuss shift symmetry and its galilean counterpart, corresponding currents and their conservations. In section VII, discussion has been done by including sources for both contravariant and covariant sectors. Finally, conclusions have been given in VIII.

II. MAPPING RELATIONS
Here we derive a certain scaling between special relativistic and Galilean relativistic quantities. As we know there exists two types of such limits for the vector quantities namely electric and magnetic limits. So first let us consider the contravariant vectors. Let us consider a generic Lorentz transformation with the boost velocity as u i : . Under such Lorentz transformations a contravariant vector changes as We can write them component-wise as (also considering u << c, so γ → 1) We next provide a map that relates the Lorentz vectors with their Galilean counterparts. 1 This particular map corresponds to the case V 0 V i = c v 0 v i in the c → ∞ limit. This yields largely timelike vectors and is called 'electric limit'. Now using eqn 5 in eqns 3 and 4 we get The above two equations define the Galilean transformations. We can write them in a single matrix equation as We now consider the magnetic limit which corresponds to largely spacelike vectors Now using 9 in 3 and 4 we get which is again a galilean transformation. We can write eqn 10 and 11 as a matrix equation We will now consider the covariant vectors. We will write first the reverse transformations of eqn 1 and 2 which is And we know covariant vectors transform as Componentwise we can again write them as Now here we take the electric limit in the following way, which will soon become clear Using 17 in 15 and 16 we get which are again Galilean transformations. We can write 18 and 19 as a matrix equation as We will now consider the magnetic limit as Using 21 in 15 and 16 we get We can write 22 and 23 as We can show that the transformation matrix in 8 and the transpose of the matrix 20 satisfies Similarly the transformation matrix in equation 12 and the transpose of the transformation matrix in 24 satisfies To justify the limiting prescriptions even further, we consider the norm preservation for both electric and magnetic limits. Let us first consider the norm in the electric limit which clearly indicates that under the scaling, the norm is preserved. Likewise, the norm in the magnetic limit is also conserved. The mapping relations, systematically derived here for both covariant and contravariant components, are essential to the subsequent analysis. Any four vector in relativistic theory will be replaced by the corresponding structure for the Galilean theory by adopting this map. These relations are summarised in table I.

III. LAGRANGIAN AND FIELD EQUATIONS
Now let us start from the relativistic Maxwell theory described by the Lagrangian where F µν = ∂ µ A ν − ∂ ν A µ and η µν is the flat space metric with signature −, +, +, + .

A. Electric limit
Now using the relations given in table I we can write the two terms in 28 as, Here A µ is the relativistic four potential while a 0 and a i are it's galilean counterpart. So in the electric limit the full lagrangian takes the following form Now we derive the equations of motion. Varying the Lagrangian 31 with respect to a 0 , a j , a 0 , a j we get the corresponding equations of motion We now derive these equations directly from the equations of motion. The relativistic equations, in component form, are given by, The Galilean version of these equations is found by using table I, followed by taking c → ∞. It reproduces 32 and 33 respectively. To get the remaining pair of equations we have to interpret the relativistic eqns given in 36 as Once again the Galilean version is obtained from table I, followed by taking c → ∞. Equations 34, 35 are reproduced. This shows the consistency of the eqn of motion in Galilean electrodynamics.

B. Magnetic limit
Here again using the relations given in table I we can write the two terms in 28 as, So the Lagrangian will take the following form Varying 40 wrt a 0 , a j , a 0 , a j we get, Here also we can show that the above equations agree with those derived directly from relativistic Maxwell equations 36 and 37 corresponding to contravariant and covariant sectors respectively, by taking the magnetic limit. The field equations for both the limits of Galilean electrodynamics are shown in table II.

IV. GALILEAN ELECTRIC AND MAGNETIC FIELDS AND DUAL TRANSFORMATIONS
Here we introduce the galilean limit of electric and magnetic fields and write down the Maxwell equations. For this purpose we will discuss Contravariant and Covariant sector separately.

A. Contravariant sector
Relativistic electric and magnetic fields are defined as First, we consider the electric limit.
Electric limit: Using the mapping relations given in table I we can write the electric field as And we can define the Galilean electric and magnetic fields as Now we write the field equations that we derived in the previous section in terms of the Galilean electric and magnetic fields. From 32 we get Similarly eqn 33 implies We can see clearly that We will now compute ⃗ ∇ × ⃗ e, So in electric limit we get the following set of equations It is now possible to obtain the equations (53 -56) directly from Maxwell's equations, by using the identification in eqn 48 and taking c → ∞.
For the electric limit, it is seen from 56, a change in the electric field influences the magnetic field. But a change in the magnetic field does not influence the electric field since the R.H.S of 55 vanishes. This implies that the electric field is considerably greater and dominates over the magnetic field, justifying the nomenclature electric limit. This is different from the relativistic case where electric and magnetic fields are treated symmetrically. This asymmetry in Galilean electromagnetism leads to physical effects, some of which have been discussed in [29].
We will now consider the magnetic limit.

Magnetic limit:
Electric field can be written in this limit from the mapping relations in I as And we can define the galilean electric and magnetic fields as Using these relations, eqns 41, 42 and two Bianchi identities are expressed in terms of electric/magnetic fields as The above equations (60 -63) also follow from Maxwell's equations 57, using the map 59 and taking the limit c → ∞. In contrast to the electric limit, here a change in the magnetic field influences the electric field but the converse does not hold. In this case the magnetic field dominates over the electric field. We can clearly see that equations in electric limit are mapped to those of magnetic limit and vice versa under the following duality transformations, This is the analogue of the electromagnetic duality in usual Maxwell's source free theory.

B. Covariant sector
Relativistic electric and the magnetic fields are defined as Now we consider the electric limit. Electric limit: In this limit the electric field looks like And we can define Galilean electric and magnetic fields as From eqn 34 we get Eqn 35 yields And finally calculation of ⃗ ∇ × ⃗ e yields, So the Maxwell equations in the electric limit are The above equations (73 -76) also follow from Maxwell's equations 57, using the map 69 and taking the limit c → ∞. Magnetic limit: In this limit electric field is scaled as And we can define the Galilean electric and magnetic fields as The equations 43, 44 and two Bianchi identities are now written as The above equations (79 -82) also follow from Maxwell's equations 57, using the map 78 and taking the limit c → ∞.
Here also we see that electric and magnetic fields satisfy certain duality relations as follows The galilean limit scalings for the fields (electric/magnetic) are shown in Table III.
Effect of the dualities at the level of Lagrangian In the electric limit the Lagrangian is represented by eqn 31. Now in this limit, the contravariant and covariant electric fields as well as magnetic fields are represented by 48 and 69 respectively. Using these definitions we can write the electric limit Lagrangian in the following form Similarly in the magnetic limit the Lagrangian is represented by eqn 40. Here electric and magnetic fields for contravariant and covaraint cases are given in equations 59 and 78 respectively. So now the Lagrangian in this limit takes the following form in terms of the fields We observe that both Lagrangians (eqn 85 and 86) are identical In other words, expressed in terms of the gauge invariant fields (electric and magnetic), the lagrangians in the two limits are same. This is to be contrasted with the potential formulation where L e and L m appear to be different.
This has been shown clearly in table IV. We like to mention that the twisted relations have not been discussed earlier.

D. Dual transformation of electric and magnetic fields under Galilean boost
Contravariant case: The Field transforms as Boost transformations are written as From 88 using 89 and 90 we get following relations Electric limit From eqn 91 using the electric limit scaling given in table I and keeping in mind that in this limit γ → 1 as c → ∞, we get Similarly, eqn 92 yields The transformations 93 & 94 manifest the same asymmetry that was observed in the Maxwell's equations 55, 56. A change in the electric field induces a change in the magnetic field but the converse is not true. For the magnetic limit, discussed right below, it is the other way round.
There is a simple group theoretical argument for the absence of any b-term in 93. For argument's sake if we retain 94 but include a term in 93 like then the group composition law fails since, and the last term spoils the transformation 95.

Magnetic limit
From eqn 91 using the magnetic limit scaling from table I we get Similarly from eqn 92 we get Adopting the same method the transformations in the covariant sector are obtained. All these results are summarised in table V. We can clearly see from this table that under duality transformation (e i → b i , b i → −e i and e i → b i , b i → −e i ) electric limit reproduces magnetic limit and vice-versa for both covariant and contravariant cases.

V. GAUGE SYMMETRY
We know in the relativistic case the Maxwell lagrangian, is invariant under the following gauge transformation, We consider the Galilean version of this gauge invarianvce.

A. Galilean version
Here we can consider a relatively more general gauge condition, In the relativistic theory the covaraint and contravariant vectors are related by a metric implying α = β. This is mot true in galilean limit. Hence we take α ̸ = β when deriving the galilean version of the gauge transformations. First we consider the electric limit.

Electric limit
From eqn 101 and using the mapping relations given in table I we deduce the following relations, Taking the variation of 31 in the electric limit, on exploiting eqns 102, 103, 104, 105. This shows the invariance of L e .

Magnetic limit
From eqn 101 and using the mapping relations given in table I and repeating the steps done for the electric limit, we obtain similar results here also. These are given in table VI. Taking the variation of the lagrangian 40 and using the results in table VI we get This shows the invariance of L m .

B. Noether current conservation
We know in relativistic classical field theory the Noether current is defined as which is conserved on-shell i.e ∂ µ J µ = 0. Specifically for the Maxwell Lagrangian, the current in eqn 108 has the form, and is on-shell conserved i.e The first term is zero because ∂ µ F µν = 0 and second term is zero because of anti-symmetry of F µν . We now consider here a suitable Galilean version of this conservation. For this we will directly start from the relativistic definition and substitute the Galilean results in proper limit (electric or magnetic). 2 Analogous conservation laws, either in electric or magnetic limit are obtained.

VI. SHIFT SYMMETRY
We know that Goldstone's theorem is a crucial input of the study of low-energy effective lagrangians implying that whenever a global symmetry is spontaneously broken, a gapless mode will appear. In relativistic theories this leads to a massless Goldstone particle described by a shift symmetry of the field where c is constant and is characterised by the scalar field action The above action is invariant under 111. Since 111 is a global transformation, the conserved currents can be found by exploiting Noether's first theorem And corresponding conservations are demonstrated as Consider a constant shift in the four potential, that leaves Maxwell lagrangian invariant. We take C and D to be different for reasons stated in section V A.

Electric limit
We can define following things Similarly, from expressions for δA 0 , δA i we find, From 31 Noether currents are found to be And current conservation can be explicitly demonstrated as The covariant components of the currents are The current conservation gives us

Magnetic limit
We can define following things Similarly, from expressions for δA 0 , δA i we find, From 40 the Noether currents are found to be So the current conseravtaions are demonstrated as Similarly the covariant current components are The current conservations give

VII. INCLUSION OF SOURCES
The relativistic Maxwell Lagrangian with source is as follows We can write the Lagrangian in the following form for convenience We know that the Maxwell theory respects the following gauge transformations The gauge invariance of the action demands the following condition Electric limit: In the electric limit the scaling of the components of the source J µ will be as follows In this limit the Lagrangian looks like Varying the lagrangian with respect to a 0 , a j , a 0 , a j will give following set of equations We now derive these equations directly from the equations of motion. The relativistic equations are given by, Using maps given in table I and 134 it is simple to verify that they reproduce eqns 136 and 137. Using the covariant counterpart of eqns 140 we can get eqns 138 and 139. This shows the consistency of the eqn of motion in Galilean electrodynamics with source.
Taking the variation of the source part of the lagrangian we get Since α, β ̸ = 0 we have two conditions The sources as given in eqns (136)-(139) satisfy the above conditions. We observe that sources are conserved off-shell.

Magnetic limit:
Here the scaling relations are as follows The Lagrangian in this limit is as follows Varying the lagrangian wrt a 0 , a j , a 0 , a j we get the following set of equations These equations may also be derived directly from the equations of motion following the same method adopted for the electric limit. The field equations for both electric and magnetic limit have been shown in table VII. Taking the variation of the source part of the lagrangian we get Since α, β ̸ = 0 we have two conditions Here also sources given by eqns (145) to (148) satisfy the above off-shell conservation equations.

VIII. CONCLUSIONS
Let us summarise, point by point, the new significant findings of the paper, comparing with existing results found in the literature.
• An unambiguous construction of the non-relativistic (NR) lagrangian, for both electric and magnetic limits, was given. We have shown that it correctly reproduces the equations of motion either in the potential or field (electric/magnetic) formulation. This lagrangian was deduced from the standard relativistic lagrangian adopting the dictionary given here. It is expressed either in terms of potentials or fields. In the later case both electric and magnetic limit lagrangians become identical having the same functional form as the usual Maxwell lagrangian.
In ref [21], a NR lagrangian has been given, also derived from the relativistic Maxwell expression, which has, however, several shortcomings. 3 • It is observed from table II that if we replace the covariant components by contravariant ones in the electric limit case we will end up with the magnetic limit case and vice-versa. This fact manifests itself only if we consider the covariant and contravariant sectors separately as we have done here.The interplay between the covariant and the contravaraint indices that leads to an interchange of the electric and the magnetic limits of the theory is a new feature observed here. The reason that it was not noticed earlier stems from the fact that various applications [14][15][16][17][18][19][20][21][22][23] only considered the contravariant components. There is a paper [28] that only gives the Galilean transformation for both covariant and contravariant components and that too confined to the coordinates and derivatives, and not for an arbitrary field. Our analysis is much more general where we provide maps, for both covariant and contravariant sectors, relating arbitrary four vectors in the Lorentz relativistic and Galilean relativistic formulations. These maps are the genesis of our analysis where we use them to obtain galilean relativistic expressions from their corresponding Lorentz relativistic counterparts. These issues are not even remotely mentioned, much less discussed, in [28].
• A central point is the formulation of a dictionary that translates four vectors in the relativistic theory to their corresponding vectors in the non-relativistic theory. Thus the formalism developed in terms of potentials was extended to field (electric & magnetic) formulation. In this set up the duality symmetry was discussed. One can clearly see that in the non-relativistic limit the duality relations are quite non-trivial. In this limit we show that apart from the usual duality relations a twisted duality relation also exists. The feature of twisted duality manifests precisely because the covariant and contravariant vectors are treated separately. This also shows that, on the lagrangian level, duality plays quite a subtle role. Duality symmetries have useful physical applications. For standard Maxwell's theory, using duality symmetry we can find new solutions from given original solutions. Here duality symmetry switches from the electric limit to the magnetic limit. Thus the solutions of the Rowland-Vasilescu Karpen's effect which is an example of the Galilean electric limit, can be exploited, using the duality relations, to find solutions of Wilson's effect which corresponds to the Galilean magnetic limit. 4 .
• Gauge symmetries play a pivotal role in the understanding of gauge theories. Since covariant (a µ ) and the contravarinat (a µ ) vectors are not connected by any non-degenerate metric, they have separate gauge transformations. While this was noticed earlier [21], its full implications were not analysed, and not just because of their problematic lagrangian 161. We show how gauge symmetries in the relativistic case naturally yield their non-relativistic counterpart, but with distinct gauge parameters. Both electric and magnetic limits were analysed. The conservation laws were derived using Noether's prescription. 3 Details are provided in the Appendix B 4 These effects and their implications have been discussed in [29], but there is no mention of duality symmetry.
• Shift symmetries, which have an important role to describe Goldstone particles in relativistic theories, were introduced in the non-relativistic context. Conservation laws, associated with such symmetries, were derived in both electric and magnetic limits.
• We have provided a completely holistic approach in terms of both potentials and fields, clearly showing the connection among them, starting from the rudimentary structures of usual Maxwell's theory. Such an analysis is lacking in the literature.

Future prospects:
This is quite a new research area and has gained attention of late as a part of the resurgence of non-Lorentzian structures in quantum field theories, holography and string theory and hence many aspects and directions are yet to be looked at. There is no consistent Hamiltonian formalism for galilean electrodynamics for example. Also it will be interesting to study the non-relativistic limits of other gauge theories for example Proca theory which describes a massive spin 1 field or Maxwell Chern-Simons theory which is a 2 + 1 dimensional gauge theory, in the same way described here. The analysis described here for vector field could be extended to include tensor fields like the Kalb-Ramond fields. Since the connection of these fields with non-relativistic fluid dynamics is known [26,27], though relatively less studied, the present formulation could find application to illuminate this connection. All these possibilities should be tractable since we have provided independent maps for both covariant and contravariant sectors. Finally, we hope to elucidate the nature of Carrollian electrodynamics [16] using the methods developed here. We expect we can address these issues in the near future.

Electric limit
The contravariant components of the current for the relativistic case are Now using the maps for electric limit given in table I we get Similarly, So we can show the conservation of the galilean currents as follows In the second line of eqn 155, the first and second term is zero from equations of motion eqn 32 and 33 respectively, third and fourth terms get cancelled and fifth term vanishes because of antisymmetry. Following identical arguments current conservation for covariant components can be shown as Magnetic limit Using the maps for magnetic limit given in table I and exploiting eqns 151 and 152 we get Using these expressions, where the second equality vanishes from eqn 42. Likewise for the covariant case, following identical arguments current conservation can be shown as

B. PROBLEMS AND INCONSISTENCIES OF THE LAGRANGIAN FORMULATION GIVEN IN [21]
Any consistent lagrangian formulation of galilean electrodynamics must yield all the equations of motion, for either contravariant or covariant vectors in both electric and magnetic limits. Simultaneously, these equations must reduce to those given in [14] using the field (electric/magnetic) formulation. This is not merely desirable, but essential, since those equations were obtained directly [14] using galilean relativistic arguments, bypassing the use of limiting prescriptions. Since the basic variables in the lagrangian are the potentials, equations of motion are obtained in the potential formulation. One has to now express the electric and magnetic fields in terms of potentials and recast the equations of motion in terms of these fields. Only then a comparison with [14] is feasible. As we explicitly show, the lagrangian given in [21] fails on all counts. The covariant Galilean relativistic largrangian given in [21] is which gives rise to following set of equations ∂ µ ∂ µ a ν − ∂ ν a µ = 0, (Electric limit) (162) ∂ µ ∂ µ a ν − ∂ ν a µ = 0, (Magnetic limit) Let us first consider eqn 162. If we re-write this equation component wise it gives the following set of equations 5 .
which are nothing but equation 32 and 33 respectively. But we cannot get eqns 34 and 35 from the lagrangian 161. In fact we cannot get any equation involving a 0 and / or a i simply because there are no covariant components.
Similarly we can open the magnetic limit equation 163 component wise as follows These two equations do not correspond to any of our equations. On top of that there are no equations for a 0 and / or a i simply because contravariant indices do not arise. Thus the lagrangian 161 fails to yield, in the electric limit, any equation involving covariant indices for potentials. Likewise, in the magnetic limit, there are no equations in the contravariant sector. It is also not possible to express equations (162, 163) in the field formulation since no map ralating potentials with fields has been given. Hence the mandatory comparison with [14] cannot be done. All these issues have been discussed successfully in our approach.
If we push the analysis of [21] further, serious inconsistencies arise. The master equations 165, from which the lagrangian 161 was written, was claimed to be derived by opening the relativistic Maxwell equation 36 in space-time components and exploiting the map given in [21], Doing this, however, instead of 165 we find Surprisingly, there is a mismatch with the first equality in 165. Thus the very construction of the lagrangian and the associated equations of motion in [21] are all riddled with inconsistencies.