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A new formulation of Galilean electrodynamics

In this paper, we discuss Galilean electrodynamics in detail. We construct the two limits of Galilean electrodynamics from 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. Next we consider the gauge symmetry, construct the Noether currents and show their on-shell conservation. We also discuss the shift symmetry under which the Lagrangian is invariant, where the corresponding currents are off-shell conserved. At the end we analyse the Gailean electrodynamics by including sources for both contravariant and covariant sectors.


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] and cosmology [14]. This formulation is tricky and markedly different from the relativistic case. Covariance in nonrelativistic 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 the Maxwellean electrodynamics which is invariant under galilean transformations. The basic construction of the Galilean electrodynamics was first given by Le Bellac and Levy-Leblond [15] back in 1970's. The more recent works on different aspects of Galilean electrodynamics and gauge theories are [16,17,18,19,20,21,22,23,24,25,26,27]. In this paper we provide a detailed analysis of galilean electrodynamics 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 [15]. 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 [15]. 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 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. 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 the 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. Recenently people have expolored shift symmetries from different aspects [28,29]. We compute correponding 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 write down the equations of motion just like the sourceless case. The paper is organised as follows in section 2 we derive a mapping relation between relativistic and non-relativistic vectors for electric and magnetic limit for both contravariant and covariant vectors. In section 3 we derive the non-relativistic lagrangian for both limits and write down the equations of motion. In section 4 we discuss Maxwell's equations in terms of fields and explore the duality relations. In section 5 we discuss the gauge symmetry, Noether cureents and their conservations. In section 6 we discuss the shift symmetry and it's galilean counterpart, corresponding currents and their conservations. Finally in section 7 we discuss galilean electrodynamics by including sources for both contravariant and covariant sectors.

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 the generic Lorentz transformation with the boost velocity as u i : Under such Lorentz transformations a contravariant vector changes as We next provide a map that relates the Lorentz vectors with their Galilean counterparts.
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 2.5 in eqns 2.3 and 2.4 we 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 2.9 in 2.3 and 2.4 we get which is again a galilean transformation. We can write eqn 2.10 and 2.11 as a matrix We will now consider the covariant vectors. We will write first the reverse transformations of eqn 2.1 and 2.2 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 We will now consider the magnetic limit as We can write 2.22 and 2.23 as We can show that the transformation matrix in 2.8 and the transpose of the matrix 2.20 satisfies 1 0 Similarly the transformation matrix in equation 2.12 and the transpose of the transformation matrix in 2.24 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 electric limit the norm is preserved. Now we consider the norm in the magnetic limit The norm is again conserved and the role of the minus sign before the scaling is also very important in this context. The mapping relations are summarised in the table 1.

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 −, +, +, + .
We can write the Lagrangian as follows:

Electric limit
Now using the relations given in table 1 we can write the two terms in 3.2 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 3.5 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 are given by, which can be written as From table 1 we get in the electric limit which reproduces eqn 3.6. From 3.12 we get which reproduces eqn 3.7. To get the remaining pair of equations we have to interpret eqns 3.11 and 3.12 as The first of these in the Galilean limit yields which reproduces eqn 3.8. Likewise eqn 3.16 yields eqn 3.9 in this limit. This shows the consistency of the eqn of motion in Galilean electrodynamics.

Magnetic limit
Here again using the relations given in table 1 we can write the two terms in 3.2 as, So the Lagrangian will take the following form Varying 3.20 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 by taking the magnetic limit. From 3.11 we have which reproduces eqn 3.21. From 3.12 we get To get the remaining pair of equations we have to start from the covariant versions (3.15, 3.16). From 3.15 we get which yields 3.23. Likewise eqn 3.16 yields 3.24 in this limit. The field equations for both the limits of Galilean electrodynamics are shown in table 2.

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.

Contravariant sector
Relativistic electric and magnetic fields are defined as First, we consider the electric limit. Electric limit: Using this relation we can write the electric field as And we can define the Galilean electric field as and magnetic field as Now we write the field equations that we derived in the previous section in terms of the Galilean electric and magnetic fields. From 3.6 we get Similarly eqn 3.7 implies We can see clearly that We will now compute ∇ × e, So in electric limit we get the following set of equations We will now consider the magnetic limit.
Magnetic limit: Electric field can be written in this limit as And we can define the galilean electric field as And magnetic field as Finally, we compute ∇ × e, So in magnetic limit we get the following set of equations 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.

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 field as And magnetic field as b i = lim From eqn 3.8 we get Eqn 3.9 yields And finally calculation of ∇ × e yields, So the Maxwell equations in the electric limit are Magnetic limit: In this limit electric field is scaled as And we can define the Galilean electric and magnetic fields as From eqn 3.23 we get, From eqn 3.24 we get So the equations we get in the magnetic limit are Here also we see that electric and magnetic fields satisfy certain duality relations as follows

Effect of the dualities at the level of Lagrangian
In the electric limit the Lagrangian is represented by eqn 3.5 which we re-write here again Now in the electric limit the contravariant and covariant electric field is represented as Similarly magnetic fields for the two cases are 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 3.20 which we re-write as In the magnetic limit the electric field is defined as And the magnetic field is same as before. So now the magnetic limit Lagrangian takes the following form in terms of the fields 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 are different. We observe there is an overall sign change (i.e L → −L) under the duality transfor- This has been shown clearly in table 4.

Dual transformation of electric and magnetic fields under Galilean boost
Contravariant case: The Field transforms as Boost transformations can be written as From 4.58 using 4.59 and 4.60 we get following relations Electric limit From eqn 4.61 using the electric limit scaling we get Similarly, eqn 4.62 yields

Magnetic limit
From eqn 4.61 using the magnetic limit scaling we get Similarly from eqn 4.62 we get

Covariant case
The field transforms as We also write the boost transformations as From equation 4.67 using 4.68 and 4.69 we get Electric limit From eqn 4.70 using the electric limit scaling for the covariant case we get Similarly from eqn 4.71 we get

Magnetic limit
From eqn 3.11 using magnetic limit scaling for covariant case we get Similarly from eqn 4.71 we get The transformed electric and magnetic field under galielan boost has been shown in table 5.
Magnetic limit

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.

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
As we discussed before in table 1, the potential map as From eqns 5.4 and 5.3 we can deduce the following relations Taking the variation of 3.5 in the electric limit, on exploiting eqns 5.5, 5.6, 5.7, 5.8. This shows the invariance of L e .

Magnetic limit
Here the potential transforms as From eqns 5.10 and 5.3 we can deduce the following relations, Taking the variation of the lagrangian 3.20 in the magnetic limit on exploiting eqns 5.11, 5.12, 5.13, 5.14. This shows the invariance of L m .

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 5.16 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 antisymmetry 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). First we consider the electric limit. 3

Electric limit
In this limit the contravariant components of the current are So we can show the galilean currents are conserved, i.e Likewise for the covariant components From this we get following components of the current j 0 = 0 (5.23) Here also we can show that they are conserved ∂ t j 0 + ∂ i j i = 0, (5.25)

Magnetic limit
In this limit the contravariant current components are j 0 = 0 (5.26) We can see that they can satisfy following conservation equation.
Similarly for the covariant components, which satisfies ∂ t j 0 + ∂ i j i = 0 (5.31)

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 6.1. Since 6.1 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 5.1.

Electric limit
We can define following things Similarly.
From 3.5 Noether currents are found to be And current conservation can be explicitly demonstrated as The covariant components of the currents are 14) The current conservation gives us

Magnetic limit
We can define following things Similarly.
From 3.20 the Noether currents are found to be δa j = 0 (6.20) So the current conseravtaions are demonstrated as Similarly the covariant current components are The current conservations give

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 Lagrangian demands following condition Electric limit: In the electric limit the scaling 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, Which can be written as From the first equation 7.12 we get which reproduces eqn 7.7 From the second equation 7.13 we have which reproduces eqn 7.8. To get the remaining pair of equations we have to interpret eqns 7.12 and 7.13 as The first of them in the galilean limit yields, which reproduces eqn 7.9. Similarly 7.17 yields eqn 7.10. 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 (7.7)-(7.10) satisfy the above conditions. We observe that sources are conserved off-shell.

Magnetic limit:
Here the scalings 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 We now derive these equations directly from the equations of motion. Eqn 7.12 will give Since α, β = 0 we have two conditions Here also sources given by eqns (7.24) to (7.27) satisfy the above off-shell conservation equations.

Conclusions
In this paper we have given a detailed map between the relativistic and galilean vectors under the electric and magnetic limits. The novelty of our approach is that we have given this map for both contravariant and covariant vectors. In previous works only the former sector was considered. We believe our analysis here gives the topic named "Galilean electrodynamics" a complete look and illuminates the underlying deeper and subtle symmetries which remains unobserved so far. We have derived the non-relativistic Lagrangians for both electric and magnetic limit using the map given in section 2. We have shown that they give consistent equations of motion. The potential formulation was extended to the field (electric & magnetic) formulation. In this set up the duality symmetry was discussed. One can clearly see that in the nonrelativistic limit the duality relations are quite non-trivial. On the lagrangian level, duality plays quite a subtle role. We briefly discuss the gauge symmetries in the relativistic case and how they give rise to consistent non-relativistic gauge symmetries. One interesting fact is for the relativistic case the gauge parameters for covarinat and contravariant case are same but that no longer holds true for non-relativistic gauge transformations as the covariant and contravariant quantities are distinct entities (since there is no non-degenerate metric to raise and lower the indices) hence the gauge parameters can be different quantities. Here we compute the non-relativistic Noether currents and show that they are on-shell conserved. Subsequently we introduced another class of symmetries called shift symmetries which play important role to describe Goldstone's particles. We have translated these symmetries in the non-relativistic limit and show that corresponding non-relativistic Noether currents are conserved on-shell. Finally we include sources in our calculations and derive the equation of motion which again give consistent results. We show that these sources satisfy off-shell conservation. This is quite new research area and many aspects and directions are yet to look at. There is no consistent Hamiltonian formalism for galilean electrodynamics. We want to explore that side and hopefully comeback soon.
δa j = f ij ∂ j β (A.9) So the conservation looks like In the conservation equation A.10, first term is zero from equation 3.9 and second term vanishes because of antisymmetry. Magnetic limit Using the maps for magnetic limit given in table 1 and exploiting eqns A.1 and A.2 we get second line of eqn A.13 vanishes from eqn 3.22. Likewise for the covariant case, So the conservations look like 16) In eqn A.16 third line vanishes from eqn 3.24.