Action principle of Galilean relativistic Proca theory

In this paper, we discuss Galilean relativistic Proca 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 Proca theory from usual Proca theory in the potential formalism for both contravariant and covariant vectors which are now distinct entities. An action formalism is thereby derived from which the field equations are obtained and their internal consistency is shown. Next we construct Noether currents and show their on-shell conservation. We introduce analogues of Maxwell’s electric and magnetic fields and recast the entire analysis in terms of these fields. Explicit invariance under Galilean transformations is shown for both electric/magnetic limits. We then move to discuss Stuckelberg embedded Proca model in the Galilean framework.


I. INTRODUCTION
Works on non-Lorentzian physics have gained considerable attention recently.Non-relativistic limit or Galilean relativistic limit of various field theories and gauge theories have been studied extensively.These non-relativistic theories have found applications in a wide variety of fields including but not restricted to holography [1], nonrelativistic diffeomorphisms (NRDI) [2][3][4][5][6], condensed matter systems [7], fluid dynamics [8,9], gravitational waves [10].This formulation is tricky and quite different from the usual 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 case poses some additional difficulties.The study of Galilean relativistic theories was initiated by Le Bellac and Levy Leblond [11] back in 1970's. 1 Their main focus was on Galilean electromagnetism which had many physical applications [13].The experiments of Rowland, Vasilescu, Karpen, Roentgen, Eichenwald and others, reviewed in [13], which were previously interpreted or understood using special relativity found a more eloquent explanation using Galilean electromagnetism since it was perfectly suited to study electrodynamics of continuous media at low velocities.It may also be recalled that the results in [11] were attained primarily by physical principles based on gauge and galilean invariances.In case of non-gauge theories such an approach would be untenable, or at least difficult to formulate.
An alternative approach would be to construct the Galilean relativistic version by taking an appropriate limit of the corresponding relativistic theory.Such attempts were done in [14], using group contraction techniques.Likewise in [15], scaling relations among potentials were introduced to obtain equations of motion in Galilean electrodynamics from Maxwell electrodynamics.Recently we [16] have developed a method where the Galilean relativistic version of any Lorentz relativistic vector theory can be constructed following a structured algorithm.It does not depend whether the vector theory is a gauge or non-gauge theory.The method was applied to the Maxwell theory [16].
In this paper, we are interested in Proca theory which describes a massive spin-1 field in Minkowski spacetime [17].The presence of the mass term actually breaks gauge invariance of the lagrangian.Proca theory plays a key role in many areas of fundamental physics, including experimental physics.For example it plays an important role in the theoretical framework for experimental studies aiming at the determination of upper bounds for the mass of the photon [18], [19], and other different areas of fundamental physics [20][21][22].Apart from these there is a recent resurgence of the study of Proca model from different aspects.For example there are studies in the context of gravitational waves [23], gravitational lensing and black hole (BH) shadows [24], alternative theories of gravity [25].There has been a prescription to generalise the Proca action [26].This generalised Proca model has various applications in different areas of fundamental physics [27,28].There is also a non-linear Proca model called Proca-Nuevo model [29].So given this importance and applications of Proca model we believe the study of non-relativistic limits of various Proca models will be interesting.Our current work is the first step towards that direction.
The construction of the cherished action and the associated action principle follows by explicitly providing maps that connect the relativistic with non-relativistic vectors.This is done for both covariant and contravariant vectors since these are distinct entities in the Galilean theory, not being connected by any metric.Also for each component, there are maps corresponding to electric and magnetic limits -the two limits in Galilean relativistic physics.Using these maps the lagrangian for Galilean relativistic Proca model is constructed from its relativistic counterpart, both for electric and magnetic limits, and its consequences are examined.
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 from an action principle.These are shown to be compatible with the equations derived directly from the relativistic Proca equations.In section IV we introduce the galilean electric and magnetic fields and write down the Galilean Proca equations.The Noether currents and their corresponding conservations are discussed in section V. We now move to discuss the gauge invariant version of Proca theory, obtained by Stuckelberg embedding, and perform its detailed Galilean analysis.This has been discussed in section VI.Finally we conclude in section VII.

II. MAPS RELATING LORENTZ AND GALILEAN VECTORS
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 take a generic Lorentz transformation with the boost velocity as u i : x where γ = . 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.2 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 These equations define the usual galilean transformations.To see this in the context of coordinates, we revert to (1, 2), consider the u 2 << c 2 limit implying γ → 1, use x 0 = ct so that, which is the exact analogue of (6,7).We next 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, although the corresponding group is not an invariance group of classical mechanics.Expectedly, the role of time-like and space-like vectors has been reversed from the earlier case.
We will now consider the covariant vectors.We will write first the reverse transformations of eqn 1 and 2 which are Covariant vectors transform as Componentwise we write them as Now here we take the electric limit in the following way, which will soon become clear Using 16 in 14 and 15 we get which are again Galilean transformations.We will now consider the magnetic limit as Using 19 in 14 and 15 we get The mapping relations are summarised in the table I.

III. LAGRANGIAN AND FIELD EQUATIONS
Now let us start from the relativistic Proca theory described by the Lagrangian where and η µν is the flat space metric with signature −, +, +, + .It gives rise to the following eqns of motion which imply Eqn 24 is a necessary condition for the relativistic Proca theory, leading to a Klein-Gordon type equation, A. Galilean relativistic theory We now discuss the derivation of the Galilean invariant Proca model using results of section 2. The two limits are now considered independently.The first step is to open the various terms in 22 as, Electric limit: Using the relations given in table I we write the two terms in 26 as, Here A µ is the relativistic four potential while a 0 and a i are it's galilean counterpart.Hence the full lagrangian takes the following form and the corresponding action is, Now we derive the equations of motion.Variation with respect to a 0 , a j , a 0 , a j yields the equations of motion We now derive these equations directly from the relativistic equations of motion 23, which can be split as From table I and eqn 36 we get in the electric limit which reproduces eqn 32.Likewise the Galilean limit of 37 reproduces 33.
To get the remaining pair of equations 34 and 35 we have to interpret eqns 36 and 37 in terms of their covariant components, Once again the Galilean version is obtained from table I, followed by taking c → ∞.Equations 34 and 35 are reproduced.This shows the consistency of the eqn of motion in Galilean Proca model.
Let us now show how 24 is manifested in the present analysis.Since covariant and contravariant components are treated separately in the Galilean construction, 24 is interpreted as, These conditions imply, and, The relation 42 follows on acting ∂ j on either side of 35.Likewise 43 follows on acting ∂ j on 33 and using 32.We will now discuss the implications of eqn 25.To do that we can write for its contravariant components, Considering the electric limits for ν = 0 & i we have following two equations Now eqn 45 is nothing but 32.We can re-write eqn 33 as which reproduces eqn 46.The term in the bracket vanishes as a consequence of 43.We can also write the other two equations i.e 34 and 35 as the the form given by 25.
TABLE II: Field equations Magnetic limit: Here again using the relations given in table I we can write the two terms in 26 as, So the magnetic action will take the following form where, Variation 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 Proca equations by taking the magnetic limit.From 36 we have which reproduces eqn 53.Similarly 37 reproduces 54 To get the remaining pair of equations we have to start from the covariant versions (39, 40) which will reproduce 55 and 56 respectively.The field equations for both limits of Galilean Proca model are shown in table II.
In magnetic limit the condition 24 implies Similarly, Here also the relation 57 follows on acting ∂ j on eqn 56 and using eqn 55.Similarly relation 58 follows on acting ∂ j on 54.We will now discuss the implications of eqn 25 in the magnetic limit.For the contravariant components the equations are From the equations of motion, 53 implies From table II we observe the dual role of electric and magnetic limits vis-a-vis their covariant and contravariant sectors.In other words the electric limit equation found by varying a 0 corresponds to the magnetic limit equation obtained by varying a 0 .The same property holds for other variables.

IV. ELECTRIC AND MAGNETIC FIELDS IN GALILEAN PROCA THEORY
Here we discuss Galilean Proca theory in terms of electric and magnetic fields introduced in analogy with Maxwell theory.For this purpose we will discuss contravariant and covariant sectors 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 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 these 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 Magnetic limit: Electric field can be written in this limit from the mapping relations in table I as We define the galilean electric and magnetic fields as Using these relations, eqns 53, 54 and two Bianchi identities are expressed in terms of electric/magnetic fields as The results for the electric/magnetic fields as well as the Maxwell-type equations (70, 71, 72, 73) and (76, 77, 78, 79) for either electric or magnetic limit agree with those given in [30].In [30] an additional mass term has been introduced to express the Galilean algebra as a centrally extended Bargmann algebra.Correspondingly, no superselection rule is needed and one works with an ordinary representation.In our analysis we are confined to the Galilean algebra but the ordinary representation has to be replaced by a projective representation.The two descriptions are completely equivalent [31].

B. Covariant sector
Relativistic electric and the magnetic fields are defined as First, we consider the electric limit.

Electric limit:
In this limit the electric field looks like The Galilean electric and magnetic fields are given by, Using these expressions the Proca equations in the Galilean electric limit are found to be,

Magnetic limit:
In this limit electric field is scaled as The Galilean electric and magnetic fields are now defined as, Finally, the equations we get in the magnetic limit are given by, C. Lagrangian in field formulation and boost invariance In both electric and magnetic limits the lagrangian will take the following form In the electric limit a 0 , a i , a 0 , a i will transform under Galilean boost as As pointed out in [16], the electric and magnetic fields transform in the electric limit Using 96 it is evident that the Maxwell part of the lagrangian (94) remains invariant δL Maxwell = 0 (97) We will show here that remaining terms are also boost invariant.To do so we write Now taking the variation of 98 and using 95 we have In the magnetic limit a 0 , a i , a 0 , a i will transform under Galilean boost as And using 100 we can write the variation of 98 as Thus the complete lagrangian 94 is boost invariant.
V. NOETHER CURRENTS Contrary to Maxwell theory there is no conserved current that directly follows from the Noether procedure since there is no gauge invariance.However it is possible to redefine the current so that it is conserved.This happens because although the Lagrangian is not invariant, the action is on-shell invariant.To see this consider the variation, which modifies the Proca lagrangian to Hence action is on-shell invariant This indicates the possibility to redefine the current that follows from the Noether procedure such that it is conserved.The usual Noether current is, Redefining the current as , it is possible to show its on-shell conservation, Finally using the equations of motion 23 and 24 we have Now we see the implications of these currents in the galilean limits.To do this we will start with the electric limit.

Electric limit:
The map from J µ → j µ or J µ → j µ just follow the corresponding map for vectors, A µ → a µ , A µ → a µ , Using the above relation we can write the Noether currents in Galilean limit as Now we define the modified currents J ′µ which in the non relativistic limit gives rise to, The conservation equation reduces to To see current conservation in the galilean limit we compute In the above equation we have used eqns 32, 33 and 43.The covariant currents are derived by adopting a similar approach.From the relativistic expression using the mapping relations given in table I, The modified currents are given by, The conservation equation reduces to Here In the above equation we have used eqns 35 and 42.Magnetic limit: In this limit, The contravariant components of the currents are Conservation of the modified currents imply where we have exploited eqns 119, 54 and 58.

VI. STUCKELBERG EMBEDDED VERSION OF PROCA MODEL
The theory is described by the Lagrangian where Φ is a scalar field.We can now see that the Lagrangian is gauge invariant under following transformations, Now varying the action wrt A ν and Φ we have the following equations of motion, We see that the two equations are self consistent.Since our aim is to construct Stuckelberg embedded Galilean Proca model where covariant and contravariant sectors are treated distinctly, it is necessary to write a more general version of 121, which bears an extended gauge invariance The equations of motion are obtained by varying We will now consider different limits, beginning with the electric case.Electric limit: In this limit we know the fields scale as The Lagrangian 124 imbibes the form, Varying with respect to a 0 , a j , a 0 , a j , φ & ψ 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 Using table I and taking appropriate limits, we reproduce 133 from 140, 134 from 141 and 137 from 142.
To get the remaining pair of equations we have to interpret eqns 140 and 141 so that the variables appear in a covariant form, Following similar steps 143 reproduces 135 while 144 yields 136 and 145 yields 138 in the Galilean limit.

Magnetic limit:
In this limit we know the fields scale as TABLE III: Field equations Electric limit Magnetic limit Now the Lagrangian looks like Varying 147 wrt a 0 , a j , a 0 , a j , φ, ψ 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.Using table I  We have already shown that the relativistic lagrangian is gauge invariant.Here we show that the Galilean counterpart is also gauge invariant.Electric and magnetic limits are treated separately.Electric limit: In this limit the relations given in 125 and 126 will give rise Using eqn 154 we can show that L SPe → L SPe (155) Magnetic limit: In this limit the relations given in 125 and 126 will give rise Using eqn 156 we can show that

B. Noether currents
For the relativistic Stuckelberg embedded Proca model the Noether current is defined as One can clearly see that this current is conserved we will now discuss the Galilean limit Electric limit: Relativistic conservation equation in the non-relativistic (electric) limit, reduces to, where using eqn 158 and the mapping relations given in table I we can write down the current components as This conservation follows by using eqns 133, 134, 137 and 161.
Following the same steps we can show current conseravation for covariant components.Magnetic limit: Relativistic conservation equation in the magnetic limit, reduces to, where using eqn 158 and the mapping relations given in table I we can write down the current components as This conservation follows by using eqns 148, 149, 152 and 163.

VII. CONCLUSIONS
Let us now summarise the new significant findings of the paper, comparing with existing results found in the literature.
• An unambiguous construction of the action principle for non-relativistic (NR) Proca model, for both electric and magnetic limits, was given.The non-relativistic analogue of massive spin-1 particle has been discussed in [32] by using a group theory point of view and a short discussion has been presented in [30].None of these is comprehensive and does not include the several features presented here.For instance in [32] only the magnetic limit is considered while [30] presents a mixed lagrangian with both electric and magnetic sectors.Nowhere a systematic construction of an action principle was presented.Here we have deduced this action from the standard relativistic lagrangian adopting the dictionary given in section II.It is expressed either in terms of potentials or fields.Explicit demonstration of boost invariance has been presented.
• We have considered contravariant and covariant vectors separately.This helps us to discover certain subtleties which were unobserved so far.For example 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.
• 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.Also, we know gauge symmetries play a pivotal role in the understanding of gauge theories.Since there is no gauge symmetry present in the Proca theory it is interesting to observe its consequence in the Galilean invariant theory which has been derived by using the dictionary presented here.We also discuss in section VI how to restore the gauge invariance by using the Stuckelberg embedding.We derive its Galielan counterpart by writing down the lagrangian for both the limits.Since covariant and contravariant components are treated separately, two scalars have to be introduced in the Galilean theory to attain gauge invariance.We show that the Galilean lagrangians for both the limits are gauge invariant.Another interesting thing to observe is the change of contravariant indices by covariant ones induces not only the electric limit to magnetic limit but the Stuckelberg scalars φ and ψ also get interchanged.
Future prospects: • There have been some observational aspects of Proca models in the context of gravitational waves [23] and black hole shadows [24].We believe a further systematic analysis of the non-relativistic Proca theory can shed some important light in this direction.
• Recently its has been observed in [29] that a certain type of non-linear Proca field theory which is sometimes called Proca-Nuevo model plays a crucial role in explaining the emergence of half degrees of freedom in Lorentz and parity invariant field theories.It will be interesting to look at the Galilean invariant counterpart and study its further implications.
• Usual Proca model can be generalised by considering derivative self-interactions with only three propagating degrees of freedom [26].This is sometimes called generalised Proca model.It will be useful to study its Galilean invariant counterpart.The generalised Proca model can be extended to curved spacetime which gives rise to Horndeski Proca model.We like to study the non-relativistic analogue (in a curved background) by adopting the approach of [33,34].
• Generalised Proca theories of gravity represent an interesting class of vector-tensor theories which can be extended to modified theories of gravity by including torsion [27].Considering the recent interests and applications of non-Lorentzian physics in different contexts, we believe the study of the non-relativistic limit will illuminate various unknown aspects of generalised Proca model as well as modified gravity theory.
• Finally we want to look at the Carrollian limit [14] of the Proca model.There have been some interesting studies of Carroll symmetries in the context of gravitational waves [35] and in the context of Hall effect [36].We would like to study the Carrollian limit for these issues, particularly the gravitational waves aspect, in the context of Proca model.
We expect we can address these issues in the near future.

VIII. ACKNOWLEDGEMENTS
61) In the first line ∂ i a i = 0 from 58.Eqn 61 reproduces 59.Similarly 54 reproduces 60 by exploiting 58.The other two equation 55 and 56 can be written in the form of 25 by exploiting 57.

TABLE I :
Mapping relations and taking appropriate limits, we reproduce 148 from 140 and 149 from 141.To get the remaining pair of equations we have to start from the covariant versions (143, 144).From table III we can observe thatContravariant ←→ Covariant =⇒ Electric ←→ Magnetic & φ ←→ ψ