Lie-Poisson gauge theories and κ-Minkowski electrodynamics

We consider gauge theories on Poisson manifolds emerging as semiclassical approximations of noncommutative spacetime with Lie algebra type noncommutativity. We prove an important identity, which allows to obtain simple and manifestly gauge-covariant expressions for the Euler-Lagrange equations of motion, the Bianchi and the Noether identities. We discuss the non-Lagrangian equations of motion, and apply our findings to the κ-Minkowski case. We construct a family of exact solutions of the deformed Maxwell equations in the vacuum. In the classical limit, these solutions recover plane waves with left-handed and right-handed circular polarization, being classical counterparts of photons. The deformed dispersion relation appears to be nontrivial.


Introduction
Non-trivial Poisson structures over spacetime may be regarded as semiclassical approximations of spacetime noncommutativity.In view of that, the latter is conveniently described in terms of an associative algebra of functions, algebraically dual to spacetime, made noncommutative by replacing the ordinary pointwise product with a noncommutative one (the star product).In such a context Poisson structures have been widely investigated since the beginning of NC geometry.The definition of a star product for symplectic manifolds goes back to [1][2][3].The generalization to Poisson manifolds has been achieved by Kontsevich [4], whose star product reduces to the previous ones for nondegenerate structures.A special family, which shall be the object of the present paper, is represented by the so-called Lie-Poisson structures, namely, linear Poisson brackets of Lie algebra type.Because of their simple, but interesting mathematical structure, they have been largely investigated by the community.The first analysis to our knowledge is performed in [5].A far from complete list of subsequent works is represented by [6][7][8][9][10][11][12].
There has been recently a renewed interest for Poisson gauge theories within a novel approach which relies on strong homotopy algebras and symplectic embeddings [24][25][26][27][28][29][30][31].Indeed, while spacetime noncommutativity represents a natural solution to reconcile general relativity and quantum mechanics in strong gravitational regimes [32], the state of the art of NC field and gauge theories is still unsatisfactory, these being generically affected by the so called UV/IR mixing [33] or not reproducing the standard commutative theories in the commutative limit, except for a few models with very special features.
The present paper represents a followup of previous work by the authors [26][27][28][29]; it therefore shares the same motivations.We briefly review our arguments along the lines of [26].
By Noncommutative Gauge Theory (NCG) it is generally intended a theory described in terms of a noncommutative algebra (A, ⋆) representing spacetime, an A-module, M, representing matter fields, a group of unitary automorphisms of M representing gauge transformations.The dynamics of fields is described by means of a natural differential calculus based on derivations of the NC algebra.Moreover, the gauge connection is the standard noncommutative analog of the Koszul notion of connection [34][35][36][37][38][39][40] (for a physically inspired perspective see the earlier approach in [41]).
Therefore, the first problem to face is the definition of an appropriate differential calculus, namely, an algebra of ⋆-derivations of A such that (1.1) For constant noncommutativity, assuming Θ non-degenerate, the latter are given by star commutators thus reproducing the correct commutative limit.For coordinate dependent Θ(x) the situation is different.Lie algebra type star products, do admit a generalisation of (1.2) according to but the limit Θ → 0 is not directly related to the correct commutative result.In particular, in [21] it is shown that, for su(2) type noncommutativity, the differential calculus defined in terms of (1.4) becomes two-dimensional in the commutative limit.A related approach consists in using a twisted differential calculus for those NC spacetimes which admit a twist formulation [42][43][44][45][46].
To summarise, for generic coordinate dependence of Θ, where one needs to employ the general Kontsevich star product [4], ordinary derivations violate the Leibniz rule, whereas twisted or star derivations, although giving rise to a well defined differential calculus, might not reproduce the correct commutative limit (see for example [21,22] where this point is analyzed in detail).
In order to deal with this and other problems of NCG, in the above mentioned papers [26-29] a different perspective is adopted: the very definition of gauge fields and gauge transformations is modified in such a way that not only the outcoming gauge theory reproduce the correct commutative limit but also, the gauge algebra close with respect to the underlying noncommutativity (namely, a homomorphism holds between the Lie algebra of gauge transformations and the noncommutative algebra of gauge parameters).
To be precise, the whole construction is only performed in the semiclassical approximation where ⋆ commutators are replaced by Poisson brackets.Strictly speaking this means that spacetime stays commutative, but it becomes a Poisson manifold with non-trivial Poisson bracket among position coordinates.Gauge parameters in turn, which are spacetime functions, inherit such a non-trivial Poisson structure.It is therefore natural to require that they close under Poisson brackets.This implies a departure from the standard picture of NCG (and corresponding Poisson gauge theory) outlined above, the latter being recovered only for Moyal noncommutativity, namely constant Poisson brackets.
Before illustrating the new results of the present paper concerning U (1) gauge theory on a Poisson manifold (the spacetime) of κ-Minkowski type, we wish to add a few comments on the motivations for considering spacetimes with a non-trivial Poisson structure and argue that such structures are conceivable and worth to be studied beyond their relationship with noncommutative geometry.One interesting instance is obtained by identifying R d with the dual, g * , of some d-dimensional Lie algebra g. g * carries the coadjoint action of the group, whose orbits are homogenous spaces.There is a natural Poisson bracket on g * , of Lie algebra type, which is non-degenerate on the orbits [the Lie-Kirillov-Souriau-Konstant bracket (LKSK)].Here , indicates the natural pairing between g and g * , f ij k are the structure constants of the Lie algebra, X k ∈ g and ℓ i their duals, identified with coordinate functions on g * .This point of view is perfectly compatible with the standard approach to translation invariant dynamics, where the carrier space of the dynamics is the homogenous space of either Poincaré or Galilei group, respectively quotiented with respect to the Lorentz or Euclidean subgroups.The induced LKSK is then trivial, being the dual of the translations commutator.
Therefore, all situations where translation invariance is not the natural symmetry of spacetime, will have a natural Lie-Poisson bracket of the kind described above.
Another interesting occurrence of nontrivial Poisson brackets on the configuration space is represented by dynamical systems defined on the manifold of some Lie group G.In that case, it is possible to endow the algebra of functions on G with a Poisson bracket compatible with the group operations (in particular, the group multiplication has to be a Poisson map).This bracket, dubbed Poisson-Lie structure,1 is quadratic in the group coordinates and gives rise to the concept of Poisson-Lie group, the semi-classical counterpart of a quantum group [47].Finally, it is worth mentioning that Poisson manifolds constitute an important family of target spaces for topological field theories (see for example [48][49][50]).
We will deal for most part of the paper with linear Poisson brackets, although more generally one could consider a smooth d-dimensional manifold M, equipped with a Poisson bracket of the form, where Θ ab (x) is a given Poisson bivector. 2t is then natural to investigate the compatibility between the Poisson structure of the carrier manifold and the algebraic structure of the set of gauge transformations.In the following we shall consider a U (1) gauge theory without matter fields, with A(x) representing the gauge one-form.
The Poisson structure on the spacetime manifold induces a non-commutative algebra of infinitesimal gauge transformations for the gauge potential A(x), whose bracket is dictated by the request that the gauge algebra be closed.We also require that at the commutative limit gauge transformations reproduce the standard U (1) gauge transformations, Table 1: Summary of the main objects and their transformation properties.ψ is an arbitrary field which transforms in a covariant manner under gauge transformations: The matrix Θ shall be often called the noncommutative structure, although this is strictly speaking the Poisson tensor.
We shall refer to field theoretical models with the gauge algebra satisfying Eqs.(1.8), (1.9) as Poisson gauge theories.As already noticed, gauge theories on Poisson manifolds have been already studied in the literature in relation with NCG and deformation quantization (see for example [13-15, 18, 20, 50]).Towards the end of the section we shall shortly comment on their relationship with our approach.
When restricting to Poisson structures of Lie algebraic type, where the structure constants 3 f ab c obey the Jacobi identity, the corresponding Poisson gauge theories will be referred to as Lie-Poisson gauge theories.
In order for gauge transformations to be compatible with the closure request (1.8) and for the corresponding field strength to be gauge covariant, it has been shown [28,29] that it is necessary to introduce a deformation of the gauge sector in terms of two field-dependent matrices, γ(A) and ρ(A), which solve the master equations and obey the classical limits, lim In parallel, a gauge covariant derivative has been introduced in terms of the same objects.In Tab. 1 we summarize the results and we refer to [28,29] for more details.We stress that, for a given Poisson structure, the matrices γ(A) and ρ(A) completely determine the fields and transformations of the related Poisson gauge model. 4 Unless otherwise specified, the forthcoming analysis shall be restricted to the Lie-Poisson case (1.10).To this, let us consider the gauge potential one-form A = A µ dx µ .For the forthcoming discussion it is convenient to introduce the following notation Two main remarks are in order before proceeding further.Firstly, the master equations (1.12) exhibit universal solutions in terms of matrix-valued functions [29,30]: (1.15) 3). 4 In general, the matrices γ and ρ may depend on x explicitly.In the present work we limit the analysis to the x-independent case.which are valid for arbitrary structure constants f ab c .In these formulae, and B − n , n ∈ Z + stand for the Bernoulli numbers.The corresponding Poisson gauge theory will be addressed as the "universal" one.
Secondly, as shown in [29], any invertible field redefinition, such that, lim gives rise to new solutions of the master equations, which respect the commutative limits (1.13).In other words, for a given Poisson bivector Θ ab one may construct infinitely many Poisson gauge theories.However, they are all equivalent: the field redefinitions map gauge orbits onto gauge orbits, thereby defining invertible maps between these theories.They have been given the name of Seiberg-Witten maps in [29], because of some analogy with the famous result by Seiberg and Witten [51] in the framework of fully noncommutative theory.
(Notice however that a semiclassical analogue of the Seiberg-Witten result, closer to the spirit of the original paper, is already discussed in [15]; more details shall be given below).Interestingly, all the Poisson gauge models which have been constructed so far within the present approach result to be connected with the universal one by means of Eqs.(1.19) [29].
As mentioned before, the case of nonconstant noncommutativity has been already addressed in the literature.As for example in the derivations based approach mentioned before, the infinitesimal gauge transformation of the potential is usually assumed to be of the form where α is a suitable constant, necessary to adjust the dimension.The semiclassical analogue is readily obtained by replacing star commutators with Poisson brackets.It is easily seen that it satisfies Eq. (1.8), but the commutative limit of the resulting gauge theory is not the standard one (see for example the comment after Eq. (1.4) and [22]).This expression, or, better to say, its semiclassical analogue, coincides with the first equation in Table 1 only for γ ν µ = δ ν µ which is a solution of the first master equation in (1.12)only for Θ = const, while differing in the first term for all other cases.The same applies to the definition of the field strength F in the second row of the table: it agrees with previous definitions (e.g.those considered in [15,22]) only if both γ and ρ are equal to the identity matrix, which is a solution for the second master equation only for Θ = const.
The Poisson, or semiclassical analogue of nonconstant noncommutativity has been addressed in [15] in relation with the Seiberg-Witten map [51].It is shown that a classical version of the latter may be established, such that, to the standard gauge theory with gauge transformation A → A + df on a classical manifold M, it is possible to associate a one-parameter family of Poisson gauge theories with gauge transformation In this formula A ρ = A ρ (A, f ) and f = f (A, f ) are respectively the gauge potential and gauge parameter of the family of Poisson gauge theories.Their results differ form the ones considered in this paper, as can be seen by comparing (1.21) with the gauge transformation in Table 1.
The paper is organised as follows.In section 2 we study the covariance of the theory under gauge transformations.We first prove the explicit covariance of the Bianchi identity previously obtained in [28].Then we present gauge-covariant expressions for the Euler-Lagrange equations of motion and the Noether identity, which exhibit correct commutative limits.Sec.2.2 is dedicated to the non-Lagrangian gauge-covariant equations of motion for a generic Poisson tensor of Liealgebraic type.In Sec. 3 we focus on the κ-Minkowski case, introducing a one-parameter family of well-behaved non-Lagrangian equations of motion.We present the exact solutions of the deformed Maxwell equations in the vacuum, which recover plane waves with the left-handed and the righthanded polarization in the commutative limit.We conclude with a summary in section 4 and five appendices which contain technical aspects.

Towards explicit covariance
The covariant field strength reported in Table 1 has been shown to satisfy the following deformed Bianchi identity [28] D with Being an identity, Eq. (2.1) must be necessarily gauge covariant.However, the transformation properties of B b de (A) were not explicitly studied; in particular, we did not address the question as to whether or not this quantity transforms in a covariant manner.It is possible to prove that this is indeed the case.More precisely, the following Proposition holds Proposition 2.1 i) In the universal Poisson gauge model (1.15)The proof is explicitly given in Appendix A. This Proposition yields essential simplifications of the Lie-Poisson gauge theories.In particular, we immediately get a manifestly gauge-covariant form of the deformed Bianchi identity, Moreover, combining the results of the Proposition 5.1 in [28] which provides the commutation relation of gauge covariant derivatives, with Proposition 2.1, we arrive at the following simple formula for the commutator of the covariant derivatives: Therefore, we shall assume in the rest of the paper that the Poisson gauge model under analysis is either the universal one, or it is connected with the universal model via a Seiberg-Witten map, i.e. the identity (2.3) applies.

Lagrangian dynamics.
Any Poisson gauge theory exhibits a Lagrangian formulation.As we shall see below, some of these formulations have a correct commutative limit, while others do not.

Admissible Lagrangian models
The classical action, M = R d , is gauge-invariant if and only if the Poisson tensor obeys the compatibility condition [27], For Θ of the Lie-algebra type this condition is equivalent to the following restriction on the structure constants, f ab a = 0. (2.9) We shall call these Lagrangian formulations the "admissible" ones.All the special Lie algebras satisfy (2.9), hence they yield admissible Lagrangian formulations of the corresponding Lie-Poisson gauge theories.Below we present manifestly gauge-covariant expressions for the equations of motion and for the Noether identity.
Proposition 2.2 For admissible Lagrangian models the Euler-Lagrange equations of motion, can be rewritten in a manifestly gauge-covariant way, Proof.Using the results of [28], accompanied by Proposition 2.1, one can easily see that, (2.12) The matrix ρ is non-degenerate, therefore Remark.In the three-dimensional case with the su(2)-non-commutativity, the combination vanishes, and one arrives to the so-called "natural" equations of motion D c F ca = 0, see [29] for details.However one can check by direct calculation, that the addition of a fourth commuting coordinate (e.g.time) breaks this property.A non-triviality due to the presence of commutative coordinates in the Poisson gauge formalism was discovered in [52].

Proposition 2.3 If an admissible Lagrangian formulation of the Poisson gauge theory is available,
the Noether identity [29], can be rewritten in a manifestly gauge-covariant form, Proof.Substituting the relation (2.12) in the left-hand side of the Noether identity (2.14), we obtain, where, on using the definition of D a as in Table 1, (2.17) In App.B we prove that, Therefore, by using the definition of F, we arrive at, which is nothing but the left-hand side of Eq. (2.15).Q.E.D.

Non-admissible Lagrangian models
In some relevant cases, such as the κ-Minkowski non-commutativity, the compatibility condition (2.8) is not satisfied.Therefore, the corresponding action is not gauge invariant.One can still define a gauge-invariant classical action, modifying the integration measure by a function µ, as follows with the measure function obeying the condition see [27] for details.For κ-Minkowski non-commutativity the general solution of Eq. (2.23) reads5 [27], where F λ (w 2 , ..., w d−1 ) is an arbitrary, sufficiently regular, function of d − 2 variables.We added the subscript λ in order to emphasise that F may also depend on the deformation parameter λ in an arbitrary manner.However, by a simple scaling argument, presented in App.C, it can be checked that this solution is not compatible with the commutative limit6 lim Θ→0 µ(x) = 1. (2.25) Thus, since the measure µ enters in the Euler-Lagrange equations of motion, the corresponding Lagrangian dynamics does not reproduce the commutative limit correctly.We shall refer to these models as non-admissible.Therefore we are strongly motivated to look for gauge-covariant equations of motion, which lie beyond the Euler-Lagrange formalism.

Non-Lagrangian dynamics
By contracting the covariant derivative D a with the covariant field-strength F ab and the structure constants f ab c , one can obtain various e.o.m., which reproduce the standard dynamical sector of Maxwell equations at the commutative limit.As we shall see below, not all these opportunities are equally good.

Natural equations of motion
The simplest possibility for gauge-covariant e.o.m. is, These are the so called "natural" equations of motion.Being manifestly gauge-covariant, these e.o.m. reduce to the Maxwell equations (2.26) in the commutative limit.However, it is possible to show that there is no one-to-one correspondence between the solutions of classical Maxwell equations and the solutions of (2.27).Indeed, on contracting the natural e.o.m. (2.27) with the covariant derivative D a , we obtain the following equality, which has to be satisfied on shell 7 , (2.28) Hence, the identity (2.6) yields In order to perform the commutative limit, it is convenient to rescale the structure constants by the deformation parameter λ according to (2.31) By construction such a limit is automatically verified off-shell.However, a priori nothing guarantees that the deformed solutions are analytic functions of λ.In other words, the assumption (2.31) is not an obvious property at all.On performing the limit λ → 0 in Eq. (2.29), we arrive at the nonlinear constraint,

Covariant equations of motion
Appropriate equations of motion, which do not suffer the problem mentioned above, can be formulated.They are indeed the covariant equations of motion introduced in the Euler-Lagrange context, namely These equations remain valid even when the compatibility condition (2.8) is not fulfilled (e.g. in the κ-Minkowski case).In such cases they can not be obtained from an action principle, but have to be stated independently.These equations enjoy the following important property.

Proposition 2.4
The left-hand-side of the covariant e.o.m. (2.33) satisfies for all non-commutativity of Lie-algebra type.
The proof is presented in App.D. As we have seen with Prop.2.3 this formula is nothing but the Noether identity, when an admissible Lagrangian formulation of the Poisson gauge theory is available.We emphasise, however, that Eq. (2.34) is valid always, no matter whether the compatibility relation (2.8) takes place or not.From now on, by somewhat stretching the terminology, we shall call Eq. (2.34) the "Noether identity" in the non-Lagrangian cases as well.
As we announced above, thanks to the latter, no extra constraints in the classical limit arise.Indeed, by contracting the covariant e.o.m. (2.33) with D c , and passing to the limit λ → 0, we get the relation which is automatically satisfied for all solutions of Maxwell equations (2.26).In contrast to the natural e.o.m (2.27), the covariant ones (2.33) do not yield any special constraints.
Let us clarify the importance of the Noether identity.In the commutative case not all of the dynamical Maxwell equations are independent.The constraint is given by the Noether identity, On one hand, the natural equations of motion (2.27) do not yield a gauge-covariant Noether identity respecting the commutative limit.From this point of view, (2.27), accompanied by the requirement of the correct commutative limits of their solutions, are overdetermined.On the other hand, the covariant e.o.m. do exhibit a gauge-covariant Noether identity, which reduces to the constraint (2.36) in the commutative limit.Therefore, the covariant e.o.m. are not overdetermined in the above mentioned sense.

Generalised equations of motion
Summarising the previous analysis, we see that the covariant e.o.m are, generally speaking, non-Lagrangian e.o.m, with the following properties: • Correct commutative limit: the e.o.m reproduce the first pair (2.26) of Maxwell equations at the commutative limit.
• Consistency with the Lagrangian formalism: the e.o.m can be obtained from the classical action (2.7), when an admissible Lagrangian formulation of the corresponding Poisson gauge theory is available.
• Minimality: The only term which contains the covariant derivative is D c F ca .The derivativefree terms are quadratic in F.
• Noether identity: Prop.2.4 holds.This property allows to avoid the pathologies discussed at the beginning of this section in the context of the "natural" e.o.m.
In what follows we rise the question as to whether it possible to generalise the covariant e.o.m, preserving the properties listed above.In other words, we wonder whether there are other non-Lagrangian e.o.m, which are as good as the covariant ones.
The following two-parameter family of generalised e.o.m. satisfies the first three requirements, When the compatibility condition (2.9) is fulfilled, the new terms vanish identically and we end up with the covariant e.o.m.The corresponding Noether identity may also contain new terms, which, of course, vanish, when an admissible Lagrangian formulation is available.In the next section, analysing the κ-Minkowski non-commutativity, we shall see that these generalisations are indeed possible.Moreover, there exist special choices of the constants α, β and ω, which simplify the analysis of the corresponding dynamics.
3 Applications to the κ-Minkowski non-commutativity Throughout this section we consider the κ-Minkowski non-commutativity (2.21) at d = 4, which is probably the most well studied example of non-commutative space with non-trivial Poisson structure [53]- [80].The corresponding structure constants read, One can easily check that the matrices, solve the master equations (1.12) 8 .For this kind of noncommutativity, Poisson gauge models, including the universal one, were already obtained in previous publications [27][28][29].They were shown to be related with each other through Seiberg-Witten maps [29].In the following we shall present a new model along the lines described above.In Appendix E we prove that it is equivalent via Seiberg-Witten maps to the previous ones as well.

Generalised equations of motion
One can check by direct substitution that the generalised e.o.m. (2.37) are compatible with the (generalised) Noether identity (2.38) iff, We have a one-parameter family of equally acceptable generalised e.o.m., whose left-hand sides, obey the (generalised) Noether identity, In order to obtain this formula, we used the equality, which is a peculiar feature of the κ-Minkowski non-commutativity.Notice that, at α = −1/12, the Noether identity takes more elegant form, As we shall see soon, this choice of α, indeed, simplifies the analysis of the nonlinear equations of motion.

κ-Minkowski plane waves
From a particle physics perspective, the most interesting solutions of Maxwell equations in the vacuum are the plane waves with the left-handed and the right-handed circular polarizations.In Quantum Electrodynamics these waves correspond to photons with the left-handed and the righthanded helicities respectively.Poisson gauge models with the κ-Minkowski non-commutativity are rotationally invariant9 , therefore, without loss of generality, we consider plane waves, travelling along the axis x 3 .First, to set the notations, we present the relevant formulae of the commutative electrodynamics.Then we focus on the non-commutative case.

3.2.1
Circularly polarized plane waves in the commutative case.
In the Coulomb gauge, the four-potential describing these waves is given by where ε stands for its amplitude, k denotes the wave vector, and the signs plus and minus specify the polarisations, see below for details.Hereafter the subscript "cl" indicates that we are dealing with the "classical" solution, which refers to the undeformed Maxwell electrodynamics.The gauge potential (3.8) solves the Maxwell e.o.m. iff the wave vector satisfies the dispersion relation, The corresponding electric and magnetic fields, If sgn(k 0 ) = sgn(k 3 ) = 1, then the fields ( E + , B + ) and ( E − , B − ) describe the right-handed and the left-handed circularly polarized plane waves respectively.

Non-commutative deformations.
By substituting the Ansatz, in the non-commutative equations of motion, we obtain, At α = −1/12 the expressions (3.11), indeed, solve the e.o.m., iff the deformed dispersion relation, is fulfilled.The structure of the deformed gauge potential is identical to the commutative one (3.8).
For other choices of α the simple Ansatz (3.11) does not work, and one has to try more involved ones, what goes beyond the scope of the present work.Note that the deformation of the dispersion relation in the scalar field theory on the κ-Minkowski space constructed from the abelian twist was obtained in [75,79], however in that approach no correction was produced for the vector fields.The corresponding deformed electric and magnetic fields, read, (3.16) In the commutative limit all the deformed quantities tend to their classical counterparts, (3.17) Though the expressions (3.16) for the deformed electromagnetic field components are very similar to the ones for their commutative analogs (3.10), there are two important differences.
• The dispersion relation is the deformed one, given by Eq. (3.14).
• The deformed magnetic field B ± has non-zero longitudinal components It is worth noticing that the absolute value of the deformed electric field and its classical counterpart is the same The deformed dispersion relation (3.14) can also be rewritten in terms of this absolute value ε, By using this formula, one can easily see that, The equalities (3.18) and (3.18) imply that, as in the commutative case,

Summary
In the paper we have addressed the problem of finding the dynamical sector of Maxwell equations for Lie-Poisson noncommutativity, which be covariant and well behaved in the classical limit.We have shown that the natural equations of motion (2.27) do not fulfil the requests.In Sec.2.2.2 we have found the appropriate equations of motion.Hence the full set of Maxwell equations with the right properties reads If an admissible Euler-Lagrange formulation is available, the dynamical equations (4.1) are nothing but the Euler-Lagrange equations of motion, rewritten in a gauge-covariant form.Moreover, the left-hand side E a C of the dynamical equations (4.1) obeys the equality, If an admissible Lagrangian exists, this formula is a gauge-covariant form of Noether identity.We have analysed in detail the κ-Minkowski case and we have found a number of interesting results, which we list below.
• One may extend the covariant e.o.m. in a reasonable way, considering a one-parameter family of generalised e.o.m., which are compatible with the (generalised) Noether identity, At α = 0 all these relations reduce to the corresponding formulae for the covariant e.o.m.
• At α = −1/12 we have obtained simple solutions of the deformed e.o.m. in the vacuum case, which recover the plane waves with the left-handed and the right-handed circular polarisations at the commutative limit, where k 0 and k obey the deformed dispersion relation, • The corresponding deformed electric and magnetic field read, The main novelty (apart from the deformed dispersion relation) is the presence of the nonzero longitudinal components of the deformed magnetic field.Therefore, in contrast with the commutative case, the fields B and E are not orthogonal to each other.However, as in the commutative case we find namely their modules are equal, and coincide with the modules of the corresponding commutative limits B ± cl and E ± cl .
A Proof of Proposition 2.1

First statement
Taking the partial derivative with respect to A j of the relation, and multiplying the resulting equality by ρ from the right, we get the identity, Therefore the definition (2.2) of B b de can be rewritten as follows, where all the derivatives act on ρ −1 .

Second statement
We now prove the validity of Eq. (2.3) for any Poisson gauge model, which is related to the universal one through the Seiberg-Witten map, discussed in the Introduction.Let the matrix ρ( Ã) be related to the universal solution ρ(A) via Eq.(1.17) and Eq.(1.19).Introducing the notation, one can easily check that the structure (2.2) remains unchanged upon the Seiberg-Witten maps: This derivation of Eq. (2.18) is quite short.
where at the last step we used Proposition 2.1.Q.E.D. The derivation of Eq. (2.19) is more sophisticated than the previous one.We first derive it for the universal Poisson gauge model.
We start from the main formula (2.3) of Proposition 2.1, representing it as follows, C Violation of Eq. (2.25) in the κ-Minkowski case From now on we assume that the measure µ(x, λ) exhibits a finite limit at λ → 0, because otherwise it would not make sense to consider Eq. (2.25) at all.Upon rescaling of coordinates, the general solution (2.24) rescales as follows, Since the identity (C.2) is valid at any λ, and, by assumption, lim λ→0 µ(x, λ) exists, this identity is valid at the commutative limit as well, If the relation (2.25) was possible for some "lucky" choice of the arbitrary function F λ , the formula (C.2) would yield the controversary equality, Thus we conclude that the κ-Minkowski non-commutativity is not compatible with the limit (2.25).Q.E.D.

D Proof of Proposition 2.4
On substituting the explicit form (2.33) of E a C in the l.h.s of the desired equality (2.34), one obtains, Applying the identity (2.6), we see that was found in [28].The Seiberg-Witten map, which connects this first Poisson gauge model with the universal one, has been constructed in [29].One has to set a 0 = λ/2, a 1 = 0, a 2 = 0, ..., a d−1 = 0, in the corresponding formulae of the quoted references.We emphasise that in the present project we are not working with the generalised κ-Minkowski non-commutativity, restricting ourselves to the standard one only.The Poisson gauge theory, based on the novel simple matrices (3.2), is related to the model, defined by γ and ρ, through the Seiberg-Witten map as well.Indeed, one can easily check that upon the invertible fields redefinition,

F
) so that f ab c be pure numbers.Let us assume that the solutions of the deformed e.o.m. (2.27) tend to solutions of the standard Maxwell e.o.m (2.26) in the limit λ → 0, namely that, on-shell, lim λ→0 ab = F ab .

1 2 [F 2 FF 2 F 4 F 2 FFFF 2 F
D a , D c ]F ca = F de f ae b D a F db , de f de b F ab + 1 de f de b D a F ab − D a F de f ae b F db .(D.3)Renaming the mute indices appropriately, one can easily get−D a F de f ae b F db = D a F bd f ab e F de = D b F da f ab e F de , =⇒ −D a F de f ae b F db = 1 2 f ab e F de D a F bd + D b F da , de D a F bd + cycl(a b d) Fac f cp b F pd +cycl(a b d) = 1 de f de b E b C + Q, (D.6)where the first term is nothing but the desired contribution, whilstQ = (I) + (II) + (III) + (IV) + (V), (D.7) with (I) = − 1 de f de b F cp f cp a F ba , (II) = + 1 de f de b F cp f bp a F ca , de F ac f cp b F pd , de F bc f cp d F pa , de F dc f cp a F pb .(D.8)In order to obtain Eq. (D.6), we used the deformed Bianchi identity (2.1) and the definition (2.33) of the covariant e.o.m.Below we demonstrate that Q = 0, what will complete our proof.The contributions (I) and (IV), being contractions of symmetric and antisymmetric tensors, vanish identically:(I) = − 1 4 • F de f de b F cp f cp a sym. in (a, b) • F ba skew-sym.in (a, b) skew-sym.in (d, e) • f ab e F bc f cp d F pa sym. in (d, e) = 0. (D.9)Renaming the mute indices, we see that the sum of the remaining three contributions equals to zero,(II) + (III) + (V) = 1 ca F de F cp f de b f bp a + f ep b f bd a + fpd b f been made of the Jacobi identity (1.11).Q.E.D. E κ-Minkowski non-commutativity: new and old solutions In the forthcoming discussion we use the notations, g(z) := z 2 + 1 + z, g ′ (z) ≡ dg(z) dz = √ z 2 + 1 + z √ z 2 + 1 .(E.1)For the κ-Minkowski non-commutativity, the first matrix γ viz,
.32)which, in general, is not automatically satisfied by the solutions of the classical Maxwell equations(2.26).Therefore, any solution of the commutative e.o.m., which does not obey this additional constraint, does not have a deformed counterpart solving the natural e.o.m. (2.27).