$\kappa$-deformed complex fields and discrete symmetries

We present a construction of $\kappa$-deformed complex scalar field theory with the objective of shedding light on the way discrete symmetries and CPT invariance are affected by the deformation. Our starting point is the observation that, in order to have an appropriate action of Lorentz symmetries on antiparticle states, these should be described by four-momenta living on the complement of the portion of de Sitter group manifold to which $\kappa$-deformed particle four-momenta belong. Once the equations of motions are properly worked out from the deformed action we obtain that particle and antiparticle are characterized by different mass-shell constraints leading to a subtle form of departure from CPT invariance. The remaining part of our work is dedicated to a detailed description of the action of deformed Poincar\'e and discrete symmetries on the complex field.


I. INTRODUCTION
It is commonly expected that the usual description of space-time as a smooth manifold is no longer reliable as we approach the Planck scale when quantum effects of the geometry can no longer be neglected. Since the pre-history of research on quantum gravity 1 non-commutativity of space-time has been advocated as a possible way to effectively model quantum gravitational effects in regimes of negligible curvature. A widely studied incarnation of this idea suggests that the scale of non-commutativity should be seen as an observer independent length scale [3], and that, in order to accommodate such fundamental scale, ordinary relativistic symmetries should be deformed into non-trivial Hopf algebras which, in the limit of vanishing non-commutativity, should reproduce the usual Poincaré algebra.
The κ-Poincaré algebra is an example of such deformations which has been intensively investigated for almost 30 years. Such algebra was originally derived by contracting the quantum anti-de Sitter algebra [4,5]. It was brought to its modern form a few years later in [6] and [7], where, in particular, the role of non-commutative κ-Minkowski spacetime was discovered and investigated. The deformation parameter κ has dimensions of mass and, in light of the possible role of the κ-Poincaré algebra in describing the symmetries of a flatspace-time limit of quantum gravity, it is usually identified with the Planck energy. Such putative relationship with a semiclassical limit of quantum gravity renders this model especially relevant for the search of possible experimental signatures of Planck scale physics [8,12]. So far most of the proposed observational frameworks having sufficient sensitivity to capture effects of quantum gravity origin [9,12] were based on purely kinematical models, like, for example, the well known case of measuring the time of flight of Gamma Ray Bursts photons of different energies [13,14]. It has however been argued that κ-deformations may have a subtle, and in principle measurable, effect on elementary particles, linked to the deformation of CPT symmetry [15]. For these reasons we believe that developing a comprehensive theory of deformed quantum fields will be beneficial for better understanding known phenomena related to κ-deformation and possibly shed light on some new ones that might 1 According to Jackiw [1] the idea of non-commuting space-time coordinates was first suggested by Heisenberg back in the 1930s. He then discussed it with Peierls who in turn told to Pauli who told Oppenheimer who asked his student Snyder to work it out in detail and thus the fist paper on non-commutative spacetime was published in 1947 [2]. be of phenomenological relevance (besides, of course, its relevance at a purely theoretical level).
In the series of papers, of which the present one is the first, we will formulate the theory of a free, complex κ-deformed scalar field. The next paper in the series will be devoted to free scalar field propagator and n-point functions. We will consider next massive higher-spin fields and then the quantum deformed abelian gauge fields. We will discuss interacting fields in the final, fifth paper of the series.
The present paper has its roots in the work [17] from which we borrow the notation and most of conventions. However there are important differences. In particular, the definition of the scalar field is different here. This change of definition is a consequence of the assumed nice behavior of the field with respect to the discrete CPT transformations and leads to one of the major results of this paper, that the mass shell relations of particles and antiparticles differ from each other, although as a manifold the mass-shell in both case is the same hyperboloid in momentum space, as anticipated in [15], [16]. Thanks to this new definition of fields also the creation-annihilation operator algebra becomes particularly simple. In the present paper we also consistently use the star product formalism instead of the equivalent formalism of non-commutative spacetime used in [17].
Various aspects of the theory of κ-deformed fields were discussed in the past. Here we mention papers that influenced us [22]- [33] in working on this project, but we would like to stress that the crucial aspects of the present construction, like the doubling of momentum space and insistence on the proper action of discrete symmetries are new.

II. PRELIMINARIES
As it is well known there are two complementary pictures of κ-deformation. One deals with the presence of non-commutative spacetime with Lie type non-commutativity, called κ-Minkowski space [6], [7], where the commutator of coordinatesx µ form the an(3) Lie with the parameter κ defining the 'strength' of non-commutativity. Another concerns the momentum space picture, in which the momentum space is curved and is a submanifold of de Sitter space with curvature 1/κ 2 [34], [35], which is constructed as follows.
Let us consider the following 5-dimensional matrix representation of the Lie algebra (1) where bold fonts are used to denote space components of a 4-vector (with the exception of the central 0 which is a 3 × 3 matrix) and ǫ is a three dimensional vector with a single unit entry, e.g., ǫ 1 = (1, 0, 0).
Let us now consider an elementê k of the Lie group AN (3), which, as we will see in a moment, represents a group-valued momentum In the representation (2) this group element is represented by a 5 × 5 matrix which acts on 5-dimensional Minkowski space as a linear transformation. One finds where 1 is the unit 3 × 3 matrix, andê k can be written in schematic form where p 0 , p i and p 4 are defined below, whilep 0 = κ sinh k 0 κ − k 2 2κ ,p 4 = κ cosh k 0 κ + k 2 2κ e k 0 /κ . To describe the manifold of the group AN(3) we choose a point in 5-dimensional Minkowski space, which becomes the momentum space origin O with coordinates (0, . . . , 0, κ) and act on it with the matrixê k (4), obtaining On the left hand side we have coordinates of a point in the 5-dimensional Minkowski space, being in one to one correspondence with the group elementê k . The coordinates (p 0 , p i , p 4 ) are related to the original parametrization (k 0 , k i ) of the group element as follows There is a natural action of the 4-dimensional Lorentz group on the 5-dimensional Minkowski space, which takes the form for infinitesimal boosts and rotations parameters λ i , ρ i . Since the Lorentian momenta components p 0 , p, transform as a vector, p 2 0 − p 2 is Lorentz-invariant and, as usual, the representations of the Lorentz group, in the spinless case that we consider here, are labelled by values of the mass m 2 and sign of energy p 0 . Therefore the representations of the Poincaré algebra are characterized by mass-shell condition p 2 0 − p 2 = m 2 . It is easy to check that 2 It follows that the group AN(3) is isomorphic, as a manifold, to a submanifold of the 4dimensional de Sitter space. This submanifold is defined by the conditions On-shell p 2 0 − p 2 = m 2 and the condition (7) takes the form Observe that this condition does not impose any restrictions on positive energy states, but provides a lower bound on the negative energy ones 0 > p 0 > − m 2 + κ 2 . This condition seemed first to be Lorentz invariance violating [36] because by acting with the Lorentz boost we can make p 0 acquire an arbitrary negative value, but was later shown to preserve Lorentz symmetry in a nontrivial way [37]. To understand how it comes about let us introduce the antipodal map S(p) defined as Notice that on-shell S(ω p ) = S( m 2 + p 2 ) is always negative.
It is worth mentioning in passing that if p 2 0 − p 2 = m 2 then S(p 0 ) 2 − S(p) 2 = m 2 and vice versa, so the former serves as an alternative form of mass-shell relation. As we will see both these mass shell conditions will arise in the theory of deformed scalar field.
One checks that this map provides a one-to-one correspondence between the 'positive energy' submanifold p 0 > 0 and the negative energy one, satisfying the constraint (8).
Indeed take a positive energy state with energy p 0 > 0 and momentum p and apply the antipode to it. We find We define the action of Lorentz symmetry on negative energy states by applying it to the corresponding positive energy one and taking the antipode of the result, schematically, With this definition the orbits of Lorentz group for both positive and negative energies belong to the momentum space. We will describe the Lorentz transformations of the antipode in to (0, . . . , 0, −κ), We defineê * and acting with this group element on (0, . . . , 0, κ), instead of (5) we get On-shell the condition (14) takes the form so that this time it does not impose any restrictions on negative energy states, but provides an upper bound on the positive energy ones 0 < p * 0 < m 2 + κ 2 . Again one solves the apparent problem with Lorentz symmetry with the help of the antipode, which has the form On-shell S(ω * p ) = S(− m 2 + p * 2 ) is always positive. To formulate the field theory we must first describe the algebra of plane waves and differential calculus. We start with the group elements (also called 'noncommutative' plane waves)ê k (3) (associated with the submanifold p 0 + p 4 > 0) andê * k (12) (for the submanifold p 0 + p 4 < 0). We use the five-dimensional Lorentz covariant differential calculus, see [17] and references therein for details. To this end we introduce the spacetime derivatives∂ µ and an additional derivative in fourth direction∂ 4 defined by their action on the plane waveŝ Following [17] we define the Weyl map 3 W that maps group elements (plane waves on noncommutative κ-Minkowski spacetime) to ordinary plane waves on commutative spacetime manifold with coordinates x, as defined by the action of the derivatives with ∂ µ being the standard partial derivative 4 . The star product presented here coincides with the one proposed in [18] and further discussed in [19], [20] and [21]. It follows that with the on-shell relations The Weyl map makes it possible to construct the star product of two commuting plane waves from the product of two group elements W(ê kêl ) ≡ e p(k) ⋆ e q(l) = e p⊕q (22) In the case of two positive energy plane waves we havê with Then acting with the group element (23) on the reference vector (0, . . . , 0, κ) we get Notice that the choice of Weyl map is not unique (see for instance [22] for a different choice, and the discussion in [38]), and from this choice depend also the star product structures. In this paper we choose to adopt the Weyl map introduced in [17], mapping "time-to-the-right" ordered non-commutative plane waves to standard exponentials of commutative coordinates, expressed in terms of "embedding" momenta p A (k) (A = 0, 1, . . . , 4). 4 An explicit realization of this star product was presented in [39] Let us use the same construction in the case of the negative energy plane waves. To this end we must first compute the product From we find (To compute this, one starts with (25), changes the overall sign, then changes the sign of p replacing it by p * , and finally changes the sign of q according to (26).) Similarly Finally, we consider the composition of two negative energy plane waves. (In this case after moving through the Q plane wave we get z 2 = 1) Notice that, remarkably, all the composition laws (25)-(30) have exactly the same form so there is no need to distinguish between them.
Let us finish this section with the definition of an adjoint of the plane wave. For the noncommutative plane waveê k its adjointê † k is defined by the condition Accordingly, in the star product formalism we express these equations as from which it follows that The analogous expressions for p * A coordinates are easy to obtain.

III. ACTION AND FIELD EQUATIONS
Having discussed all the necessary technical tools in the preceding section we can now turn to the construction of the theory of free complex scalar field. As customary in noncommutative field theories, we define a notion of integral on non-commutative space-time via the Weyl (or quantization) map (18). In particular we set Fields on κ-Minkowski can be defined in terms of a suitable "noncommutative" (or, for some authors, quantum group-) Fourier transform [17,38,[40][41][42][43]. In accordance with our choice of Weyl map, we adopt the noncommutative Fourier transform introduced in [17]: and its inverseφ where the measure dµ(p) is the AN(3) left-invariant measure and the coordinates p are intended as the "embedding" coordinates p(k) given by (5). The definition can be thus extended to fields of commutative coordinates through Weyl map Explicitly Notice that the φ(x) defined by (39) and (40)  From (22) and (35) it follows that the inverse noncommutative Fourier transform can be expressed asφ and that the noncommutative product extends to a star product of fields of commutative coordinates In particular we have the following useful identity The star product here coincides with the one defined in [18] (generalized to 4d), which can be checked by calculating that it gives the identical result for the coordinate functions x µ .
However it is not clear if the construction of the integral/twisted trace presented in that paper coincides with our definition of the integral.
Using the non-commutative Fourier transform and the star-product, we can formulate the action of free fields on κ-Minkowski space-time as a standard integral action in terms of (properly defined as above) fields of commutative coordinates. In particular, we define the action to be an integral of the bilinear hermitian expression, in fields and derivatives, obtained with the help of the star product. The integral satisfies the exchange properties for the plane waves [17] R 4 d 4 x e † p ⋆ e q = R and the most general expression for the hermitian action is In order to compute the variation of the action and to derive field equations, we have to make use of the ⋆-integration by parts, which is described in detail in Appendix B. Writing and we find which can be rewritten as where and, analogously, which can be rewritten as where Therefore the field equations have the form which, as we will see below, lead to two non-trivially related mass-shell conditions, describing the same orbit of the Lorentz group on the momentum manifold.

IV. THE COMPLEX SCALAR FIELD
Now we are in position to formulate the theory of the deformed free complex scalar field.
In what follows we will use the strategy adopted in [17] of developing the non-commutative field theory in terms of fields on commutative Minkowski space-time equipped with a noncommutative star-product. Using the identity (cf. (31)- (34)) to define the adjoint of the plane wave we can write the adjoint field as and one can define Changing integration variables in the last expression, and using that S (S (p)) = p, we can rewrite it as where we used 5 as one can easily check.
It follows, by comparing (62) with (40), that the condition for φ (x) to be real is 6 where we considered that S (p + ) = κ 2 p −1 + . We will discuss real fields in the forthcoming paper and here we will concentrate on the complex fields only.
According to the properties of the momentum space manifold described in Sec. II (see especially Eq. (7)), the left-invariant Haar measure on AN (3) can be rewritten as the ordinary Lebesgue measure on a restricted 5-dimensional momentum space with (the factor 2κ here is included is for dimensional reasons) Let us now consider a field on the mass shell defined by m, that we can write as (A = 0, 1, . . . , 4) One way of splitting the δ p µ p µ − m 2 into "positive and negative energy" solutions, is to rewrite it as 5 Notice in passing that the r.h.s. of (63) coincides with the right invariant on AN 3 , as one can check from the multiplication of two group elements. If we denote the left invariant measure we are using as dµ L (p), one thus have the property that under antipode, dµ L (S (p)) = dµ R (p). This property is indeed a manifestation of the fact that the antipode map on the manifold corresponds to the inversion on the group elements. 6 The same result was obtained in [44] working with the k parametrization.
Using this, we can rewrite the field as where φ + (x) and φ − (x) denote the "positive and negative energy" components of the onshell field. Consider the "negative energy" part φ − (x). From the properties of the antipode map that imply also S (p A ) S(p A ) = p A p A , if we change the integration variables as p → S (p), and use that S (S (p)) = p and (63), we can rewrite φ − (x) as where we take into account the property θ (S (p + )) = θ p −1 Now, notice that (accordingly to the discussion of Sec. II), The proof is straightforward, since, on the mass shell, thus, The proof that p 0 > m implies S (p) 0 < −m, is also straightforward. This shows that, for p + > 0 and p 4 > 0, i.e. on the AN(3) submanifold we are interested in (i.e. on that section of the de Sitter hyperboloid selected by the measure dµ (p)) the antipode acts indeed as a bijective map that splits the positive and negative energy parts of the manifold belonging to the same mass shell, as argued in Sec. II, and in agreement with the observations reported in [37]. Since the map is bijective (one-to-one), we can then interchange the θ (−S (p) 0 − m) with the θ (p 0 − m) in the integral, and rewrite finally φ − (x) as If the field is real, condition (65) holds, and we have obtained the following result: on the AN(3) measure the on-shellness condition naturally splits the field into positive and negative energy components, that are conjugate with each other, with the antipode playing the role of conjugation for the plane wave, i.e.
For a complex field, it will be convenient to define the antiparticle states, i.e. the ones associated to the negative energy part of the field, as the ones associated to the dual (starred) copy of momentum space. We first substitute, for φ − (x), p → −p = p * , so that (since where p * 4 = − m 2 + κ 2 . Thus, using (69), we have the expansion Since the mass-shell condition has the standard classical form, we would like to define the Fourier components of the complex field as close as possible as the classical expression [45] in terms of creation and annihilation operators We postulate where we include an additional factor (p + has to be considered onshell, that makes the form of the momentum space action, which we will make use of later, particularly simple.
Finally, we have, for the on-shell complex field and its adjoint, the expansions Since ω p > 0 and S(ω * p ) > 0 the field (84) is a combination of positive energy particle states and negative energy antiparticle ones, while in (84) we have the opposite arrangement, as it should be. This particular definition of the field and its adjoint, contrary to earlier approaches where to define the field and its adjoint only one portion of de Sitter space was used, allows for simple action of discrete symmetries, see Section VI below for the details.
From eq. (49) one sees that the equations of motion (EOM) for the field φ are indeed the expected ones. Furthermore, one can get the EOM also for the a p , a † p , b p * and b † p * by applying the EOM to the fields in eq. (84), (85). We get Notice that (86) is equivalent to (88) because one can show that S(S(p)) µ S(S(p)) µ = p µ p µ , and analogously (89) is equivalent to (87) because S(p * ) µ = −S(p) µ .
We find that with the definition of the fields (84) and (85) the particle, characterized by creation (annihilation) operator a p (a † p ) has the mass shell condition p 2 − m 2 = 0, while the antiparticle characterized by creation (annihilation) operator b p (b † p ) follows the mass-shell condition S(p) 2 − m 2 = 0. These mass-shells are identical, so that both the particle and the antiparticle have the same rest mass, and the mass-shell manifold is in both cases the same, but when we apply a Lorentz boost to a particle and an antiparticle at rest with the same boost parameter, they would end up carrying different momenta and energies. This leads to subtle deformation of CPT symmetry, discussed in [15] and [16].

V. SYMMETRIES OF THE ACTION
Let us now check that the above-defined fields transform properly under Poincaré and discrete symmetries, rendering the action (45) invariant.

A. Poincaré symmetry of the action
In order to check the Poincaré invariance of the complex scalar field action 7 (45) it is convenient to rewrite it in the momentum space where such invariance can be easily checked.
As for the space-time action the procedure is much more involved and it is reported in Appendix C.
Let us note that in order to turn the space-time action (45) to momentum space one we cannot use the on-shell field decomposition (84), (85), because the resulting momentum space action would contain the mass shell conditions as coefficients, which will make the action identically equal to zero. Therefore we use as a starting point the off-shell field where we include the additional factor (83) to make the momentum space action as simple as possible. In (90) we used the left-invariant measure (38) on the group manifold AN (3), and we restricted the range of integration in the first term to the positive energy p 0 > 0 subspace J + and to the energy p * 0 < 0 subspace J − in the second term. This arrangement is analogous to the introduction of the θs in (69), but without the mass-shell restriction. The decomposition (90) can be further simplified observing that since p * is a dummy variable we can instead use the variables p = −p * in the second integral, so that we have The adjoint field has the form Plugging these expressions to the action integral (45) after tedious computations, adjusting the free functions we obtain the momentum space action in the form The paper [18] provides a general abstract proof of Poincaré invariance of the κ-deformed complex scalar field action in 2 spacetime dimensions; here we show explicitly that the same holds in the particular of the theory considered here, in 4 dimensions.
It is clear from the action in the form (93) above that the mass-shell of the 'particle' is p 2 = m 2 , while for the 'antiparticle' it has the form S(p) 2 = m 2 , as discussed above.
Moreover it is straightforward to check its Poincaré invariance. The translations act on a p , b p as phases; for the translation parameter ε we have This completes the proof of Poincaré invariance of the action (93).

VI. DISCRETE SYMMETRIES
There are three discrete symmetries: parity P, time reversal T , and charge conjugation C. In each case we will first shortly recall their action on the undeformed field with decomposition and then generalize it to the case of the deformed fields (84) and (85). For parity and time reversal we have spacetime concepts to guide us, and therefore we consider these two first. A.

Parity
The parity operator P acts on space coordinates as an inversion x = (t, x) → x ′ = (t, −x).
For the complex scalar quantum field, we define the parity operator as and using (95) we see that for the creation/annihilation operators 8 Turning to the deformed case we notice first that the spacetime transformationx → −x leaves the defining commutator (1) invariant and therefore is compatible with the form of κ-Minkowski non-commutativity. Further, the positive and negative energy fields φ (±) (x) can be considered separately. For the positive energy part we can use exactly the same considerations as in the case of the undeformed field above. Since this is also true for the negative energy fields and thus we can readily define and

Time reversal
Next we consider the time reversal T , which changes the time direction It should be remembered that the operator T is anti-hermitian T iT −1 = −i, and we have We find that Let us now discuss the deformed case. We start noticing that as a consequence of antihermiticity of T the defining algebra (1) is again invariant, so that we see that κ-Minkowski space is both parity and time reversal invariant. Turning to fields we again see that the classical reasoning can be verbatim repeated in the case of time reversal as well and we end up with and C.

Charge conjugation
The symmetry that exchanges particles with antiparticles does not have any spacetime counterparts and since it changes the charge it is called charge conjugation. The charge conjugation operator C acting on the field produces its conjugation therefore and we have Let us now consider the deformed field. Take the φ (+) component first On the other hand we have so that we can conclude that Analogously, for the φ (−) component we have and So that It should be stressed that this simple transformation rules of the field φ with respect to charge conjugation is a result of the use of the second (starred) copy of momentum space and of the particular arrangement of the components φ (±) (x) and φ † (±) (x). In particular, the field constructed in [17] and many other papers on this topic does not transform nicely under charge conjugation. It should be added also that the deformed action of discrete symmetries P, T , and C leads to the form of the CPT operator Θ anticipated in the paper [15], although the action of parity and time reversal differ from that proposed in [46].

VII. CONSERVED CHARGES AND SYMPLECTIC STRUCTURE
In this section we derive the conserved charges and symplectic structure associated with our free complex scalar field theory defined by the action Both are given in terms of the appropriate boundary integrals, and reflect, respectively, the symmetries of the theory (charges) and its kinematics (symplectic structure). Our starting point here will be the variations of the actions computed above, eqs. (48)-(57). Assuming field equations in the bulk these variations are just the boundary terms, which become conserved charges in the case of field variations corresponding to symmetries of the action and Liouville form, for generic variations.

A. Conserved charges
On-shell the variation of the action reduces to the boundary term and we define the conserved charges associated with the field transformation that leaves the action invariant δ S φ, δ S φ † as usual as an integral over the constant time surface In the case of translational symmetry, for which we find where the relevant components of the energy-momentum tensor are and action preserving all the relavant symmetries. To compute the symplectic structure we must return to (49) and (54). Defining the Liouville form θ as a boundary term in the variation of the action on-shell, for generic variation of the field δφ, δφ † we find To find the symplectic form, which will lead to the Poisson bracket of field coefficients a and b and in turn to the creation/annihilation operators commutators, we have to compute δθ and express the result using the momentum space decomposition (84), (85). We find which implies the following Poisson brackets

VIII. TOWARDS QUANTUM THEORY
In this section we will construct the one particle states in quantum field theory. At this stage we cannot go any further, in particular we cannot construct many-particles states and investigate their properties, because this would require knowing details of the coproduct properties of creation and annihilation operators, i.e. how they act on tensor product of states.
In quantum theory the Poisson brackets (125), (126) become commutators (from now on we stop distinguishing p from p * ) We define the vacuum |0 that satisfies the condition Then we define the one-particle and one-antiparticle states Now we are ready to present the most important result of this investigations. Consider the state |p a , (130). Its momentum can be computed by acting with the momentum operator P i , (121) on it. Using the commutational relation (127) we find Analogously, for the one-antiparticle state |p b , (130), using the commutational (128) we get In exactly the same manner we can use the Hamiltonian (120) to compute the energy of the one particle states, obtaining and Therefore, one-particle and one-antiparticle states belong to the same mass-shell manifold, since but p and S(p) are, in general different points on this manifold, with a single exception being the case p = S(p) = 0, ω p = −S(ω p ) = m.
Finally, the momentum P 4 measures, essentially, the deformed charge of the state and Therefore the one-particle state carries the momentum −S(p) i , while the one-antiparticle state has the momentum p i . But according to (113) the latter is the C (and also CPT ) of the former Therefore, as anticipated in Section IV the charge conjugation (and CPT ) transforms a particle into an antiparticle with different momentum. This transformation has the remarkable property that the rest mass of the particle and antiparticle is the same. The phenomenological consequences of this have been recently discussed in [15], [16].

IX. SUMMARY AND CONCLUSIONS
We laid down the basic ingredients for the construction of a complex field theory on creation and annihilation operators. With these we were able to characterize the energy and momentum of one-particle and anti-particle states and write down the action of discrete symmetries on them which showed that the CPT operator maps particle states into antiparticle states with a different momentum. This important result could have non-trivial phenomenological consequences which might be relevant for experimental searches of Planck scale effects [15,16].
There are several open issues that we are going to address in the future publications.
First it does seem that the particle state and its associated charge conjugated anitiparticle one have different momenta, and it is not trivial to define the real scalar field. We will return to it in the forthcoming publications. The main open issue at the quantum level concerns the construction of a Fock space on which the commutators that we derived for creation and annihilation operators can act, mapping multiparticle states given by appropriately symmetrized tensor products of one-particle states consistent with the non-trivial co-product and covariant under the action of the κ-Poincaré algebra. This is notoriously a thorny issue which has not yet found a satisfactory answer [24,27,30,[50][51][52] and which we hope we will be able to successfully address within the approach to field theory proposed in this work.
The satisfactory solution of this problem is the major prerequisite for the construction of the interacting κ-deformed quantum field theory and κ-deformed standard model, which is our ultimate goal in the research project that the present paper is the first step of.
Thus the Lorentz transformation of the zero component of the antipode is an ordinary Lorentz transformation, with parameter ζ i .
For the spacial component we have a more complicated expression.
The first term here is again the standard Lorentz transformation with parameter ζ i . The second term is an infinitesimal rotation of S(p i ) with the parameter Since under Lorentz boost transformation of momenta the components of the antipode transform under a combination of boost and rotation it is clear that the components of the antipode satisfy the same mass shell condition as the components of the original momenta.

Appendix B: Integration by parts
In this Appendix we derive the ⋆-integration by parts formula, which is necessary do derive field equations from the action (45).
The starting point is provided by the coproduct rules for the κ-Poincaré algebra in the classical basis (p 0 , p i , p 4 ) [17], [53] Notice that the coproduct relations are an immediate consequence of (25). The coproducts tell us how the momentum operators act on star products of two functions. Since momenta are spacetime derivatives p 0 = i∂ 0 , p i = i∂ i these equations tell us how derivatives act on the star products of functions on Minkowski space, defining in this way the modified Leibniz rules. In the calculation below we use the short-hand notation p + → ∆ + = i∂ 0 + p 4 = i∂ 0 + (κ + i∂ 4 ), where the nonlocal operator p 4 is expressed in terms of the corresponding derivatives as p 4 = κ 2 − ∂ 2 0 + ∂ 2 i . Equations (B1), (B2), (B3) then imply Furthermore defining the adjoint derivative and using equation (9) we have We now use eq. (B4), (B5), (B6), (B8) to obtain the expressions needed for the integration by parts of expressions of the form (∂ µ φ) † ⋆ ∂ µ ψ and (∂ µ ψ) ⋆ (∂ µ φ) † . With some algebra we find Similarly Notice that using this convention, equations (B9) and (B10) are still fine substituting φ † with any other quantity (because the above derivations do not use in any way the presence of the † over φ), and therefore can be used regardless of the combination of fields to which they can be applied.
The hermitian conjugates of equations (B9), (B10) take the form Finally, we will also need the following identity and its hermitian conjugate For the opposite ordering we have instead Appendix C: Poincaré symmetry of the action -spacetime approach We want to discuss the invariance of the action (45) under κ-Poincaré transformations.
As a first step, let us notice that it is equivalent to This is easy to see using (B15) and (B16) for integrating by parts the second term, and (B11) and (B12) for the first term, in the action (45), since the Lagrangians are the same up to a total divergence. Let us consider infinitesimal transformations. The basic assumption is that a scalar field transforms as where d is the differential operator corresponding to κ-Poincaré transformations. In order to show the invariance of the Lagrangian appearing in (C1), it is enough to prove that where δL is the functional variation L [φ + δφ] − L [φ].
The invariance of the Lagrangian is ensured if the differential satisfies the Leibniz rule with respect to the ⋆-product, which is a standard requirement for the definition of a differential calculus. Two different prescriptions have been proposed in the literature [17,23,54,55]. We adopt here the one proposed in [17] that is based on a differential calculus that satisfies the "bicovariance" property [56]. In this case the differentiald, generating infinitesimal κ-Poincaré transformations in κ-Minkowski spacetime, takes the form where P A and L µν are respectively the κ-Poincaré translation and Lorentz generators (in classical basis). These are defined through their action on noncommutative plane waves as κ P 0 +P 4 . It can be proved however (see [17]) that the action of the Lorentz generator on the field, through the Weyl map (18), reduces to the standard action The parametersǫ A andω µν must obey commutation relations withx µ so thatd satisfies the Leibniz rule in Minkowski spacetimê The commutation properties ofǫ A andω µν are reported in appendix D, and the corresponding relations (D3) and (D4) between the images of the parameters under Weyl map and the associate ⋆-product, lead to and where the matrices K and Ω are also defined in appendix D.
Two additional properties of the transformation parameters (see [17]) are that and that The field variation δφ = −dφ implies that formally we can state The last relation is very similar to its classical analogous, which is given by Notice thatX µ and K (p (k)) matrices coincide respectively with the 5D representations of x µ andê k given in (2) and (4) and e p ⋆ ω µν = Ω µν ρσ (p) ω ρσ ⋆ e p , ω µν ⋆ e p = Ω −1 µν ρσ (p) e p ⋆ ω ρσ . (D4)