On the energy-momentum tensor in Moyal space

We study the properties of the energy-momentum tensor of gauge fields coupled to matter in non-commutative (Moyal) space. In general, the non-commutativity affects the usual conservation law of the tensor as well as its transformation properties (gauge covariance instead of gauge invariance). It is known that the conservation of the energy-momentum tensor can be achieved by a redefinition involving another star-product. Furthermore, for a pure gauge theory it is always possible to define a gauge invariant energy-momentum tensor by means of a Wilson line. We show that the latter two procedures are incompatible with each other if couplings of gauge fields to matter fields (scalars or fermions) are considered: The gauge invariant tensor (constructed via Wilson line) does not allow for a redefinition assuring its conservation, and vice-versa the introduction of another star-product does not allow for gauge invariance by means of a Wilson line.


Introduction
Groenewold-Moyal (or θ-deformed) space [1,2] represents one of the simplest models for quantized spaces, and has been extensively studied over the past twenty years, e.g. see [3][4][5] and references therein for a review. The theories on this space are formulated in terms of ordinary functions by means of a deformed product, the so-called star-product which implies that the coordinates fulfill This commutation relation is invariant under translations of the space-time coordinates and under the so-called reduced Lorentz transformations (or reduced orthogonal transformations in the Euclidean setting), see for instance reference [6]. In general, field theoretic models on such spaces suffer from a new type of divergences which arise due to a phenomenon referred to as UV/IR mixing [7,8] and which render the models non-renormalizable. This problem can be overcome in the case of some special scalar field models [9][10][11], one of them having been shown to be solvable even non-perturbatively [12]. The present work is devoted to a basic aspect of classical field theories on Moyal space, namely the energy-momentum tensor (hereafter referred to as EMT ) and its properties at tree level. In earlier studies [13][14][15][16][17][18] some modifications to the conservation law of the EMT due to the non-commutativity parameters θ µν were found in φ 4 and in gauge theory (without matter couplings). Here, we wish to investigate more generally complex scalars and fermions coupled to U (1) gauge fields. In view of the infamous time-ordering problems in quantum field theory on Moyal space [19], we restrict ourselves to the Euclidean version of Moyal space.
The present notes are organized as follows. Since the EMT is already a fairly subtle concept for classical field theories in ordinary Minkowski space, we first summarize the facts which are known for this case: this summary may be of independent interest and may be skipped by those familiar with all of these facts. In Section 3, we examine the gauge invariance and conservation properties of the EMT for a gauge field in Moyal space. In Sections 4 and 5 we then extend the discussion to include various couplings to matter.
2 Energy-momentum tensor(s) in classical field theory on Minkowski space The EMT (which is also referred to as stress-energy tensor or stress tensor for short) is an important physical notion in classic relativistically invariant field theories (as well as in the corresponding quantum field theories, but we will not address the latter theories here). However it comes along with various technical and conceptual subtleties, and is not a hazard that the historical road to the EMT in relativistic dynamics has been a long and bumpy one [20]. Einstein's long struggle for constructing a relativistic theory of gravitation was also related to the proper definition and incorporation of the EMT which is by no means evident from the physical point of view, e.g. see Ref. [21]. Therefore it is worthwhile to review briefly the basic facts and results for Minkowski space since further complications arise in non-commutative space. Some general references for these topics are [22][23][24], see also [25] for some more topical discussions. The pioneering works on the improvement of the canonical EMT are references [26][27][28][29]. For the sake of generality we consider the case of n ≥ 2 dimensions and only specialize to n = 4 for the results which are specific to four dimensions.

Canonical EMT and its improvements
Definition of the canonical EMT and its conservation: Consider a collection ϕ of classical relativistic fields for which the dynamics is specified by an action functional S[ϕ] ≡ d n x L(ϕ, ∂ µ ϕ) which involves a Lagrangian that does not explicitly depend on the space-time coordinates x, i.e. the x-dependence of L is only due to the x-dependence of ϕ and ∂ µ ϕ. By virtue of Noether's theorem, the invariance of the action S[ϕ] under translations x ν → x ν = x ν + a ν implies the existence of current densities (j µ can ) ν satisfying the conservation law ∂ µ (j µ can ) ν = 0 for ν ∈ {0, 1, 2, 3}. The conserved second-rank tensor T µν can ≡ (j µ can ) ν is referred to as the canonical EMT field and is given by 1 The local conservation law ∂ µ T µν can = 0 holds by virtue of the equations of motion: If we assume as usual that the fields vanish sufficiently fast at spatial infinity, then the boundary term d n−1 x ∂ i T iν can vanishes and we have conserved "charges" P ν ≡ d n−1 x T 0ν associated with the conserved densities (3). The four-vector (P ν ) represents the total energymomentum of the fields and generates space-time translations of the fields ϕ in classical and in quantum theory. In particular P 0 coincides with the canonical Hamiltonian function H.
On the symmetry of T µν : In general the expression (3) is not symmetric in its indices, e.g. it is not for the free Dirac field ψ and the free Maxwell field A µ , though it is for the free scalar field φ for which one has We suppose that the action for the fields ϕ is also invariant under Lorentz transformations (rotations and boosts) so that the resulting equations of motion are Lorentz invariant in accordance with the principle of relativity. By virtue of Noether's theorem we thereby have a locally conserved canonical angular momentum tensor If the EMT T µν can is not symmetric on-shell (i.e. for fields ϕ satisfying the classical equations of motion), then the tensors M µνρ can and T µν can are not related in the same way as angular momentum and momentum are related in classical mechanics, i.e. by a relation of the form "the components of M are the moments of T ", Indeed for a tensor M µρσ of this form, the conservation law ∂ µ M µρσ = 0 only holds if the tensor T µν is symmetric on-shell: where we used the fact that ∂ µ T µν = 0 holds for solutions of the equations of motion.
On the gauge invariance of T µν : Furthermore, for the free Maxwell field (i.e. L = and thus it is not gauge invariant. One may note however that under a gauge transformation A µ → A µ + ∂ µ f , this expression becomes T µν can − ∂ ρ (F µρ ∂ ν f ) (upon using the equation of motion ∂ µ F µν = 0) and thereby the associated conserved charges P ν are on-shell gauge invariant. Nevertheless this lack of gauge invariance of the components of the classical EMT is embarrassing since these physical quantities are in principle measurable.
On the localization of energy and momentum: Quite generally, the physical interpretation of a Noether current like (3) is a priori subtle due to the fact that, for a given field equation, the Lagrangian density is not uniquely defined, e.g. the addition of a total derivative to L yields the same field equation, but a different tensor T µν can . We may also note that for any conserved Noether current j µ can we can always add a so-called superpotential term, i.e. the divergence of an antisymmetric tensor, This operation yields a current which is still conserved by virtue of the antisymmetry of B µρ (which implies ∂ µ ∂ ρ B µρ = 0) and for which the associated charge is still the same. In view of this non-uniqueness of the EMT and its unpleasant or problematic properties (lack of symmetry or of gauge invariance), some improvements of the canonical expression (3) have been looked for. In the following we will present two such improvements. The first one is due to Belinfante [26] and represents a procedure for constructing a symmetric EMT out of the canonical expression. A further improvement, which is due to Callan, Coleman and Jackiw [29], consists of constructing a traceless EMT for certain classes of theories out of Belinfante's symmetric tensor.
All of these expressions represent different localizations for the energy and momentum, and one may wonder whether there is a preferred or a correct localization. An answer to the latter question can be provided by coupling the matter fields (including gauge fields) to the gravitational field. In fact, in general relativity the EMT represents the source for the gravitational field in Einstein's field equations very much as the electric current density represents the source for the electromagnetic field in Maxwell's equations. From this view-point that we will review in the next subsection, T µν represents the derivative of the matter field action S[ϕ, g µν ] with respect to the metric tensor g µν and thereby it is a symmetric, gauge invariant and generally covariant expression. As shown by Rosenfeld [27] and Belinfante [28], this conceptually quite different approach yields, in the flat space limit, Belinfante's symmetric EMT whose construction does not make any reference to gravity. (In particular one recovers the canonical EMT if this one is already symmetric. We note that Belinfante's symmetric EMT admits a natural geometric formulation if gravity is viewed as a gauge theory of the Poincaré group, see the work of Grensing [22].) One can also recover the traceless Callan-Coleman-Jackiw EMT in the flat space limit of a conformally invariant field theory on curved space. In summary, if one takes for granted that Einstein's general relativity is the correct theory for the gravitational field then the EMT in Minkowski space should not only be conserved and gauge invariant, but it should also be symmetric and naturally generalizable to a generally covariant tensor in curved space. Yet, we should note that within certain alternative theories of gravity, the canonical EMT appears naturally.
Improved EMT: Following Belinfante, the canonical EMT can be symmetrized by the addition of a particular superpotential term so as to obtain the following expression which is referred to as improved EMT or Belinfante(-Rosenfeld) tensor: The antisymmetry of χ µρν in its first two indices ensures that T µν imp is conserved and the derivative in the superpotential term entails that one has the same conserved charge as for T µν can . More precisely, one considers where s µρν represents the spin density tensor of the fields which appears in the canonical angular momentum tensor (5). By construction this procedure yields an improved angular momentum tensor M µρσ imp which is conserved and of the form (6): Henceforth, as argued concerning expression (6), T µν imp is symmetric on-shell. Since M µρσ imp = M µρσ can + ∂ ν (x ρ χ µνσ − x σ χ µνρ ), the improved tensor M µρσ imp only differs by a total derivative from the canonical expression and therefore provides the same conserved charge as M µρσ can . For the free Maxwell field, the described improvement procedure (in particular the use of the equation of motion ∂ µ F µν = 0) yields the tensor which is not only conserved and symmetric, but also gauge invariant. In n = 4 dimensions it is traceless. This tensor encodes the familiar expressions [30] for the energy and momentum densities of the electromagnetic field (i.e. T 00 imp = 1 2 ( E 2 + B 2 ) and T 0i imp = ( E∧ B) x i ) as well as for Maxwell's stress energy tensor density (given by the spatial components T ij imp ). The result (8) expresses in a remarkable manner the union of space and time, energy and momentum as well as electricity and magnetism as achieved by the masters of electrodynamics and special relativity (Maxwell, Einstein, Lorentz, Poincaré and Minkowski).
The improvement procedure is more subtle when matter fields are coupled to gauge fields. For instance, when the Dirac field ψ is coupled to the Maxwell field (A µ ), the canonical EMT of the gauge field is 'improved' following Belinfante's redefinition (7) whereas the improved tensor for the Dirac field not only involves a total derivative as in equation (7), but also an additional gauge dependent term J µ A µ with J µ ≡ eψγ µ ψ: this term amounts to an implementation of the minimal coupling procedure ∂ µ → ∂ µ + ieA µ at the level of the EMT.
For some approaches to Noether's theorem which directly yield the improved version of the EMT, we refer to the works [31].
Scale invariance and tracelessness of T µν : The tracelessness of the EMT can be related to the invariance of the action under rescalings. A scale transformation (or dilatation or dilation) of the space-time coordinates is defined by x → x = e ρ x where ρ is a constant real number. The induced change of the Minkowski metric is also a rescaling with a positive factor: A generic classical relativistic field ϕ transforms under such a rescaling according to Here, the natural number d ϕ denotes the so-called scale dimension of the field ϕ. If one chooses this dimension to coincide with the canonical (engineering) dimension of the field ϕ in n space-time dimensions (i.e. d ϕ = n−2 2 for a scalar field φ or for a vector field A µ , and d ϕ = n−1 2 for a spinor field ψ), then the action for free massless fields ϕ in n dimensions is scale invariant, However mass terms and in general also interaction terms involving non-dimensionless coupling constants violate scale invariance so that one is not simply dealing with dimensional analysis. From the invariance of the action under infinitesimal scale transformations, and δ ρ L = −ρ(x · ∂ + n)L = −ρ∂ µ (x µ L), it follows by virtue of Noether's theorem that we have a conserved canonical dilatation current density of the form This result is reminiscent of expression (5) for the canonical angular momentum tensor: for non-scalar fields the latter not only involves the moments of the canonical EMT, but also an additional term, namely the spin density tensor. This motivated Callan, Coleman and Jackiw [29] to search for a superpotential term such that its addition to j µ dil,can eliminates the second term in expression (12). For simplicity, we consider the case of a free massless scalar field φ in n space-time dimensions for which we have Following Callan, Coleman and Jackiw [29] who considered the four dimensional case, we add a particular derivative term to the canonical EMT of the free massless scalar field (i.e. to expression (4) with m = 0) so as to obtain the so-called new improved EMT or CCJ tensor This tensor is still symmetric, conserved and yields the same conserved charge as T µν can . We have labeled it by 'conformal' since it is traceless (T µ conf µ = 0 by virtue of the equation of motion φ = 0) and directly related to the conformally invariant coupling of scalar fields to gravity, see expression (29) below. With equation (13) and the redefinition we get the expression for j µ dil that we looked for, i.e. j µ dil,conf is conserved, yields the same conserved charge as j µ dil,can and is simply given by the "moments of the EMT": For a generic field theory in a space-time of dimension n, the scale invariance implies that one can define a traceless symmetric EMT provided a particular condition for the fields ϕ is satisfied 2 (and this is actually the case for a large class of theories) [23,29,32]. The new improved EMT then represents a generalization of (13), where T µν imp is the improved EMT and where X ρσµν is symmetric in µ, ν. In this case, the result (15) also holds. Although this is often taken for granted, there is no general proof that the EMT for a scale invariant two-dimensional field theory can be made traceless [23].
To conclude, let us briefly spell out the physically important case of the free Maxwell field in four dimensions. Starting from (12) we have the symmetric EMT (8) which we presently denote by T µν conf : it is related to T µν can by and we have Finally we emphasize that a mass term for free fields violates the tracelessness of the EMT T µν conf . Henceforth this trace (rather than the one of T µν can ) can be directly related to the mass of the fields.

Einstein-Hilbert's EMT
EMT from the point of view of gravity: The study of the interaction of matter with gravity led Einstein and Hilbert [34,35] to the conclusion that matter fields ϕ couple to the gravitational field (which is described by the metric tensor g ≡ (g µν )) by virtue of a linear coupling of g with the EMT (T µν ) of matter: for the corresponding action functional (involving g ≡ det g) we have Thus, the local field T µν (x) plays a fundamental role if matter fields are coupled to gravity while the formulation of field theory in flat space essentially relies on the conserved charges P ν ≡ d n−1 x T 0ν . Concerning the global sign in (18) we note that g µν g νλ = δ µ λ implies that δg µν . The result (18) serves as a motivation for defining Einstein-Hilbert's EMT for bosonic matter fields ϕ in flat space as follows 3 : Since the metric tensor is symmetric in its indices, the so-defined EMT is identically symmetric, i.e. we have a symmetric expression without using the equations of motion of ϕ. Moreover, by construction the so-defined expression for T µν in flat space admits a natural generalization to curved space. The case of fermionic matter fields which requires the consideration of vielbein fields will be addressed towards the end of this section.

Conservation law:
The local conservation law of T µν follows from the invariance of the action under general coordinate transformations (diffeomorphisms). More precisely, if we assume that g is a fixed background metric and ϕ a collection of dynamical bosonic fields (scalar and/or vector fields), then the invariance of the matter action S[ϕ; g] under diffeomorphisms generated by a vector field ξ ≡ ξ µ ∂ µ means that If we assume that the matter field equations δS/δϕ = 0 hold, the first term vanishes. The metric transforms as δ ξ g µν = ∇ µ ξ ν + ∇ ν ξ µ and the symmetry of g µν thereby implies the following result involving T µν as defined by (18): From the arbitrariness of ξ ν one concludes that T µν is covariantly conserved, i.e. ∇ µ T µν = 0, for the solutions of the matter field equations. In flat space we thus recover the usual conservation law ∂ µ T µν EH = 0 for T µν EH defined by (19). If the metric represents a dynamical field, then the conservation law for T µν follows from Einstein's field equations very much as the conservation of the electric charge in electrodynamics follows from Maxwell's field equations ( Gauge invariance and conservation law: For the Maxwell field (A µ ), the action yields, by virtue of definition (19), the EMT (8) in flat space: thus the result coincides with the improved canonical EMT T µν imp for the Maxwell field and it is gauge invariant by construction since the Lagrangian density in the action (22) has this property.
If scalar fields φ are coupled to an external gauge potential (A µ ) and a fixed background metric g, then the gauge invariance of the total action S[φ; A, g] and the use of the matter field equations of motion δS/δφ = 0 imply the conservation law ∇ µ j µ = 0 for the electric current density vector j µ ; indeed, for an infinitesimal gauge variation we have The invariance of the total action S[φ; A, g] under diffeomorphisms then leads to the continuum version of the Lorentz force law, and δS/δφ = 0, δ ξ g µν = ∇ µ ξ ν + ∇ ν ξ µ as well as Once the external gauge field is promoted to a dynamical field by adding the functional (22) to the action, the total EMT of both matter and gauge fields is conserved as we noted already in (20) and (21).
Case of spinor fields: The coupling of spinor fields to gravity requires the consideration of vielbein fields e a µ (x) related to the metric by g µν = η ab e a µ e b ν . The components E µ a of the inverse of the matrix (e a µ ) define the frame vector field E a ≡ E µ a ∂ µ : E µ a e a ν = δ ν µ . We have |g| = e with e ≡ | det (e a µ )|. The EMT for the spinor fields is then defined by differentiating the action for the spinor fields with respect to the vielbein or frame fields, whence the symmetric tensor For instance the variation of the action for the free Dirac field coupled to gravity, with respect to the frame fields E µ a and use of the matter field equations of motion (which imply that the term in δS which is proportional to δe vanishes) yields T a µ = iψγ a ∇ µ ψ. The associated symmetric tensor (24), i.e. presently is covariantly conserved and coincides in the flat space limit with the improved canonical EMT for the free Dirac field.
Weyl invariance and tracelessness of T µν : In conclusion we consider Weyl rescalings of the metric and of the matter fields. These rescalings are sometimes also referred to as "conformal transformations", but strictly speaking conformal transformations are diffeomorphisms for which the induced change of the metric is a Weyl rescaling. (For the relationship between scale invariance and conformal invariance in classical and quantum field theories we refer to the recent work [36] and to the references given therein.) The Weyl transformations of the metric, vielbein and scalar field in a space-time of dimension n ≥ 2 are defined bỹ g µν = e 2σ g µν ,ẽ a µ = e σ e a µ ,φ = e −σd φ φ , where σ denotes a smooth real-valued function and where the so-called conformal weight d φ of φ coincides with its canonical dimension. For g µν = η µν and σ constant, one recovers the rescalings (9), (10) of the space-time line element and of a scalar field in flat space-time.
The conformally covariant Laplacian acting on scalar fields is defined by on scalar fields and R denotes the Ricci scalar curvature. The Weyl transformation law Aφ = e −σ n+2 2 Aφ implies the Weyl invariance of the conformally invariant wave equation Aφ = 0 which follows from the following Weyl invariant action for a massless scalar field: Here, the contribution ξRφ 2 is the only possible local scalar coupling with the correct dimension [24]. The EMT following from the action (28) reduces in flat space to the expression This conserved and symmetric expression is identically traceless. Upon using the equation of motion φ = 0 it reduces to the new improved EMT (13) for a massless scalar field in n dimensions.
For a free massless spinor field ψ in a space-time of dimension n, the Weyl transformation lawψ and the transformation (27)  For the free Maxwell field in four dimensions we assign a conformal weight zero to the field A µ and to the field strength tensor F µν with lower components 4 : this implies that the sourceless Maxwell's equations (i.e. g µν ∇ µ F νλ = 0 and ∇ λ F µν +cyclic permutations = 0) are Weyl invariant and so is the action (22) which yields these equations of motion. The associated EMT T µν ≡ F µρ F ρ ν + 1 4 g µν F ρσ F ρσ is identically traceless in four dimensions and in the flat space limit this result reduces to the one obtained from scale invariance, i.e. relation (16). It should be stressed that these results hold exclusively in four dimensions and that the conformal weight of A µ and F µν in curved space differ from their scale dimension in flat space, see equation (10) and remarks thereafter. For the case of conformally covariant differential operators acting on tensor fields in an arbitrary number of dimensions we refer to [37] and references given therein.
We note that for Weyl invariant actions we can also argue with the infinitesimal variations, e.g. for the Maxwell field in four dimensions, Similarly, for the free massless Dirac field ψ in n dimensions described by the action (25) with m = 0, the use of the transformation lawψ = e − n−1 2 σ ψ and of the fermionic field equations δS/δψ = 0 = δS/δψ imply that the EMT (26) is traceless:

EMT for a gauge field in Moyal space
We consider a U (1) gauge field (A µ ) coupled to an external current (J µ ) in four-dimensional flat Euclidean 5 Moyal space: the action yields the equation of motion The

functional S[A] is invariant under the infinitesimal gauge transformations
provided the current (J µ ) does not transform and is covariantly conserved, i.e. D µ J µ = 0. This is also consistent with the equation of motion in the sense that Notice, however, that gauge invariance of the equation of motion (32) requires that J µ transforms covariantly, i.e. δ λ J ν = −ig [J ν , λ], but that would destroy gauge invariance of the action unless ∂ µ J µ = 0. This inconsistency was already noticed in Ref. [38] and is due to the non-Abelian nature of non-commutative gauge theory. In fact, a similar inconsistency occurs in Yang-Mills theory on ordinary commutative space when coupling the gauge field to an external current [39]. This problem can be overcome by coupling the gauge field to dynamical complex scalar and/or fermion fields so that the external current is replaced by the corresponding matter current, see next section. For now, we keep in mind that the action (31) is not the complete action. Concerning the transformation laws (33) we emphasize that, by contrast to a U (1) gauge theory in ordinary Minkowski space, the field strength F µν is not a gauge invariant quantity as in electrodynamics. This non-Abelian nature of the theory in Moyal space is due to the non-commutativity of space-time coordinates which implies that the field strength "feels" the non-commutativity of the space in which it lives. (This even applies to the simplest case of a constant field strength [40].) The transformation law of F µν implies that the Lagrangian density L = 1 4 F µν F µν is not invariant under gauge transformations since δ λ L = −ig [L , λ]: it is only the integral which plays the role of a trace which ensures cyclic invariance of factors and thereby gauge invariance. Henceforth, the lack of gauge invariance of the EMT will not come as a surprise and contrasts the situation for non-Abelian Yang-Mills fields in Minkowski space.
The improved EMT for a free (i.e. not coupling to a current) gauge field in Moyal space was already computed in Ref. [16][17][18]: It is symmetric and traceless, and it transforms covariantly under gauge transformations: From the Bianchi identity D µ F νρ + D ν F ρµ + D ρ F µν = 0 and the equation of motion (32) with J µ = 0, it follows that the covariant divergence of the gauge field EMT vanishes, i.e. T µν is covariantly conserved.
In the Minkowskian version of Moyal space with non-commutativity parameters satisfying θ 0i = 0, the integral d 3 x of a star-commutator vanishes (assuming as usual that fields vanish sufficiently fast at spatial infinity), hence equation (37) implies that Thus, the four-momentum (P ν ) of the gauge field represents a conserved quantity. Moreover, this quantity is gauge invariant by virtue of (36) and the definition of P ν in terms of T 0ν . Let us now come back to the local transformation law (36). In Ref. [41] (see also [42]) it was explained how to construct gauge invariant objects in Moyal space out of gauge covariant ones. In fact, this task is achieved by folding the quantity in question with a straight Wilson line defined by a length vector (l µ ) with l µ = θ µν k ν ≡ (θk) µ . Using this procedure, the authors of reference [16] obtained a standard local conservation law for the so constructed EMT.
In the following we will also follow this strategy for gauge fields, scalars and fermions, and therefore we briefly outline the procedure here.
The non-commutative generalization of a straight Wilson line with the appropriate length is given by where P denotes path ordering with respect to the contour parameter σ. The expression (39) transforms as W (k, x) → U (x) W (k, x) U (x + θk) † under a gauge transformation U (x). Hence, d 4 xW (k, x) exp(ikx) is a gauge invariant object because the length vector of the Wilson line is adjusted to be θ µν k ν and exp(ikx) induces a translation of U † by −θk, cf. [41,43]. One may now construct (Fourier transforms of) gauge invariant objects from gauge covariant ones by star-multiplication with W (k, x) and exp(ikx) and integrating over d 4 x. The choice of a straight Wilson line is the most natural one because for such a line it makes no difference if the operator is attached to an endpoint of the Wilson line or somewhere in the middle [41]. Furthermore, in the commutative limit (θ → 0) the Wilson line's length goes to zero. For the EMT of a gauge field in Moyal space this means that is a gauge invariant quantity 6 (which reduces in the commutative limit to the ordinary EMT due to lim θ→0 W (k, x) = 1). However, it is not conserved [16], where the star-commutator term arising from ∂ µ T µν was canceled by part of the contribution coming from ∂ µ W . The factor ik β can be pulled out of the integral by rewriting it as ∂ y β , thus allowing for the definition of a gauge invariant, conserved (but no longer symmetric or traceless) EMT T µν : The fact that this modified EMT is not traceless is actually not surprising since θ µν is not dimensionless and thereby introduces a scale into the theory. However, sacrificing the symmetry of the EMT will only be worth the price, if the construction above also works when couplings to matter are considered. In the following sections, we show that this is not the case.

Coupling to matter 4.1 Complex scalar field
Scalar field action: We study the minimal coupling of a complex scalar field φ to an external U (1) gauge field (A µ ) as described by the action where Due to the invariance of the integral under a cyclic permutation of the factors in the star product, the star-anticommutator in the action (43) has no effect, but we choose to keep it in order to make manifest the symmetry under the exchange φ ↔ φ * in all expressions to be considered in the sequel. The equations of motion for the scalar field read and we have This matter current is covariantly conserved, D µ J µ = 0, as a consequence of the equations of motion (44) for φ and φ * .
The gauge transformations imply that the covariant derivatives of φ and φ * also transform covariantly, i.e. δ λ (D µ φ) = −ig [D µ φ , λ] and analogously for D µ φ * . It follows that the Lagrangian density L in the action integral (43) transforms as δ λ L = −ig [L , λ] so that the action is gauge invariant. A short calculation using the Jacobi identity shows that the matter current (45) also transforms covariantly, Next we turn to the EMT of the model described by the action (43): after coupling to an external gravitational field g ≡ (g µν ) we obtain the Einstein-Hilbert EMT in flat Moyal space: From the equations of motion (44) it follows that For non-Abelian Yang-Mills theory in commutative space there would be a trace on the right hand side and the cyclic invariance of this trace would enable us to rewrite it in terms of the matter current J µ as Tr (F νµ J µ ) (i.e. we have a continuum version of the non-Abelian Lorentz-force equation). In the present case, however, all we can do is add and subtract the missing terms to arrive at In commutative space, the second term would vanish under the trace. In Moyal space, the cyclic invariance is only present under the integral d 4 x (which in fact corresponds to a trace). However, integrating the above equation is not very helpful, since d 4 xD µ T µν = 0 for the left hand side, i.e. it is a surface term and we would not get any new information.
In fact, as argued concerning equation (38), the second term on the right hand side of (49) also vanishes upon integration with d 3 x in Minkowskian Moyal space with θ 0i = 0: we will come back to this point after adding the gauge field action. Another observation is that, just like the Lagrangian density, the EMT is not gauge invariant for the same reason, i.e. lack of a trace (hence of the cyclic invariance). Instead, T µν transforms covariantly (as did its free gauge field counterpart discussed in the previous section): Addition of the gauge field action: If we add to the action (43) a kinetic term for the gauge field so as to obtain the total action then the equation of motion of A µ represents the non-commutative version of Maxwell's equations, where the expression of J ν in terms of φ and φ * is given by equation (45). From (52) and the argumentation in equation (34) it follows that the current J µ is covariantly conserved. Again, all fields and their covariant derivatives transform covariantly under gauge variations. The associated Einstein-Hilbert EMT in flat Moyal space is a sum of expressions (35) and (47), and using (49), (52) we obtain In commutative space, there would be a trace on the right hand side so that this expression would be zero and T µν would be conserved. (In fact, in that case we would also have a trace on the r.h.s. of (53), and D µ T µν tot = ∂ µ T µν tot .) In the present case, however, we are once more lacking a trace to get rid of the r.h.s. In this respect we emphasize that according to the non-commutative generalization of Noether's theorem (see [46] and references therein), a continuous symmetry of the action does not generally imply a standard local conservation law for interacting theories: additional "source" terms (star-commutator terms which ultimately vanish under the space-time integral) generally appear. Actually, integration of (54) with d 4 x yields trivially zero on both sides (since integration corresponds to a trace). In the Minkowskian version of Moyal space with non-commutativity parameters satisfying θ 0i = 0, it suffices to integrate over d 3 x to render the r.h.s. zero. In this case we have which means that the four-momentum P ν ≡ d 3 x T 0ν tot is a conserved quantity (which is also gauge invariant by virtue of (36), (50)). Of course the different contributions of P ν ≡ P ν φ +P ν A are not conserved: Restoring gauge invariance: In the present setting, we may follow the same strategy as in Section 3 and define an EMTT µν which is gauge invariant in analogy to expression (40): For its divergence one obtains The first term can be taken care of in the same way as in equation (42), but not so the second term. Thus, we are stuck with a small (θ-dependent) breaking of ∂ µT µν , which of course vanishes in the commutative limit.
In Ref. [16] a redefinition of the EMT for a φ 4 theory which restored its conservation was discussed. However, in the present context the same strategy would destroy gauge covariance of T µν tot making the construction of its gauge invariant counterpart via Wilson line impossible, as we will now show.
Since the additional terms on the r.h.s. of (54) are star-commutators it is generally possible to pull out one derivative by making use of the so-called -product (introduced in references [44,45]), so that we may write Thus, a shift in the EMT restoring its conservation can be made in principle, but at the cost of destroying its gauge covariance, i.e.
where a ∈ [0, 1] is a free real parameter. Similarly, a redefinition achieving ∂ µ T µν tot = 0 could be made, but it would not gain us anything with respect to gauge invariance.
Thus, the best one can do is the gauge invariant expression (57) above with its modified conservation law (58). On the operator level, this means that only the trace over the divergence of the EMT (which is an operator in quantized space) is conserved, a fact which is obscured by the star-product prescription where the trace becomes an integral over space(-time). Therefore, it is not surprising that we have the local equation ∂ µT µν = 0, as was already observed in the case of the φ 4 -theory in reference [15]. We note that d 4 y∂ y µT µν tot (y) = 0 = d 4 x∂ x µ T µν tot (x) as can be checked explicitly by using the cyclic properties of the star product under the integral, as well as d 4 k exp(iky) = (2π) 4 δ 4 (k) and W (0, x) = 1.
In the next section, we show that one finds similar results when coupling to fermions instead of (or in addition to) scalars.

Fermions
The action for a fermionic (Dirac) field coupled to an external gauge field in Moyal space is given by and D µψ ≡ ∂ µψ − ig A µ ,ψ , and where the square matrices γ µ fulfill the Clifford algebra relation {γ µ , γ ν } = 2δ µν 1. The components ψ α of the spinor field ψ are considered to be anticommuting (i.e. Grassmann) variables.
Under the gauge transformations which imply the Lagrangian density L of the model transforms as δ λ L = −ig [L , λ], hence the action is gauge invariant. The equations of motion for ψ andψ read and the fermionic matter current is given by where α, β denote the spinor indices. This current is covariantly conserved due to the equations of motion above (i.e. D µ J µ = 0) and it transforms covariantly under gauge transformations: The action (62) may also be written in the following more symmetric form by virtue of an integration by parts: In analogy to the commutative case we obtain the EMT for the Dirac fields, 7 where the second line holds on-shell due to the equations of motion. Note that this tensor is hermitian as well as traceless on-shell. Furthermore, it represents a sum of products of bilinear quantities which transform covariantly, hence it also transforms covariantly under the gauge transformations (63), as can be checked straightforwardly.
Using the equations of motion, the covariant divergence of this EMT can be evaluated, and we again have additional terms which would vanish under a trace in Yang-Mills theory, resp. an integral in Moyal space. It is interesting to note that similar additional terms were found in the context of matrix models 8 (cf. Appendix A.3 of reference [50]). Once combined with the gauge field EMT as discussed in the previous subsection, one may again define the gauge invariant counterpart of the total EMT via Wilson line as in equation (57). Once more ∂ µT µν tot involves breaking terms which depend on the non-commutativity parameters θ µν and which cannot be absorbed into a redefined EMT in a gauge invariant way.

Coupling to matter: a simpler version
In this section, we wish to present an alternative way of coupling gauge fields to scalars and fermions for which the EMT is gauge invariant so that one does not have to resort to Wilson lines. Nonetheless we will not have the standard local conservation law for this EMT and the latter cannot be achieved while maintaining gauge invariance.

Complex scalar field
A complex scalar field in Moyal space can also be coupled to a gauge field via the covariant derivativeD µ φ ≡ ∂ µ φ − igA µ φ and its hermitean conjugate expression, i.e.
(where the latter derivative amounts to an action of ∂ µ + igA µ from the right). Thus, by contrast to the coupling discussed in the previous section the basic covariant derivatives presently do not involve a commutator of fields. For a given external gauge field A, the action is now defined by and the associated equations of motion read The Lagrangian density L in the action integral (72) is invariant under the gauge transformations which imply the transformation laws The matter current J µ transforms covariantly, δ λ J µ = −ig [J µ , λ], and is covariantly conserved by virtue of the equations of motion of φ and φ * : The transformation property of J µ also leads to δ λ (D µ J µ ) = −ig[D µ J µ , λ] = 0 which shows that the conservation law (77) is gauge invariant as well, and thus provides a consistency check.
Let us now turn to the EMT of this model: We note that we did not explicitly write any anticommutator in the action (72) in contrast to the scalar field model discussed in the previous section (where we had only one type of covariant derivative involving a commutator). This is a choice that we may make (because the anticommutator can be dropped under the integral in the action) and the motivation for the present choice is that the Lagrangian density and the EMT are now gauge invariant (in contrast to our previous scalar field model), as follows readily from the transformation laws (76). Finally, using and the covariant divergence of the EMT (78) can be evaluated: From the last term we see that a simpler result is obtained by considering only the divergence 9 , Once more, we have non-vanishing commutator terms due to a missing trace/integral (typical of non-commutative space) even in this simpler scalar field model. By adding the gauge field action and integrating with d 3 x in Minkowskian Moyal space with θ 0i = 0, we again find a conserved and gauge invariant four-momentum (P ν ) by virtue of equations (55) and (79).
The advantage of the current simpler formulation is that no Wilson line construction is necessary to define a gauge invariant EMT, but in the present case the local conservation law of T µν is still broken by θ-dependent terms. As mentioned in the previous section, in Ref. [16] a trick was used to eliminate similar (commutator) terms from ∂ µ T µν in φ 4 -theory in Moyal space (but without coupling to a gauge field). In that construction, one needs additionally the -product (59). We presently have and the expression with a ∈ [0, 1] a free parameter, is a locally conserved quantity. But the latter is no longer gauge invariant, nor is it symmetric.
The equations of motion of the present model read and the fermionic matter current is given by It is covariantly conserved by virtue of the equations of motion (89), i.e. D µ J µ ≡ ∂ µ J µ − ig [A µ , J µ ] = 0. Furthermore, it transforms covariantly under gauge transformations, δ λ J µ = −ig [J µ , λ]. Thus, we also have δ λ (D µ J µ ) = 0. The (on-shell) expression for the EMT of fermion fields reads It is traceless on-shell and (just like the Lagrangian density) it is invariant under the gauge transformations (87). Thus, no Wilson line construction is needed here either. The divergence of this EMT can be determined by using the equations of motion: Once again, there is an additional term which would vanish under a trace, resp. an integral.
Since the fermions are Grassmann variables, the additional term may be written in terms of a -product as This allows for a redefinition of T µν tot which is conserved but not gauge invariant.

Conclusion
According to the non-commutative generalization of Noether's theorem [46], some extra θ-dependent terms ("source"/star-commutator terms) generally appear in the local conservation law for the EMT T µν for interacting theories. In the present paper, we have shown (for complex scalars as well as for fermions coupled to gauge fields) that the standard local conservation law of the EMT T µν is always modified due to non-commutative effects and that T µν can always be redefined so as to be conserved, but that the so defined EMT is not gauge invariant. (Yet, for dynamical matter and gauge fields we always have a conserved and gauge invariant four-momentum with components P ν = d 3 x T 0ν .) More specifically, we discussed two possible couplings of scalars and fermions to gauge fields: one where the basic EMT transforms covariantly and its gauge invariant counterpart could be constructed by using the non-commutative generalization of a Wilson line, and one where the EMT is gauge invariant. For the latter case, we found that the consideration of the -product allows to achieve the standard local conservation law for the EMT, but at the expense of losing gauge invariance (and symmetry). We note that the tools employed here are also those which are generally considered for the quantization, e.g. see references [41,44,53].