Field Theory with Coordinate Dependent Noncommutativity

We discuss the formulation of classical field theoretical models on $n$-dimensional noncommutative space-time defined by a generic associative star product. A simple procedure for deriving conservation laws is presented and applied to field theories in noncommutative space-time to obtain local conservation laws (for the electric charge and for the energy-momentum tensor of free fields) and more generally an energy-momentum balance equation for interacting fields. For free field models an analogy with the damped harmonic oscillator in classical mechanics is pointed out, which allows us to get a physical understanding for the obtained conservation laws. To conclude, the formulation of field theories on curved noncommutative space is addressed.


Introduction
Over the last twenty years a great amount of work has been devoted to the study of structural aspects and phenomenological applications of field theories on the simplest quantized space, namely the Groenewold-Moyal (or θ-deformed) space [1,2], e.g. see [3][4][5][6][7][8][9][10][11][12] and references therein for a review. On this space the theories are formulated in terms of ordinary functions by means of a deformed associative product, the so-called Groenewold-Moyal star product Here, the noncommutativity parameters θ µν = −θ νµ are real constants and expression (1.1) implies that the space-time coordinates x µ fulfill a Heisenberg-type algebra, One refers to this case as the canonical deformation. Quite generally, the interest in this and more general deformed spaces was triggered by their link with quantum gravity, quantum geometry, string theory and D-branes, matrix models, the quantum Hall effect as well as other physical systems (e.g. see [3][4][5][6][7][8][9][10][11][12][13][14] and references therein).
In the present work, we are interested in classical field theories defined on a space-time for which the noncommutativity parameters θ µν appearing in the algebra (1.2) are space-time dependent. These models, which appear to be more natural from the point of view of gravity and which may have some interesting physical applications, have been much less investigated in the literature 1 . Our work is based to a large extent on the mathematical results [23,24] obtained by V. G. Kupriyanov (and his collaborators) and applied mostly in the context of quantum mechanics on noncommutative space [25][26][27][28][29]. We should also mention the special instances where the noncommutativity parameters are linear in the coordinates (the so-called linear or Lie algebra case which is related to fuzzy spaces and κ-deformations) and the case where they are quadratic in the coordinates which is related to quantum groups, both cases having been the subject of various studies in the literature -see the reviews mentioned above as well as [29].
A basic issue in the construction of Lagrangian models in classical field theory consists in the study of the underlying symmetries and of the conservation laws in differential or integral form. These questions which are related to Noether's first theorem have been investigated in the case of constant noncommutativity parameters by numerous authors, e.g. see [30][31][32][33][34] and references therein. Already for this simplest instance of a noncommutative space, technical complications arise, in particular for the energy-momentum tensor in gauge field theories -see reference [34]. The main concern of the present work is to address these questions in the simplest possible manner for a space-time with x-dependent noncommutativity, both for free and interacting classical field theories. Thus, we formulate a simple and general procedure for deriving local conservation laws (or balance equations) which is based on the equations of motion and which follows the familiar line of reasoning in classical or quantum mechanics. For bosonic and fermionic matter fields, we then derive the local form of charge conservation laws and of energy-momentum conservation laws for free fields as well the energy-momentum balance equation for interacting fields. For those models which have already been discussed in the context of quantum mechanics [28] or of field theory on a space with a noncommutativity of Lie algebra-type [29], we recover the same results.
The text is organized as follows. In Section 2, we present the set-up of noncommutative spacetime defined by a generic associative star product as well as the characteristics of the latter. In Section 3, we discuss the formulation of classical field theory on such a space and we point out an analogy of the free field models with the damped harmonic oscillator in classical mechanics. In Section 4, we consider the derivation of conservation laws in non-relativistic classical and quantum mechanics as a motivation for a simple derivation of local conservation laws in relativistic field theory. This procedure (relying on the equations of motion) as well as the properties of the star product then allow for a straightforward derivation of local conservation laws in noncommutative space-time (along the lines followed by V. G. Kupriyanov in noncommutative quantum mechanics). In Section 6, we address the formulation of classical field theories on curved noncommutative space-time by generalizing the description considered in flat space. More precisely, we follow the star product approach to gravity which was recently put forward by M. Dobrski [35] and which appears to fit nicely into the framework discussed in flat space-time. To conclude, we comment on the quantum theory in Section 7. The aim of the appendices is to complement different parts of the text. In Appendix A, we outline the passage from the operatorial approach to the star product formulation as well as the construction of a closed star product. We have deferred to Appendix B several mathematical comments on star products and on the deformed Leibniz rule, as well as on two-dimensional space-time. Appendix C summarizes the Lagrangian and Hamiltonian formulations of the damped harmonic oscillator in view of its similarities with free field theoretical models on noncommutative space-time.
In the present paper we do not follow the operatorial approach based on the commutation relations (2.1), but rather the equivalent star product approach which we will outline in the sequel (see references [38][39][40][41] for the pioneering work, references [42,43] for an introduction to the general theory, and references [16-18, 23, 24] for an explicit and constructive approach). However, the operatorial approach is quite useful for describing some basic aspects and for a simple construction of the star product as we outline in Appendix A.
Star product approach: An equivalent description of the deformed space above can be given by passing (by virtue of an isomorphic map) from the noncommutative algebra generated by the operatorsX µ to the commutative, associative algebra C ∞ (R n ) of smooth functions (depending on real variables x µ ) equipped with an additional so-called star product which is associative, but noncommutative: The star-commutator is then given by where the real-valued function θ µν with θ µν = −θ νµ is the afore-mentioned symbol of the operator θ µν (X) which satisfies the Jacobi identity (2.2). Relation (2.3) for the variables x µ is satisfied if the ordinary product f g (of any two smooth functions f, g of the variables x µ ) is deformed according to As is customary in the theory of formal star products or, more generally, in Gerstenhaber's theory of algebraic deformations [42], a constant real formal deformation parameter h can be factored out of θ µν by writing θ µν = hΘ µν so that expression (2.4) may be viewed as a formal power series in h. By a slight abuse of terminology, we refer to the order of h in the series expansion as the order of θ.
The correspondence between the operator and star product formulations is such that we have (in terms of the notationŴ [f ] ≡f (X) for the operators) where • denotes the product of operators. The operatorŴ [f ] ≡f (X) acts on the unit function 1 according tof (X) 1 = f , hence relation (2.6) implies that its action on a smooth function g is given byf According to this equation, an explicit expression for f g can be determined perturbatively [23] by looking for an expansion ofX µ (and thereby of the Weyl ordered functionf (X)) as a differential polynomial in the standard, commuting position operators X µ and in ∂ ν (see Appendix A): more precisely, one may easily check that the expansionX µ = X µ + i 2 θ µν ∂ ν + O(θ 2 ) (which is familiar from the definition of the Groenewold-Moyal product) satisfies the commutation relation (2.1) and that equation (2.7) then yields the expansion (2.4) to first order in θ. The expression that one finds [23] for the star product to second order in θ is given by This result (which is also referred to as the Weyl star product [29]) coincides with the explicit expression which has been given for R n by Kontsevich up to second order [41] and which is generally referred to as the Kontsevich formula in the literature (e.g. see reference [43] for a nice introductory review). For constant noncommutativity parameters, the star product (2.8) reduces to the Groenewold-Moyal star product at the given order. The first three terms may be referred to as the "Groenewold-Moyal-like part" [44] since they have the same form as the Groenewold-Moyal star product up to this order. Quite generally the terms of even order in θ are symmetric with respect to the exchange of the functions f and g while the odd order terms are antisymmetric [23,29]. The series (2.8) can be interpreted physically as the perturbative expansion of the path integral for a non-linear sigma model for the world-sheet description of bosonic strings [4,[44][45][46].
Closed star product: For the formulation of quantum mechanics or field theory we are interested in so-called closed star products [47], i.e. star products for which the integral of a function f over R n is the trace of the operatorf (X): Here, the standard integration measure d n x has been modified 2 by a weight factor w so as to ensure the cyclicity of the trace, i.e. the validity of the so-called closedness or closure relation More precisely, substitution of the expansion (2.8) into relation (2.10) and the assumption that f and g are smooth functions of fast decay at infinity with respect to the considered integration measure (so that all boundary terms in the integral vanish) yield Since this condition has to hold for arbitrary functions f, g, we obtain the condition This relation means that the tensor (θ µν ) is divergenceless with respect to the integration measure w d n x. (We note that for a curved n-brane in a flat background space-time, the divergence condition (2.12) admits the physical interpretation of a Born-Infeld equation of motion on the brane [4].) Thus, for non-constant functions θ µν , the factor w cannot be chosen to be a constant. Given the noncommutativity parameters θ µν (x), this partial differential equation determines the function w in terms of θ µν . By using the usual formula for the infinitesimal variation of the determinant of a matrix as well as the Jacobi identity (2.2), one can readily check that a solution [18,24,28] of Eqn. (2.12) is given as follows if the matrix (θ µν (x)) is invertible for all x ∈ R n (which requires n to be even): In this respect, we recall that the determinant of an invertible, real, antisymmetric matrix Θ ≡ (θ µν ) of even order has a strictly positive determinant; in fact [37], this determinant is the square of the so-called Pfaffian of the matrix, det Θ = (Pf Θ) 2 . If the rank of the matrix (θ µν ) is not even and maximal for all x ∈ R n , then one has to look for the corresponding solutions of Eqn. (2.12) (e.g. see reference [20] for the two-dimensional case). In the following we will assume that the matrix (θ µν ) is invertible whenever needed, the weight function w being then given by expression (2.13). The quadratic terms in the closure relation (2.11) are discussed in Appendix A and we will address them in equations (2.15), (2.16) below.

Some mathematical remarks:
The antisymmetric tensor θ µν satisfying the Jacobi identity represents a Poisson tensor (also referred to as Poisson bivector field [42]) and relation (2.5) identifies the star-commutator as a deformation of the Poisson bracket. If we assume that the matrix (θ µν ) is invertible at all points, then it admits an inverse matrix (ω µν ) ≡ (θ µν ) −1 . The latter matrix is non-degenerate, antisymmetric and, by virtue of the Jacobi identity for θ µν , its components satisfy the relation 0 = ∂ ρ ω µν + cyclic permutations of the indices .
Thus ω ≡ 1 2 ω µν dx µ ∧dx ν is a symplectic two-form and we have w = det (ω µν ). We note that the latter factor is reminiscent of the density √ g ≡ det (g µν ) which appears for the integration on a Riemannian manifold with metric tensor (g µν ). Indeed, the canonical volume form (or so-called Liouville measure) on a symplectic manifold of dimension n = 2m reads [38,40,42] dV where we did not spell out the exterior product symbols. For a symplectic manifold M , the integral M d n x w f of a function f : M → C with respect to the Liouville measure is also qualified as a Poisson trace [42]. Concerning our considerations on star products, we emphasize that Minkowski space R n is considered (for n even) as a symplectic manifold or as a Poisson manifold rather than a symplectic vector space [49] or a Poisson algebra [50] since the tensor (θ µν ) is not constant. Thus, in general (ω µν ) and (θ µν ) cannot be cast into canonical form (Darboux or canonical coordinates) by a linear change of coordinates: This contrasts the case of constant deformation parameters (Groenewold-Moyal star product). As for the star products, they are to be viewed as non-trivial associative deformations of the associative algebra C ∞ (R n ) [42]. For the case of a general Poisson manifold M (generalizing M = R n ), relation (2.10) is known as the generalized Connes-Flato-Sternheimer conjecture and was proven in reference [48] for any Poisson tensor which is divergenceless.
Explicit expression of closed star product: Due to the non-trivial weight factor w which has to be present in equation (2.10), the terms of order θ 2 in the Kontsevich star product (2.8) do not satisfy the closure relation. However, as pointed out in reference [24] and as outlined in Appendix A, one can pass over to a gauge equivalent [41] star product which satisfies this relation. The corresponding gauge transformation of the Weyl star product is given by and it readily leads to the following expression to order θ 2 of a closed star product [24] (in which we dropped the prime on ): (2. 16) In summary, for a given antisymmetric tensor field (θ µν ) satisfying the Jacobi identity (2.2), the star product (2.16) satisfies the closure relation (2.10) with an integration measure w d n x with respect to which the Poisson tensor (θ µν ) is divergenceless. By construction, the expressions w f g and w f g only differ by a total derivative so that equality (2.10) holds. For later reference, we spell out the explicit expression of this derivative [28] (for which the Jacobi identity has again been used): In the case of constant noncommutativity parameters θ µν , the star operation is not x-dependent in the sense that the Leibniz rule ∂ µ (f g) = ∂ µ f g +f ∂ µ g holds. This is no longer true for nonconstant functions θ µν where one has a modified rule for the differentiation, see equations (3.4)-(3.5) below. It is worthwhile to note the behavior of the star product of complex-valued functions under the operation of complex conjugation [28]: This relation can be explicitly checked for the expansion (2.16), the verification for the last term making use of the Jacobi identity and of relation (2.12) which is satisfied by the function w. We refer to Appendix B for some mathematical comments on the potential relationship between the star products considered here and the approach of A. Connes to noncommutative geometry.

Some particular cases:
As pointed out in reference [23], some simplifications in the perturbative expansions underlying the star products occur for the special case of linear Poisson structures, i.e. for a commutator algebra of Lie algebra type, [X µ ,X ν ] = i f µν ρX ρ . The latter instance (and in particular the case of the Lie algebra su(2) where [X i ,X j ] = i ε ijkX k and where the weight function w can be chosen to be constant, w ≡ 1) has been studied in detail in reference [29] (see also [51] for some elaborations and applications). Remarkably, some closed expressions can then be determined in the case of su(2) (and even for more general Lie algebras) for the operatorsX i , for the star product as well as for the gauge transformation relating the Weyl star product and the closed star product. Moreover, the latter product can be identified [29] with the one following from the so-called Duflo quantization map for su(2) which appears to be [29] the mathematically preferred quantization in this context (see also reference [52]).
We will briefly come back to the su(2) case in the concluding Section 7 with some comments on the properties of quantum field theories on such spaces. The case of two space-time dimensions, which also represents an instance of particular interest, is commented upon in Appendix B.

Field theory on noncommutative space
Generalities: In field theory the action functionals are generally expressed in terms of an L 2 -type scalar product ·, · of complex-valued fields f, g. In the present context, this scalar product involves the weight factor w = 1/ det (θ µν ) (the latter being strictly positive with the assumption that the matrix (θ µν (x)) is non-singular for all x ∈ R n ): Any derivatives appearing in the Lagrangian should represent anti-Hermitian operators with respect to this scalar product, very much as the momentum operator in relativistic quantum mechanics should be Hermitian with respect to the scalar product of wave functions f, g. As remarked by Kupriyanov [28] following the works [25,53,54], this implies that the momentum operator in relativistic quantum mechanics is no longer given by p µ ≡ −i∂ µ , but presently involves an additional term,p µ ≡ −i∂ µ − i∂ µ (ln w 1/2 ), so as to have f,p µ g = p µ f, g for all wave functions f, g belonging to the domain of definition of the operator 3p µ . We note that and f, D µ g = − D µ f, g for smooth functions f, g. Here, the derivative D µ has the form of a covariant derivative in Abelian gauge theory with the particular gauge potential Since the latter is pure gauge, the covariant derivatives commute with each other (i.e. [D µ , D ν ] = 0) and they are related by a gauge transformation to the ordinary derivatives (namely D µ = w −1/2 • ∂ µ • w 1/2 ). Actually, the fact that the coefficient w = 1/ det (θ µν ) = det (ω µν ) resembles the metric coefficient |g| ≡ | det (g µν )| 1/2 in general relativity, indicates that the gauge potential (3.3) may be viewed as the analogue of the contraction Γ ν µν = ∂ µ (ln |g|) of the Christoffel symbols Γ λ µν which appear in the covariant derivatives on a pseudo-Riemannian manifold (equipped with the Levi-Civita connection).
with ω( p ) ≡ p 2 + m 2 represents the Lorentz invariant integration measure over the mass hyperboloid p 2 = m 2 . The usual expression for the position operator in momentum space, i.e. X ≡ i ∂ p ≡ i ∂/∂ p, is not hermitian with respect to the given scalar product, but the Newton-Wigner position operator is and readŝ Before considering some field theoretical models, we mention a general identity [28] for star products involving derivatives D µ which is useful for the derivation of conservation laws in field theory. By differentiating Eqn. (2.17) with respect to x µ we obtain If we now apply Eqn. (2.17) to the first contribution on the right hand side, we conclude that where the last term is given by Here, the Jacobi identity (2.2) was again taken into account for deriving the expression in the last line. The derivative (3.4) can also be expanded by using the Leibniz rule for partial derivatives: Comparison of this result with expression (3.2), i.e. D µ f ≡ ∂ µ + 1 2 ∂ µ (ln w) f , then suggests to attribute a weight (or degree or charge) 1/2 to a function f and a weight 1 to the star product of two such functions, so that (3.7) represents the covariant derivative of f g. With this notation, relation (3.5) reads i.e. the Leibniz rule for covariant derivatives with a correction term ∂ ρ b ρ µ (f, g). Some mathematical considerations concerning the deformed Leibniz rule (3.8) are presented in Appendix B. Here, we only note that some alternative approaches are based on the introduction of derivatives D µ which satisfy the Leibniz rule, but which do not commute with each other and which generally represent infinite power series in the elements of the matrix (θ µν ) and of its inverse, e.g. see reference [20]. The Leibniz rule can also be imposed within a more abstract approach based on Hopf algebras and Drinfeld twists or by following an approach based on L ∞ -algebras, i.e. generalized DGLA's (differential graded Lie algebras) with a 'mild' violation of associativity, see reference [56].
Lagrangian models: With the ingredients introduced above, field theoretical Lagrangian models on noncommutative space with a given θ-tensor can now be defined by starting with models on ordinary space, replacing the integration measure R n d n x by R n d n x w, ordinary derivatives ∂ µ by the covariant derivative D µ and ordinary products by star products. For a complex scalar field φ with a quadratic self-interaction we thus obtain the action functional [24,29] where we dropped one star in each term in accordance with the general property (2.10). For the Dirac spinor ψ coupled to an external U (1)-gauge field (A µ ), we have the action [28] For a U (1)-gauge field (A µ ) in four dimensions, the action functional reads For an action functional S depending on a bosonic field ϕ (i.e. a scalar field φ or a gauge field (A µ )), the functional derivative is defined by The components of a Dirac spinor ψ are supposed to be anticommuting variables and the corresponding functional derivatives are defined by The equations of motion associated to the action functionals (3.9)-(3.11) are respectively given by the vanishing of the following functional derivatives: Concerning the Dirac field action (3.10), we recall that a classically equivalent (though nonreal) expression for the kinetic term can be obtained by partial integration and presently reads Under an infinitesimal gauge transformation parametrized by a function x → α(x), the U (1)gauge field (A µ ) transforms with the gauge covariant derivative, By virtue of relation (3.8), this induces the following transformation law of the field strength tensor F µν : This variation leaves the gauge field action (3.11) invariant. Concerning the field strength we remark that the modified Leibniz rule (3.8) for the derivatives D µ implies a modified Bianchi identity: Since the Bianchi identity for the field strength generally reflects the Jacobi identity for the gauge covariant derivatives, the right hand side of Eqn. (3.13) (which expresses the deviation from the Jacobi identity for star-commutators of gauge covariant derivatives) may be qualified as 'gauge' Jacobiator [42,58] for these commutators. The noncommutative U (1)-gauge theory as well as the Chern-Simons theory in three dimensions have been discussed in terms of L ∞ -algebras in reference [56].
Limiting cases: If the noncommutativity parameters θ µν are constant, then the weight factor is constant too. Then, it can be factored out of the integrals (3.9), (3.10), (3.11) and simply be dropped: thus we recover the canonical deformation case described by the Groenewold-Moyal star product. The commutative theory is obtained from the latter case (which does not involve w anymore) by letting the constant parameters θ µν go to zero.
Free field models: Let us have a closer look at the free scalar and Dirac field actions. From the definition (3.2) of the covariant derivative, it follows that Thus, contrary to the case of constant noncommutativity parameters, the free field actions presently do not have the same form as in commutative space: they only do so after having expressed these actions in terms of the θ-dependent fields Φ, Ψ. In terms of the original fields φ, ψ, the free field equations have the form These equations have a formal analogy with the equation of motion for a damped harmonic oscillator in non-relativistic mechanics (see Appendix C): The latter equation reads and can be rewritten in an undamped form,Q + Ω 2 Q = 0, by virtue of the redefinition Q ≡ √ wq which is analogous to the redefinition (3.14) of fields. In field theory, the time variable t becomes the space-time variable x and the function w(t) ≡ e γt becomes w(x) ≡ 1/ det θ µν (x) . This analogy is of interest in view of the fact that the damped harmonic oscillator is known to possess conserved charges involving an explicit time-dependence, see Appendix C. The generalization of the latter result to the energy-momentum tensor in field theory will be discussed in Section 4.2.
Symmetries: If the field θ µν transforms under Poincaré transformations x µ → x µ = Λ µ ν x ν + a µ as a classical relativistic field, i.e. as then the weight function w = 1/ det (θ µν ) is Poincaré invariant and the derivative D µ transforms covariantly. The given action functionals are then Poincaré invariant (as noted in reference [28] for the case of a Dirac field).
The action (3.9) for a scalar field of charge e is invariant under global U (1)-gauge transformations whose infinitesimal form is given by where ε is a constant real parameter. For the Dirac field action (3.10), we also have such an U (1)-invariance for the fields ψ,ψ. By virtue of Noether's first theorem, one thus expects the existence of locally conserved current densities associated to the global U (1)-invariance of the models describing the scalar and Dirac fields: the corresponding expressions will be derived in Section 5, see equations (5.1) and (5.4) below.
Due to the coupling of fields to the x-dependent external fields θ µν , one does not expect the energy-momentum tensor of matter or gauge fields to be locally conserved, even for the case of free scalar or Dirac fields. However, the analogy of the corresponding free field models with the damped harmonic oscillator in mechanics and the existence of a conserved charge for the latter dynamical system indicate that a local conservation law also holds for these models on noncommutative space-time. The corresponding expression will be derived in Section 5 along with the energy-momentum balance equation which holds for interacting fields.

Simple derivation of conservation laws
Before deriving conservation laws for the field theoretical models on noncommutative space-time discussed above, we outline a simple derivation of conservation laws in Minkowski space which can be generalized to noncommutative space. We proceed as in non-relativistic mechanics (where the conservation of energy is obtained by multiplying the equation of motion by d x/dt) or in nonrelativistic quantum mechanics (where the continuity equation for the probability current density is obtained by multiplying the wave equation for ψ by ψ * and then subtracting the complex conjugate expression). The application of this procedure in relativistic field theory amounts to a simple derivation of Noether's first theorem in this setting.

General procedure
General procedure in relativistic field theory: Consider a collection ϕ ≡ (ϕ r ) r=1,...,N of classical relativistic fields in Minkowski space and suppose that their dynamics is described by an action functional S[ϕ] ≡ d n x L(ϕ, ∂ µ ϕ, x) which involves a Lagrangian density L which may explicitly depend on the space-time coordinates x. The associated equations of motion are given by δS/δϕ = 0 where the functional derivative has the following form if the Lagrangian depends at most on the first order partial derivatives of ϕ: Let us now consider the case of an x-independent Lagrangian density and some active symmetry transformations δϕ(x) ≡ ϕ (x)−ϕ(x) which depend continuously on one, or several, real constant symmetry parameters. We suppose that the Lagrangian density is quasi-invariant under these transformations ("divergence symmetry"), i.e. δL = ∂ µ Ω µ for some (possibly vanishing) vector field (Ω µ ) depending on ϕ. Let us multiply the functional derivative (4.1) by δϕ and apply the Leibniz rule to the partial derivative term: Thus we have i.e. the general form of Noether's first theorem in relativistic field theory. This suggests a similar procedure to be followed in the next section for noncommutative space: we multiply the functional derivative by an appropriate variation of fields and then express the product as a total derivative ∂ µ j µ , hence this derivative vanishes for the solutions of the equations of motion.
After factorizing the symmetry parameters, j µ = a ν T µν can , we conclude that the local conservation law for the canonical energy-momentum tensor (EMT) T µν can holds for all solutions of the equations of motion: If the Lagrangian density L depends explicitly on the space-time coordinates x, then the last term in Eqn. (4.2) reads a ν (∂ ν L − ∂ ν expl L) so that we obtain the energy-momentum balance equation where ∂ ν expl L reflects the explicit x-dependence of L. A simple illustration which is relevant for the coupling of matter fields to a given symplectic structure (that we address in the next subsection) is given by the linear coupling of a real scalar field φ to a fixed x-dependent external source J, i.e. the Lagrangian density The energy-momentum balance equation (4.5) then reads Another illustration of the general procedure (4.2)-(4.3) is given by internal symmetries for charged fields, e.g. for ϕ = (φ, φ * ) where φ represents a complex scalar field. The real-valued Lagrangian density is invariant under internal symmetry transformations labeled by a constant real parameter ε (and the electric charge e of the field φ), In this case, the procedure (4.2)-(4.3) yields the off-shell identity The local conservation law ∂ µ j µ = 0 (which holds for any solution of the equation of motion) now expresses the conservation of electric charge.

Scalar field coupled to a symplectic structure
It is instructive to apply the general procedure outlined above to the action for a self-interacting complex scalar field coupled to a given symplectic structure (ω µν ), this action being given by expression (3.9) without the star products: We presently multiply the functional derivatives δS ω /δφ and δS ω /δφ * by the covariant derivatives of fields and apply the Leibniz rule for these derivatives as well as the relation [D µ , D ν ] = 0: For the covariant derivatives, we have to keep in mind the weight of the fields on which they act, e.g. φ, φ * have weight 1/2, φφ * has weight 1 and (φφ * ) 2 has weight 2, hence D µ (φφ * ) 2 = ∂ µ (φφ * ) 2 + 2∂ µ (ln w) (φφ * ) 2 . From this fact we deduce that equation (4.9) reads as follows for the solutions of the equations of motion (for which the left hand side of (4.9) vanishes): This result means that the EMT T µν ω of matter fields is conserved for c = 0, i.e. in the absence of a self-interaction, though it is not in the presence of the latter. This result can be traced back to the fact that the matter fields are coupled to a fixed external field (ω µν ) (by means of the variable w = det (ω µν )) and is reminiscent of the coupling of a scalar field to an external scalar source described above, see Eqn. (4.6).
The previous conclusions can be further elucidated by rescaling the matter fields as we did in equation (3.14): with Φ ≡ √ w φ and Φ * ≡ √ w φ * , the action (4.8) reads i.e. the external field w now only appears in the last term of the Lagrangian 5 . In terms of Φ and Φ * , the EMT (4.10) reads and one can readily verify the energy-momentum balance equation (4.10) by using the equation of motion 0 = ( +m 2 )Φ+c 1 w Φ * Φ 2 and its complex conjugate. For c = 0, we have the analogy with the damped harmonic oscillator in classical mechanics pointed out in equations (3.14)-(3.16). The expression (C.12) of the conserved charge for the latter dynamical system then allows us to get a physical understanding of the local conservation law ∂ µ T µν ω = 0 for matter fields which holds for c = 0 despite their coupling to an external field w: in the course of the temporal evolution of matter fields φ, the presence of w in T µν ω is compensated by the dependence on w of the solutions φ of field equations, thus ensuing the existence of conserved charges R n−1 d n−1 x T 0ν ω for ν ∈ {0, 1, . . . , n − 1}. 5 We will come back to the factor 1/w (and the introduction or more general functions of w) at the end of Section 6.1.

Conservation laws for noncommutative field theories
In the previous section, we presented a derivation of the local conservation laws for the charge and for the energy-momentum which only relies on the equations of motion. The application of this procedure for Lagrangian field theories on noncommutative space simply consists of replacing the multiplication by a field ϕ (or by its derivative ∂ ν ϕ) by the star product with this field (or by its covariant derivative D ν ϕ). Furthermore, for the differentiation of star products, one has to consider the product rule (3.5) which applies to this kind of products. (We note that Noether's first theorem for constant noncommutativity parameters has been discussed by numerous authors, e.g. see [30][31][32][33][34] and references therein.)

Charge conservation law
For the complex scalar field described by the action (3.9) and the associated functional derivatives (3.12a), the procedure (4.7) (generalized to the noncommutative setting [29]) yields upon application of the product rule (3.5) We note that by symmetrizing these expressions with respect to φ and φ * , we can eliminate the terms of b µ ν which are linear in θ since b µ ν (f, g) is antisymmetric in f, g at this order: In the particular case where n = 3 and where the noncommutativity is given by θ ij = ε ijk x k with i, j, k ∈ {1, 2, 3} (in which case one can choose w ≡ 1 [29]), the result (5.1) coincides, upon exchange φ ↔ φ * , with the one derived in reference [29]. For the Dirac field described by the action (3.10) and the associated functional derivatives (3.12b), the same procedure gives (in agreement with reference [28]) Once more, the terms in b µ ν which are of order θ can be eliminated by symmetrization. These conservation laws (which hold for all solutions of the equations of motion) reflect the invariance of the underlying models under the global U (1)-gauge transformations (3.18).

Energy-momentum conservation law
Free field case: We start with the case of a free complex scalar field φ described by the action (3.9) with c = 0, and by the associated functional derivatives (3.12a). By following the procedure (4.2)-(4.3) for the collection of fields ϕ = (φ, φ * ), we have By virtue of the product rule (3.5), the term proportional to m 2 reads For the second order derivative terms, we add and subtract the terms which are missing in order to apply relation (3.5) with respect to the indices µ and ν: For the second order derivatives in the last line, we use the fact that the covariant derivatives commute with each other. Thus, relation (3.5) allows us to rewrite each line of the previous expression as a total derivative. Altogether we obtain the off-shell identity involving the (on-shell conserved) EMT of the scalar field given by where b µν ≡ η νσ b µ σ . For the case of constant noncommutativity parameters, this tensor reduces (up to a multiplicative constant) to the expression For real-valued fields φ we thus recover the well-known result which has been obtained by other arguments in the literature [30]. In the commutative limit, the expression T free,θ=const reduces to the familiar result from Minkowski space [59].
As for the derivation of the local conservation law of electric charge, one can start from a symmetrized expression in Eqn. (5.5), i.e. replace the star products by star anticommutators: this again allows us to eliminate in the final result the terms of b µ ν which are linear in θ. For the free Dirac field, the same line of reasoning [28] yields an equation which is completely analogous to Eqn. (5.6): In the commutative limit, we again recover the familiar expression T µν free,θ=0

Case of interacting fields:
For the self-interacting complex scalar field φ, we have an additional contribution on the right hand side of Eqn. (5.5): This term can be rewritten as a sum C ν + B ν where C ν is a sum of star-commutators, and where B ν is given by A quartic star monomial has weight 2, hence From this relation and from the modified Leibniz rule (3.8) it follows that where A ν represents a star-anticommutator, A ν ≡ w c 2 φ * φ , w −1 ∂ µ b µν (φ * , φ) . Thus, the EMT for the self-interacting theory reads and, for the solutions of the equations of motion, we have the energy-momentum balance equation If we integrate this relation over space-time, then the integral over C ν (i.e. a sum of starcommutators) vanishes due to the cyclicity of the trace (2.10). However, this is not the case for the other terms on the right hand side of (5.12). The non-conservation law of T µν [φ] for interacting fields is related to the fact that the matter field φ is coupled to the external tensor field (θ µν (x)) and its derivatives as noted already in Section 4.2 for the coupling of a scalar field to a symplectic structure.
For constant noncommutativity parameters, we have w = const and D µ φ = ∂ µ φ: the result (5.12), with C ν given by (5.10), can then be checked readily by using the equations of motion. For real-valued fields, the latter result reduces to c 2 [φ φ , [φ , ∂ ν φ]], i.e. the result which was first obtained in reference [30] by other methods (and which has been further discussed in reference [32]).
For the gauge field (A µ ), an energy-momentum balance equation can be obtained by starting from the product −w δS δAν F νµ . We will not expand further on this point since it is already fairly involved in the case of constant noncommutativity parameters, see [34] and references therein.

Field theory on curved noncommutative space-time
With the description of gravity in mind, the formulation of noncommutative field theories (and in particular of gauge theories) on generic symplectic manifolds with curvature and/or torsion has been addressed by various authors using diverse approaches, e.g. see references [4,11,35,38,40,[60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77] as well as [78,79] for some nice introductions and overviews of the literature up to the year 2010. In relationship with the main subject of the present work (in particular the conservation laws for field theories on flat noncommutative space-time) we note that it should also be possible to obtain the energy-momentum tensor (EMT) of matter fields in flat space-time by coupling these fields to a metric tensor field: the EMT is then given by the flat space limit of the curved space EMT defined as the variational derivative of the matter field action with respect to the metric tensor (see [59] and references therein for a justification of this procedure).
Here we outline the approach to curved noncommutative space which was recently put forward by M. Dobrski [35] who discussed the case of pure gravity following a series of related works by the same author, notably [73]: this formulation appears to fit nicely with the one that we considered here for flat noncommutative space-time. In a separate work [80], we further discuss star products on curved manifolds and in particular different approaches to the description of tensor fields and differential forms on noncommutative manifolds.

Curved space-time and symplectic structure
So far we discussed the star product on R n (with n even) where R n is considered as a flat symplectic manifold, i.e. as a flat smooth manifold equipped with a symplectic two-form ω ≡ 1 2 ω µν dx µ ∧ dx ν . In Einstein's theory of gravity, the gravitational field is described by an xdependent metric, i.e. the space-time manifold M is endowed with a symmetric tensor field g ≡ g µν dx µ ⊗ dx ν . Whatever the manifold under consideration, the definition of a parallel transport of vectors (and more generally of tensor fields), requires the introduction of a linear connection ∇: Its action on tensor fields (V µ ) or (V µ ) is locally defined in terms of the connection coefficients Γ λ µν of ∇, namely [81] (with the notation ∇ ∂µ ≡ ∇ µ ) To this connection one associates its curvature and its torsion given by the tensor fields (R ρ σµν ) and (T λ µν ) defined by the relation Linear connections exist on any smooth manifold M and if no further assumption is made they are independent of other structures on M like the metric structure or the symplectic structure.
In the following, we successively consider the cases where M is endowed with a metric structure, with a symplectic structure and with both structures.

Metric structure:
We recall that on a pseudo-Riemannian manifold (M, g), there exists a unique linear connection ∇ (referred to as the Levi-Civita connection, its connection coefficients being referred to as the Christoffel symbols) which is characterized by the following two properties: Thus, the connection coefficients Γ λ µν are symmetric in the indices µ, ν and the metric is covariantly constant with respect to ∇. These relations imply the well-known expression for the Christoffel symbols, i.e. Γ λ µν = 1 2 g λρ (∂ µ g ρν + ∂ ν g µρ − ∂ ρ g µν ) which implies Γ ν µν = ∂ µ ln |g| where g ≡ det (g µν ). This connection is used in Einstein's theory of gravity and we will also consider it here for the pseudo-Riemannian manifold (M, g) while denoting it as above by ∇ with the connection coefficients Γ λ µν .
Symplectic structure: Since we want the generalize our description of noncommutative field theory in flat space to a more general manifold M , we suppose that the latter manifold is endowed with a symplectic two-form ω ≡ 1 2 ω µν dx µ ∧dx ν . Like the metric (g µν ), the symplectic tensor (ω µν ) is given by a non-degenerate matrix, and it is thus natural to consider a linear connection (which we denote by We note that in the literature [42,82,83] a 'symplectic connection' is generally required to be torsionless, but we do not include this condition in our definition of 'symplectic' (hence we should rather use the terminology 'almost symplectic' [67,83]). A symplectic manifold (M, ω) equipped with a torsionless, symplectic connection is referred to as a Fedosov manifold, e.g. see references [82,83] for a general study. We remark that the antisymmetry of (ω µν ) and the relation • ∇λ ω µν = 0 imply the closedness relation dω = 0 since • ∇λ ω µν +cyclic permutations of (λ, µ, ν) = ∂ λ ω µν + cyclic permutations of (λ, µ, ν).

Metric/symplectic compatible structure:
For the formulation of gravity on the evendimensional space-time manifold (M, g, ω), it is natural to relate (ω µν ) and (g µν ), or at least to ensure their compatibility for the parallel transport of vectors. Different conditions or relations for the connection coefficients • Γ λ µν and Γ λ µν can be envisaged [35,84]. The strongest condition which consists of equating both connections is very stringent since this condition entails that ω is covariantly constant with respect to the Levi-Civita connection ∇: in four dimensions this implies that the metric locally decomposes into a sum of two-dimensional metrics [35,85]. A weaker compatibility condition was considered by M. Dobrski who referred to it as a weakly compatible metric/Fedosov structure: Since the symplectic condition (6.3b) yields

relation (6.4) is equivalent to
Thus, the divergenceless condition ∂ µ w θ µν = 0 that we imposed for M = R n in equation (2.12) in order to have a closed star product amounts, on a general manifold M , to the vanishing of expression (6.6).

Dynamics of fields for a weakly compatible metric/Fedosov structure:
Let us now consider a space-time manifold (M, g, ω) of dimension n = 2m together with a weakly compatible metric/Fedosov structure, i.e. connections ∇, • ∇ satisfying the set of relations (6.2a)-(6.4). The dynamics of matter fields (e.g. of a scalar field φ) which are coupled to the gravitational field described by the metric tensor field (g µν ) can presently be described [35] by the standard action functionals though involving the integration measure ω m /m! (see Eqn. (2.14)). Thus, the variables ω µν become dynamical fields which only couple to the gravitational fields g µν by means of their respective determinants. The set of anholonomic constraints and . (6.11) The set of equations (6.9a)-(6.9c) can be combined in the following manner [35]. By applying δ λ ν • ∇µ to Eqn. (6.9b) and then substituting Eqn. (6.9c), we get an expression for • ∇µs µ = ∇ µ s µ : substitution of the latter into Eqn. (6.9a) leads to the result Here, we recognize Einstein's tensor G µν ≡ R µν − 1 2 Rg µν . By applying the covariant derivative ∇ µ to Eqn. (6.12) and by taking into account that ∇ µ G µν = 0 = ∇ µ T µν EH as well as ∇ µ g µν = 0, one concludes that Λ is necessarily a constant. Henceforth, relation (6.12) is Einstein's field equation (for the metric tensor) including a cosmological constant Λ. More precisely [35], the solutions of the equations of motion (6.9a)-(6.9c) must include a metric g which solves Einstein's field equation (6.12) and, conversely, for each solution of Eqn. (6.12) there exists a weakly compatible metric/Fedosov structure and Lagrange multipliers satisfying the set of equations (6.9a)-(6.9c). The symplectic data are determined by the set of equations (6.3a), (6.3b), (6.4) which imply relation (6.5), i.e. the proportionality of w and |g|. As noted in reference [35], the action S g + S L characterizing pure gravity (with a cosmological constant) might be of interest for the canonical quantization since it does not involve the square root of the determinant of the metric. Scalar field coupled to a weakly compatible metric/Fedosov structure: Let us assume that the complex scalar field Φ is coupled minimally to the metric tensor (g µν ). The matter field action functional then reads Under a general coordinate transformation (diffeomorphism) x → x (x), the fields Φ and L M transform as scalar fields, e.g. Φ (x ) = Φ(x). The argumentation presented in equations (6.9a)-(6.12) then applies. In particular, the tensor • T µν (as given in Eqn. (6.11)) now only involves the derivative with respect to g µν , i.e. − 1 2 • T µν ≡ ∂L M /∂g µν , and we obtain the usual, covariantly conserved EMT for a complex scalar field: Let us presently assume that we rescale the scalar field Φ as we did in Eqn. (3.14), i.e. Φ ≡ √ w φ. While the weight factor w = det (ω µν ) was invariant under the Poincaré transformations (3.17) in flat space (due to the fact that |det (Λ µ ν )| = 1), it is no longer invariant in curved space under general coordinate transformations: it rather transforms like the density |g|, i.e.
Thus, the field φ = Φ/ √ w transforms like a scalar density of weight 1/2 (i.e. the weight attributed to matter fields in Section 3): By rewriting the scalar field L M given in Eqn. (6.13) as well as the associated, covariantly conserved EMT (6.14) in terms of the densities φ, φ * , we obtain the result In the Lagrangian (6.17a) we note the appearance of a global factor w (which implies that the action M ω m m! L M = M d n x w 2 [· · · ] involves a global factor w 2 ) as well as of an extra factor w in the quartic term.
If one considers the weakly compatible metric/Fedosov structure (i.e. w ∝ |g|), one cannot directly recover a flat space model for the coupling of the field φ to the symplectic tensor since |g| reduces to unity in the flat space limit (g µν η µν ) while w is an x-dependent function in flat space 7 . Let us ignore for the moment being the relation w ∝ |g| and consider the following procedure to obtain a Lagrangian model in flat space: Then, the curved space Lagrangian (6.17a) yields the flat space action functional Due to the extra factor w in the self-interaction term, this flat space model is different from the one that we discussed in Eqn. (4.8) (see also the equivalent expression (4.11)) and, more generally, in Eqn. (3.9) for flat noncommutative space. Indeed, for the self-interaction term of a noncommutative model associated to the action (6.19) one could consider As a matter of fact, the factor w being invariant under Poincaré transformations in the flat space case, one could then consider some general functions of w as coefficients in L M . These alternative flat space models lead to energy-momentum balance equations which differ from those we encountered, e.g. the EMT associated to the matter field functional (6.19) is locally conserved by construction.

Curved noncommutative space-time
The Fedosov star product deformation of the field theoretic model (6.7)-(6.8) has been studied by M. Dobrski [35] for the case of pure gravity so as to obtain a theory of noncommutative gravity which is generally covariant and independent of the symplectic background, the latter being dynamical. The first modification brought about the noncommutativity (which appears at order θ 2 ) is already quite complex and has been determined by using a Mathematica package for tensor calculus. The incorporation of matter fields is beyond the scope of the present work, but in view of the previous discussion concerning the coupling of matter fields to gravity, the matter action functionals that we studied for M = R n should appear naturally. Here, we only outline the formulation for pure gravity in terms of the notation used so far and we comment on matter fields.
One of the pioneering works on deformation quantization is the one of B. Fedosov who constructed a star product for a generic symplectic manifold (see references [40,42] for an introduction to these topics). As a matter of fact, any star product on such a manifold is equivalent to a Fedosov star product, i.e. the equivalence class of the latter star product comprises all other ones. The approach of Fedosov amounts to a geometric extension of the Groenewold-Moyal quantization, the latter applying only to a symplectic vector space. Quite generally, the deformation quantization on a given space M is a deformation of the product of functions defined on this space in the direction of the Poisson bracket of these functions. Since the Poisson bracket involves derivatives, the formulation of star products on manifolds calls for the introduction of a linear connection, the latter defining a covariant derivative 8 . On a symplectic manifold, a natural choice for such a connection • ∇ is the one which is torsion-free and symplectic, see equations (6.3a) and (6.3b). Indeed, the Fedosov scheme (and other constructions which it inspired like the globalization of the local expression for star products on Poisson manifolds) relies on the introduction of such a connection. The final results are usually presented in a somewhat abstract form, but an explicit (iteratively determined) expansion for the Fedosov star product has recently been elaborated, see [35] and references therein. For smooth complex-valued (i.e. scalar) functions f, g on the Fedosov manifold (M, ω, • ∇), the Fedosov star product reads (6.20) where the brackets denote symmetrization. The higher order terms in θ involve explicitly the curvature tensor • R ρ σµν associated to the connection • ∇, e.g. see [35] for the third order term.
and analogously for the total connection∇. The Fedosov star product of x-dependent matrices (endomorphisms) F ≡ (F µ ν ) and G ≡ (F µ ν ) now reads [35,73] where {, ·, ·} denotes the anticommutator of matrices. For a flat connection E ∇, i.e. for E Rµν = 0, and the choice E Γµ = 0, the star product (6.21) reduces to a star product of matrices for which the multiplication of entries is given by the star product of functions (6.20).
For the formulation of field theories and more precisely of action functionals, it is again necessary to introduce an appropriate cyclic trace functional tr i.e. tr (F G) = tr (G F ) for compactly supported endomorphisms F , G. Such a functional has also been introduced by B. Fedosov and an explicit expression for it has been worked out by this author in reference [88] (see also reference [73] where a Mathematica package for tensor calculus is applied): For the formulation of pure gravity, the Fedosov manifold (M, ω, • Γ ) endowed with a given metric structure, one considers the vector bundle E ≡ T M and the Levi-Civita connection on this bundle, i.e. E ∇= ∇ (see equations (6.2a) and (6.2b)). Then, the natural choice for an endomorphism F ∈ End T M in an action functional of the type (6.22) is given by the Ricci tensor with the first index raised, i.e. F ≡ R with R ≡ (R µ ν ) ≡ (R ρµ νρ ). Indeed, with this choice one has the real, diffeomorphism invariant action functional Here, the first term is the functional S g considered for pure gravity in commutative space-time (see Eqn. (6.7)) and S nc represents the noncommutative corrections. The latter are of order θ 2 and higher order in θ since the term in (6.22) which is linear in θ vanishes for F = R due to the symmetry properties of the curvature tensor: Tr ( E Rµν R) = R ρ σµν R λσ ρλ = 0. This appears to be a general feature of noncommutative gravity [35].
The calculation (6.7)-(6.12) can now be generalized by starting from the complete action functional S ncg + S L with S ncg given by (6.23) and S L given by (6.8). This leads to a set of equations of the form (6.9a)-(6.9c) with • T µν = 0 = L M and with a noncommutative correction term in each equation. Explicit expressions for the latter terms up to order θ 2 have been given in reference [35]. The elimination of the Lagrange multipliers is presently more complex than in the commutative case due to the appearance of integrability conditions. In view of the complexity which already underlies pure noncommutative gravity, the incorporation of matter fields (which also requires the introduction of tetrad fields in the case of Dirac spinors) is beyond the scope of the present work.

Particular examples of curved noncommutative space-time
Rather than studying the dynamics of space-time, we can also choose a given curved space-time (e.g. four-dimensional space-time endowed with the Schwarzschild metric) and study deformed field theories on such a background. A simple, but non trivial example for a Riemannian manifold is given by an orientable surface. For such a space-time of dimension n = 2, the Poisson tensor (θ µν ) only involves a single independent component θ 12 (x 1 , x 2 ) ≡ ϑ(x 1 , x 2 ) which implies that w ≡ det (ω µν ) = 1/ det (θ µν ) = 1/|ϑ|. By virtue of Eqn. (6.5), the compatibility of the metric and Poisson structures then implies that 1/|ϑ| is proportional to √ g where g denotes the determinant of the metric tensor (g µν ). For simplicity, we consider ϑ to be positive and equal to 1/ √ g [75], i.e.
Let us now suppose for concreteness that the orientable surface under consideration has constant curvature, i.e. it is (up to a homeomorphism) a 2-sphere for the case of positive curvature or the hyperbolic plane for the case of negative curvature. We will elaborate briefly on the example of the unit 2-sphere [75]. For the latter the upper hemisphere can be parametrized by Cartesian coordinates (x 1 , x 2 , x 3 ) ≡ (x, y, z) with x 2 + y 2 < 1 and z = z(x, y) ≡ 1 − x 2 − y 2 (and similarly for the lower hemisphere). In terms of these coordinates, the standard line element ds 2 = dθ 2 + sin 2 θ dϕ 2 of the 2-sphere is given by This entails that √ g = 1/z, hence we have coordinate dependent components for the Poisson tensor which are given by θ 12 = z(x, y) = −θ 21 . The Kontsevich formula (2.8) or the closed star product (2.16) then imply At the first order in h, these star-commutation relations are the ones which characterize the fuzzy sphere. The Levi-Civita connection (associated to the standard metric) and the symplectic two-form considered to be the (opposite of the) Riemannian volume form give the structure of a Fedosov manifold to the 2-sphere [82]. The case of the hyperbolic plane modeled by the Poincaré upper halfplane {(x, y) ∈ R 2 | y > 0} endowed with the metric ds 2 = y −2 (dx 2 + dy 2 ) can be studied along the same lines and leads to the results w = 1/y 2 and [x , y] = i h y 2 . These considerations again fit into the general framework discussed before, yet the dynamical study of matter fields on these spaces obviously requires some further work.

Concluding remarks
About the quantum theory: For R 3 θ , i.e. R 3 with su(2)-noncommutativity (in which case the commutator algebra [X i ,X j ] = i ε ijkX k which may be realized by the Pauli matrices), the one-loop quantization of self-interacting scalar field theory has recently been investigated [51], in particular for the scalar field model described by the classical action (3.9) (with w ≡ 1 for R 3 θ ). The main results may be summarized as follows [51] (see also reference [89] for some further recent work). The 2-point function does not involve infrared singularities in the external momenta (even in the massless case) which indicates the absence of the infamous UV/IR mixing problem for these models. This result appears to have its origin in the Lie algebraic nature of the underlying noncommutativity. Moreover, the 2-point function is finite in the ultraviolet regime where the deformation parameter corresponds to an ultraviolet cut-off Λ ∝ 1/θ.

Conclusion:
The mathematical framework for field theories on a space-time defined by generic noncommutativity parameters is more complex than the one for constant parameters. Nevertheless the classical theory can be formulated to a large extent along similar lines. In this context, some interesting mathematical structures appear which may be worthwhile to explore further. The simple approach to the conservation laws that we considered here should also be of interest in other contexts. The presented analogy of free field models with the damped harmonic oscillator is intriguing and may also be useful for the investigation of some aspects of the quantum theory. For the latter one also has to tackle the subtleties of time ordering in the noncommutative setting, see e.g. [90] and references therein for a discussion on this point in Moyal space. A scheme for describing the dynamics of the fields θ µν (x) was outlined in curved space-time, but an elaboration and better understanding of this point definitely requires further work. The case where the matrix (θ µν (x)) does not have maximal rank for all x (e.g. θ 12 vanishing on a line in the two-dimensional case) also requires extra work involving a regularization of integrals, e.g. see reference [20]: this instance appears to be of interest for the study of boundary effects occurring in condensed matter systems.
where the operatorθ µν (X) acts as usual as multiplication by the function θ µν (x). Moreover, one readily checks that the Jacobi identity for commutators (i.e. the relation 0 = [X λ , [X µ ,X ν ]] + cyclic permutations of the indices λ, µ, ν) is satisfied to order θ if the antisymmetric tensor θ µν satisfies the Jacobi identity (2.2) characterizing a Poisson tensor. Finally, the linear term of the star product also follows straightforwardly: For the terms of higher order in θ, one has to expandX µ andθ µν to higher order than first, which yields [23,24] as well as an expression forθ µν (X) (as a differential polynomial in the Poisson tensor θ µν ) which extends the first order result (A.4). For the definition of a Weyl-ordered functionf (X), one applies the general formula [3,29,36] wheref (k) = R n d n x f (x) e ikµx µ denotes the Fourier transform of f . The latter relation defines the Weyl symbol of the operatorf (X). The expression for the star product which results to order θ 2 from this procedure is given by equation (2.8) and is commented upon in that context.

Closed star product:
For physical applications, we are interested in a closed star product [24], i.e. such that relation (2.10) holds for some integration measure w d n x. As we noted in equation (2.11), the closure relation for the star product yields (at the first order in θ) the divergenceless condition ∂ µ (wθ µν ) = 0. By taking into account the latter as well as the Jacobi identity for the Poisson tensor and by performing some integrations by parts, the closure relation leads, to the second order in θ, to the result Here, the matrix (B ρν ) whose elements are given by B ρν = ∂ µ (wθ ρσ ∂ σ θ µν ) is symmetric by virtue of the divergenceless condition and the Jacobi identity for the Poisson tensor (θ µν ). The result (A.7) means that the Kontsevich star product does not satisfy the closure relation. However, this result suggests to make a judicious equivalence (or so-called gauge) transformation [41] of the star product so as to obtain a closed star product (see reference [24] for the treatment of (A.7) and reference [29] for the general procedure to all orders in θ): by considering a linear differential operator of the form , one readily finds that the gauge transformed star product induced by D has the following form (to order θ 2 ): Thus the particular choice 2d ρν = 1 24w B ρν allows to eliminate the nonvanishing term on the right hand side of equation (A.7), i.e. to obtain a gauge equivalent star product which satisfies the closure relation R n d n x w f g = R n d n x w f g. Its explicit expression (up to order θ 2 ) is spelled out in equation (2.16) where we suppressed the prime on the star product. The basic operatorX µ now becomesX µ and its θ-expansion follows fromX µ g ≡ x µ g: in comparison to the operator X µ given by expression (A.6) it thus involves, at order θ 2 , a supplementary contribution. The latter ensures [24,28] that the operatorX µ is Hermitian with respect to the inner product (3.1).

B Some comments on Section 2 and Section 3
In this appendix, we gather some mathematical remarks concerning Section 2 and Section 3, respectively.
Linear operators on associative algebras which do not satisfy the Leibniz rule appear in various contexts in physics and in mathematics, e.g. in the Batalin-Vilkovisky anti-bracket formulation of gauge field theories [102][103][104] or in general relativity, see [105] and references therein. The deviation from the Leibniz rule of a linear operator (acting on an associative algebra like A[[h]] equipped with the star product) is known in mathematics as the Hochschild differential δ of this operator [42]: for the operator D µ , the bilinear map δD µ : According to relation (3.8), we thereby have Thus, the Hochschild differential of D µ looks like a coboundary term. In the physics literature, the deviation from the Leibniz rule has also been qualified as the Leibniz bracket [105]. More precisely, for the algebra A[[h]] equipped with the star product and the grading introduced after Eqn. (3.7), the Leibniz bracket of the linear operator D µ with respect to the star product is defined (in terms of the notation of reference [105]) by We will not elaborate on these mathematical aspects here, but their application should be worth exploring in greater detail. We only mention that the fundamental property of the Leibniz bracket is given by

C Damped harmonic oscillator
The models of free Lagrangian field theories on noncommutative space discussed in this paper and the corresponding energy-momentum conservation laws admit close analogies with the Lagrangian formulation of a damped harmonic oscillator in non-relativistic mechanics and with a corresponding conserved quantity. Therefore, we describe the latter system in this appendix by stressing the analogies using an appropriate choice of notation. For a mechanical system with one degree of freedom, i.e. a single second order differential equationq = f (q,q, t) for the particle's position t → q(t), it has already been shown by Jacobi and Darboux that a Lagrangian function always exists. A simple example for a dissipative system in one dimension is given by the damped harmonic oscillator, i.e. a dynamical system governed by a differential equation with constant coefficients m > 0, k > 0, γ ≥ 0: mq + kq + γmq = 0 , (C.1) or, with ω ≡ k/m,q + ω 2 q + γq = 0 . (C.2) To simplify the notation, we will consider a unit mass in the following. As we just stated, a single differential equation of second order like (C.2) can always be obtained as an Euler-Lagrange equation, eventually after multiplying it by an integrating multiplier, i.e. a non-vanishing function t → w(t) in the present one dimensional case. Indeed [106][107][108] i.e. the equation of motion (C.2). Since γ = ∂ t ln w, the last equation can also be written as 0 = 1 w ∂ t (wq) + ω 2 q , (C. 5) and thus has the same structure as the equations of motion (3.15) of our field theoretical models for free scalar and Dirac fields in noncommutative space. For the discussion of the conservation law below, we spell out the solution of the equation of motion (C.2) which satisfies given initial conditions q(0) = x 0 andq(0) = v 0 : with Ω 2 ≡ ω 2 − γ 2 2 , we have We note that the canonical momentum associated to q is given by p ≡ ∂L/∂q = wq, hence the canonical Hamiltonian reads This function may also be expressed in terms of q,q and t, where H 0 represents the total energy of the undamped oscillator. However, due to the dissipation, the Hamiltonian H is not a conserved quantity: we have an energy balance equation which can be determined straightforwardly by using the equation of motion for q: Nevertheless, a conserved charge can be constructed by different methods and in particular as follows by performing some redefinitions [109]. By virtue of the time-dependent rescaling q Q = w 1/2 q (C.10) (which is analogous to the x-dependent rescaling of matter fields encountered for our field theoretical models in Eqn. (3.14)), the equation of motion for q takes the form of the one for an undamped oscillator:Q For the latter dynamical system parametrized by Q, the total energy E is obviously conserved and is given by 2E ≡Q 2 + Ω 2 Q 2 = e γt q 2 + ω 2 q 2 + γqq . (C.12) Here the last expression is the rewriting of the conserved charge in terms of the original variable q. It represents an explicitly time-dependent conserved quantity for the dissipative system under consideration. Its physical interpretation can be elucidated by considering its value at the time t = 0: 2E(t) = 2E(0) = v 2 0 + ω 2 x 2 0 + γx 0 v 0 . Thus, the conserved quantity is simply a particular combination of the initial conditions, the first two terms representing the energy of the undamped oscillator. In the course of the motion, the exponentially increasing factor in the charge (C.12) is compensated by the exponential decrease of the solution (C.6) of the equation of motion.
Within the Hamiltonian formulation, the conserved quantity E (expressed in terms of the phase space variables q, p) generates local transformations δq, δp of the phase space variables by means of the Poisson brackets and, conversely, the latter transformations give rise to the conserved charge E by virtue of Noether's first theorem. More precisely, with the standard Poisson bracket of functions F, G on phase space, Hence the variation of the Lagrangian (C.3) under these transformations reads δL = dL dt . (C.14) Thus, we have a divergence symmetry of the action, namely δL = df dt with f = L. According to Noether's first theorem the conserved charge associated to such a divergence symmetry reads where the last factor describes the divergence symmetry (C.14): Substitution of the expression for L yields E = E. We conclude with two comments concerning the conservation laws in a dissipative dynamical system like the damped harmonic oscillator. First, we note that non-autonomous Hamiltonian systems like the one given by the explicitly time-dependent Hamiltonian (C.7) can be described in a symplectic extended phase space, see reference [110] for a general study. In that framework, a time-dependent Hamiltonian can be mapped by a generalized canonical transformation into a time-independent Hamiltonian. Indeed, our conserved charge (C.12) coincides with the invariant (46) of reference [110] (upon considering n = 1, ω = constant, F (t) = γt, and ξ = 1 in [110]).
Second, we remark that for a linear, explicitly time-dependent dynamical system like the damped harmonic oscillator, one can perform a so-called Arnold transformation (section 1.6.A of reference [111]) which maps the equation of motion of the system into the one of a free particle by virtue of a mapping (q, t) → (ξ, τ ) of the underlying non-relativistic space-time into itself, see [112] and references therein. Indeed, by decomposing the general solution (C.6) of the equation of motion subject to the initial conditions q(0) = x 0 andq(0) = v 0 as q(t) = v 0 u 1 (t) + x 0 u 2 (t) with u 1 (t) = e − γ 2 t 1 Ω sin Ωt u 2 (t) = e − γ 2 t cos Ωt + γ 2Ω sin Ωt , we obtain the solution τ → ξ(τ ) of the free particle equation of motion, An extension of the two-dimensional space-time, (q, t) (q, t, s), to a three-dimensional space (which is referred to as Bargmann space or Eisenhart lift) and the related extension (q, t, s) → (ξ, τ, σ) of the Arnold map then allows to show [112] that the damped harmonic oscillator has the same symmetries as the free particle (and thereby also admits corresponding conserved quantities whose expression can be derived by means of the extended Arnold map).