Nonlocal Lagrangian fields and the second Noether theorem. Non-commutative U (1) gauge theory

This article focuses on three main contributions. Firstly, we provide an in-depth overview of the nonlocal Lagrangian formalism. Secondly, we introduce an extended version of the second Noether’s theorem tailored for nonlocal Lagrangians. Finally, we apply both the formalism and the extended theorem to the context of non-commutative U(1) gauge theory, including its Hamiltonian and quantization, showcasing their practical utility


Introduction
Nonlocal Lagrangian field theories are characterized by their formulation through a Lagrangian density that functionally depends on the field.This functional dependence is manifested either in the form of infinite series of derivatives or integral operators, such as the convolution one.In the scope of this article, we will focus exclusively on the nonlocality defined by integral operators.This choice stems from observations made in [1,2], which point out that the functional domain in which infinite derivative operators are defined is more restricted compared to integral operators.
In [3,4], a novel Lagrangian and Hamiltonian formalism for nonlocal field theories is presented, representing an enhancement over previous methods like the (1+1)-dimensional Hamiltonian formalism [5,6].This novel nonlocal Lagrangian formalism, represented by integral operators instead of an infinite series of derivatives, yields two significant results: the development of a Hamiltonian framework for these Lagrangians and an extension of Noether's first theorem, which is particularly valuable in the derivation of the canonical and Belinfante-Rosenfeld energy-momentum tensors [7,8].
An outline of the general ideas inspiring the generalization of the Legendre transformation to the nonlocal scenario is as follows: In the context of a standard first order Lagrangian, the role played by the momenta p j (q, q) = ∂L(q, q) ∂q j is twofold.Initially, they appear as the prefactors of the variations δq j within the boundary terms emerging from the application of the variational principle.Then, and as a consequence of this fact, these momenta also manifest as the prefactors of δq j in the conserved quantities derived from Noether's theorem.A remarkably similar scenario occurs for a higher order Lagrangian, L q, . . ., q (n) , where there is a conjugate momentum p l q, . . .q (2n−l) for each q (l) , 0 ≤ l < n , contributing to the conserved current through n−1 l=0 p l δq (l) .In the regular case, the relations, p l q, . . .q (2n−l) , can be inverted for the higher velocities, q (l) , n ≤ l ≤ 2n − 1 , starting from the highest-order momentum p n−1 q, . . .q (n) and so on.
In the context of Lagrangians that depend on derivatives of infinite order, the canonical momenta are typically formulated by substituting n with ∞ and transforming finite sums into infinite series.However, the absence of a highest-order derivative prevents the straightforward inversion of the generalized Legendre transformation -specifically, the substitution of one half of the derivatives with the canonical momenta becomes infeasible.To put it bluntly, because "half of infinity is infinity".Consequently, the Legendre transformation inevitably becomes singular in the context of infinite order.Moreover, the Euler-Lagrange equations form a differential system of infinite order, for which there is no established theorem guaranteeing existence and uniqueness; this leaves the issue of initial conditions highly ambiguous, specifically regarding the quantity of initial data required to uniquely determine a solution.
Given these complexities, a paradigm shift is required: the dynamic space cannot be the space of initial data, as it is not even clear that such a concept is applicable or meaningful in this context.Instead, our attention shifts towards the space of dynamic trajectories, constituting a subspace of K, the class of the kinematic trajectories.This specific subspace is based on those trajectories that satisfy the Euler-Lagrange equations.As a consequence of this fact, it establishes these equations as constraints that define the dynamic space.Simultaneously, canonical momenta also play the role of constraints that, together with the Euler-Lagrange equations, define the dynamic space D as a submanifold -maybe infinite dimensional -within the cotangent space T * K .
The standard canonical structure on T * K , characterized by the canonical coordinates q (l) and p k , l, k = 0, 1 . . ., can be translated into a canonical formalism on D → T * K , the submanifold defined by the Euler-Lagrange equations and their derivatives, along the momenta, p l = p l q, . . .q (k) . . ., taken as constraints.This can be achieved via two methodologies: by employing the Dirac brackets for the aforementioned constraints [6], or equivalently, by pulling the symplectic form Ω = ∞ l=0 dp l ∧ dq (l)   back onto D [8].A quite similar scenario occurs with Lagrangians that are functionally dependent on the entire trajectory, q(τ ), τ ∈ R (or a finite segment of it), where the canonical momentum 1 , P ([q], τ ) , is a functional that depends on the whole q, i.e. nonlocal.Once more, the Euler-Lagrange equations and the canonical momentum p(σ) = P ([q], σ) must be taken as constraints to define the dynamic space D as a submanifold on T * K , where K represents the space of all possible trajectories.The technicalities of this procedure are developed in detail in [3,4].
Most infinite-order Lagrangians in literature are obtained from Lagrangians that depend functionally on the whole trajectory.This transformation involves substituting q(τ ) by the formal Taylor series As kinematical trajectories are not necessarily real analytic, this relationship possesses only heuristic significance.Nevertheless, it serves as a bridge connecting the treatment involving infinite derivatives with the functional approach.Hence, in light of this consideration, throughout this manuscript, we will consistently employ this novel formalism based on integral operators, rather than the infinite series approach.
In [9] Tomboulis sets up an alternative Hamiltonian framework for nonlocal Lagrangians characterized by integral operators.This framework is expressly limited to kernels with compact support, referred to as quasilocal.This is a noteworthy distinction between his proposal and our current approach, based on [3,4], which does not suffer from this limitation.Particularly, the case study we consider here, from Section 4 onwards, is fully nonlocal.
A relevant illustration of nonlocality can be found in the framework of non-commutative (NC) spacetime [10], particularly in the context of U(1) gauge theory [11][12][13][14].The fundamental motivation behind introducing non-commutativity into this theory (apart from String Theory's motivation) is to enhance renormalizability at short distances and potentially establish finiteness within these scales [15].See, for instance, [16][17][18] (and references therein) for other non-commutative theories.
The paper is structured into two main parts.In the first part -Sections 2 to 4-, we extend the second Noether theorem to nonlocal Lagrangians and apply it to NCU(1) gauge theory for testing purposes.In the second part -from Section 6 to the end-, we practically implement the Hamiltonian formalism developed in [3] within the framework of NCU(1) gauge theory.
The article is organized in seven sections.Section 2 is an outline of the nonlocal Lagrangian formalism already developed in previous works [3,4,8]: we present the variational principle for such a Lagrangian and derive the nonlocal Euler-Lagrange field equations 2 .We also pay some attention to these field equations when the nonlocal Lagrangian is a total divergence.As proved in [4] and contrarily to what happens in the local case, these field equations do not identically vanish.We here outline a sufficient condition [3] for the nonlocal Euler-Lagrange field equations derived from a nonlocal total divergence do identically vanish.In Section  [11,19].
In Section 5, we deepen the study of the dynamical space in the context of the NCU(1) gauge theory and propose a solution for the field equations using a perturbative approach.Furthermore, we somehow revisit the Seiberg-Witten map within this specific context.In Section 6, we implement the Hamiltonian formalism introduced in [3] into the NCU(1) gauge theory.This leads us to derive the symplectic form, identify the canonical variables, establish the elementary Poisson brackets, determine the Hamiltonian up to the second order in some nonlocality parameter, and adress the canonical quantization.Additionally, for readers who are more interested in reproducing the calculations, we provide detailed steps in Appendices A and B to guide them through the process.

An outline on nonlocal Lagrangian field theories
In this section we review the nonlocal Lagrangian formalism proposed in [3,8] to which the interested reader may be referred for further details.To begin with, we introduce the notions of kinematic space, dynamic space, nonlocal Lagrangian density and spacetime translation.
The kinematic space is the class of all possible fields C m R 4 , R n that may not necessarily be on shell, whereas the dynamic space D is the subclass of fields fulfilling the field equations 3 .
A nonlocal Lagrangian density is a real-valued functional on the kinematic space K : 2 In the present work we focus on nonlocal Lagrangians that do not depend explicitly on the spacetime point because this is the case of the NCU(1) gauge theory to which we intend to apply our mathematical methods. 3For the sake of conciseness, we consider a four-dimensional spacetime R 3+1 but the results here derived are also valid for any number of dimensions.
NC1-Main JLLC.texApril 8, 2024 5 which may depend on all values ϕ A (y), A = 1 . . .m, of the field variables at any point y b .This is why it is called nonlocal.Notice that the functional dependence is not emphasized with square brackets, as is common elsewhere.We will also dispense with superscripts for both field variables and point coordinates, unless the context explicitly requires it.
Given y ∈ R 4 , the spacetime translation is the transformation The class of all these transformations is an Abelian group Notice that all the information about the field evolution is already contained in ϕ(z) .
As presented in ref. [3], the variational principle is based on the one-parameter family of finite action integrals where |y| = 4 j=1 (y j ) 2 is the Euclidean length, and it reads for all variations δϕ(z) with compact support.The Lagrange field equations are and the dynamic fields are the solutions of these equations.From these definitions it easily follows that

The Lagrange equations for a total divergence
A common feature of local theories is that, if the Lagrangian density is a total divergence, then the Lagrange equations vanish identically.In our notation, L(x) and W b (x) must be respectively understood as L(T x ϕ) and W b (T x ϕ, x).
In the nonlocal context things are not so simple because, once the requirement of locallity is relaxed, equation ( 6) has infinitely many solutions.Indeed, its general solution is: where x = (x 1 , x 2 , x 3 ) and Ω bc +Ω cb = 0 .Nevertheless this does not imply that the Lagrange equations for any L are identically null, as it is implied in the context of standard local Lagrangians and local W b -see an example in [4].
The family of actions (2) for such a Lagrangian where Gauss' theorem has been applied and dΣ b (y) is the hypersurface element on |y| = R .Thus, the variational principle (3) yields the field equations which do not necessarily vanish.However, as dΣ b (y) scales as |y| 3 , they do vanish identically provided that which yield a sufficient condition for the Lagrangian ∂ b W b (x) to yield identically null Lagrange equations.This condition is obviously met if W b (T y ϕ, y) is local.

Noether's theorem
Consider the infinitesimal transformation The transformation of the Lagrangian density is defined so that the action integral over any four-volume is preserved, namely, where V ′ is the transformed of the spacetime volume V .Therefore, L ′ T x ′ ϕ ′A = L T x ϕ A ∂x ∂x ′ , and we define From ( 9) it follows that , where we have replaced the dummy variable x ′ with x .Neglecting second order infinitesimals, we get that Then, including (10), we can write where δϕ A (y) := ϕ ′A (y)−ϕ A (y) and we have used that, up to second order infinitesimals, δL(T x ϕ ′A ) = δL(T x ϕ A ) .Therefore, where λ A (ϕ, x, y) is defined in (4).By introducing the variable z = y − x in the latter equation and then substituting it into (11) while applying Gauss' theorem, we arrive at that, including (4), can be rewritten as Using now the identity in equation (13), we obtain where L and δL are shorthands for L(T x ϕ) and δL(T x ϕ) , and where (5) has been included.Now, as equation ( 14) holds for any spacetime volume V, it follows that where The identity ( 16) holds for any Lagrangian and any infinitesimal transformation, regardless of whether the field equations are invariant or not.
The transformation ( 8) is said a nonlocal Noether symmetry if that is δL is the four-divergence for some W b fulfilling the asymptotic condition (7).Hence for a Noether symmetry equation ( 16) becomes which is known as the Noether identity and it holds for any kinematic field ϕ A .Only for dynamic fields, i. e. on-shell, this identity implies the local conservation of the current:

Finite dimensional Lie groups of transformations. First Noether theorem
Let the transformations (8) belong to an N -parameter Lie group of Noether symmetries, where ξ b α (x) is the infinitesimal generator for the constant parameter ε α , α = 1 . . .N and δϕ A might depend functionally on ϕ B .Then, the conserved current J b (T x ϕ, x) can be written as Hence, for each group parameter there is a conserved current given by Recall that L and W b α respectively refer to L(T x ϕ) and W b α (T x ϕ, x) .

Hamiltonian formalism
Assuming the Lagrangian remains invariant under the variation δϕ A , i. e. W b = 0 , the local conservation law (20) reads ∂ b J b = 0 and, provided that the fields ϕ A (x, t) decay fast enough at spatial infinity, the total charge is a constant.Therefore, using (17), we get that As highlighted in the Introduction, our focus in defining the canonical momentum centers specifically on the prefactor of δϕ A , drawing an analogy with the local case.By introducing a set of new variables z, u = x + sz, ζ, and ρ = sζ, we can re-express the previously equation in terms of these variables as follows: and, using that we arrive at where u b = (u, ρ) and z b = (z, ρ) .
Finally, since the charge is conserved: with which we shall adopt as the definition of the canonical momentum.
The Hamiltonian, representing the generating function for time evolution, corresponds to the conserved charge derived from the variations δx b = −εδ b 4 and δϕ A = ε ∂ 4 ϕ A , namely, the total energy.The energy density, identified as the time component of the current (21), is: By following the same steps described above, or by directly applying the variations mentioned earlier to equation (24), we obtain that the Hamiltonian (total energy) h := Q/ϵ is: where As proven in [3,4], the Hamilton equations (in geometric form) are i H ω = −δh , where is the (pre)symplectic form on the dynamic space D, is the generator of time evolution in D.

Gauge groups. Second Noether theorem
When dealing with a gauge group, the transformations (8) depend on a number of arbitrary functions ε α (y) , α = 1 . . .N , maybe in a nonlocal manner, For the sake of simplicity, we will focus only on the case δx b = 0 ; thus, we find that where we have taken y = x + z and, to avoid an overelaborated notation, we have written ψ A (x) and becomes with and Again, for the sake of a simple notation, we have not explicited the functional dependence of N α and K b on ϕ .
Upon introducing the variable y = x + sz , the last expression becomes where x and n is the number of dimensions of the space of x b .Finally, substituting (30)(31)(32) into ( 16), we arrive at the Noether identity where We have used the Noether identity in the form ( 16) rather than (19), i. e. we have not replaced δL(x) with ∂ b W b (x) because we are interested in covering a wider class of transformations than just Noether symmetries.The motivation of this will be seen in Section 4.2.
Furthermore, by using (28) and including that δx b = 0 , the current (17) can be written as: where the variable u = x + sz has been used and The Noether identity (33) applies for all kinematic fields and the terms N α and K b exhibit a linear dependence on ψ A (ξ) .Therefore, this identity serves as a functional constraint on the field equations.

The non-commutative U(1) gauge theory
Now the concepts developed in previous sections will be applied to the non-commutative U (1) gauge theory.Please notice our outsider perspective: we focus on displaying the application of our mathematical methods rather than analyzing the results, a task we leave to the experts on this field.
Throughout this section we will refer to the results derived in Appendix A, where we have extensively detailed the transformation of the Moyal product and its bracket into integral forms, as well as the expression of the Moyal bracket in Fourier space.In addition, we provide a list of eleven properties to which we will refer the reader interested in reproducing in detail the developments in the present section.
The dynamical variable is a 1-covariant real field A b (x), b = 1, . . ., 4 , x ∈ R4 .For simplicity, we will focus on the four-dimensional scenario, but extending it to n dimensions is straightforward.The nonlocal action integral is where 4 and the commutator is the one associated with the Moyal * product where θ ab is a constant, skewsymmetric, contravariant 2-tensor.In Appendix A, we show that if θ ab is non-degenerate 5 , then and v θu := v a θab u b , θab is the inverse matrix for θ ab and |θ| = det(θ ab ) .As a result, the quantities F ab (x) depend nonlocally on the field A b .Moreover, the Moyal bracket of two real functions is an imaginary number and, as a consequence, F ab is real.
Introducing the covariant derivative equation ( 36) becomes and the identity arises from the definition and properties (123) and (128).
The nonlocal Lagrangian density is easily inferred from the nonlocal action (35) and it is Next, using the definitions (36) and (39), and taking δ x (y) := δ(y − x) , we obtain that which, substituted into equation ( 4), yields For simplicity, we do not make the dependence on A b explicit and use λ c (y, x) instead of λ c (A b , y, x) .
The field equations arise from substituting (45) into equation ( 4) and they are where ( 44) and ( 124) have been included.It is worth noticing that this result coincides with that obtained in [11], albeit by alternative methods.

Gauge transformations
Let us consider the (infinitesimal) nonlocal gauge transformation, which only affects the potential.The fields transform according to whereas the field equations (46) transform as In turn, the variation of the Lagrangian (43) is Notice that the field equations are not gauge invariant but transform into a set of equivalent equations, therefore the solution space is.Moreover, as δψ c ̸ = 0 , the transformation (47) cannot be a Noether symmetry and the variation δL is not a total divergence of W b that fulfills the asymptotic condition (7) discussed in Section 2.1.
It is well-known that gauge invariance imposes that the field equations must satisfy certain off-shell identities.As noted above, the current nonlocal Lagrangian lacks gauge invariance due to (50), but it can be shown that such relations do exist even for this case.In particular, Indeed,

Application of the second Noether theorem
We have derived the relation (51) by mere inspection; however, it could also be derived from the second Noether theorem (see [20] for a review of the local case), in particular, from the identity (33).Indeed, from ( 47) and ( 28), it follows that and, substituting this into (31) and including the properties (129) and (133), we easily reach Next, using properties of the Moyal bracket (129) and (130) in equations ( 32) and (34), we arrive at where and where (131) has been included.From the definitions, it immediately follows that and using (54), we get Recalling (55), this leads to where (125) has been included.Now, using (53) and (56), we can represent the central term on the left-hand side of the Noether identity (33) as that, using Leibniz's rule, equations ( 44) and (41), and the Jacobi identity (123), we obtain Finally, substituting this and (50) into the Noether identity and using (52), we arrive at that is the offshell relation (51).

The First Noether theorem. The canonical energy-momentum tensor
We will now discuss the invariance of the Lagrangian density (43) under infinitesimal Poincaré transformations where ω ab is skewsymmetric, i. e. ω ab + ω ba = 0. Since the nonlocal Lagrangian (43) does not depend explicitly on x and is invariant under infinitesimal Poincaré transformations, we have W b = 0 in equation (20).Consequently, the Noether conserved current ( 17) is with y = x + sz .Using now (45) and the fact that δy c = δx c + s ω c e z e , we easily arrive at where is the canonical energy-momentum tensor (or current), and is the internal angular momentum current, which captures both the pure spin and a contribution from This allows us to express them as and where y := x + sz , and The similarity between the initial terms on the right-hand side of both expressions and their homologous in the local theory is noteworthy.The integral terms, on the other hand, represent nonlocal corrections in integral form.Notice also that T cb is not symmetric.Achieving symmetrization would require resorting to the construction of the Belinfante-Rosenfeld energy-momentum tensor [21,22], as was done in [7], but this is beyond the scope of the present paper.
On the contrary, it is important to mention the exploration of an infinite series representation for the energy-momentum tensor, as discussed in [23,24], offering an alternative perspective.However, it is also worth mentioning that in [25], the summation of this kind of infinite series was undertaken to attain a more concise representation of the energy-momentum tensor.This summation was exemplified within the context of classical electrodynamics in dispersive media, and the result was an integral expression (quite close) to the one presented in this manuscript (or in [3,4]).
5 The dynamic space.Solving the field equations Seeking a suitable parametrization for the dynamic space D , i. e., the class of solutions of the field equations ( 46), we write them as with We then define where □ −1 means a chosen particular solution of the D'Alembert inhomogeneous equation.We easily obtain that Now, from the identity (51) and the fact that ∂ c E c a ≡ 0, we get that and, therefore, Whence it follows that E c a Z a = 0 whenever ψ c = 0 .Thus the definition (67) transforms the nonlocal system ψ c = 0 into a local one E c a Z a = 0 , which is nothing but Maxwell equations.Were it invertible, we could obtain the general solution of the nonlocal field equations (46) as the inverse image of the general solution of the Maxwell field equations.
A way to clarify under what conditions it is possible to reverse the relation (67) could be to take it as a C 1 map, A a → Z a , connecting two Banach spaces.Since A a = 0 is mapped into Z a = 0 and the Jacobian map, δZ a (x) is obviously an isomorphism, then the inverse function theorem [26] assures that the inverse map, A a = A a (Z b ) , exists and is C 1 in some respective neighbourhoods of A a = 0 and Z a = 0 .However, working it out in detail is beyond the scope of the present paper.
This argumentation ensures the existence and uniqueness of perturbative solutions, i. e. in a neighbourhood of A a = 0, but says nothing about other regions in the dynamic space.Now, given a solution Z a of the Maxwell equations E c a Z a = 0 , let us consider the field A a (Z) derived by solving (67) in that neighbourhood of Z a = 0. Is it a solution of the nonlocal equations From equation (69) and the fact that Z a is a solution we have that which is a homogeneous linear equation on ψ a .The Jacobian map (with respect to ψ a (y)), is non-degenerate, provided that A a is small enough (in the sense of a suitable norm).Therefore, the homogeneous linear equation admits only the trivial solution in a neighbourhood of A a = 0 .

Connecting the non-commutative nonlocal gauge with a commutative local one
The To begin with, (67) implies that δZ a = δA a + □ −1 δΦ a and from the definition (66) it follows that where (48) has been included.Now, after a little algebra, we obtain that and that where we have used that ψ a = 0 because A a belongs to the dynamic space D. Substituting these in (70) we obtain which, combined with (67) and δA a = D a ε , leads to Since f depends on A a as well as ε, this relation does not define a homomorphism group, and it could not be otherwise because one gauge group is not abelian and the other is.
We have managed to find a relationship between the two Gauge transformations at the level of the dynamic space.This proposal is different from the one of Seiberg and Witten [19] which establishes, only to the first order of approximation in the series of derivatives formalism, a correspondence between the non-commutative gauge and the commutative gauge in the kinematic space.

The solutions
The system of equations is linear and the given of a set of initial data does not determine a unique solution.As a matter of fact, the equations are fulfilled by ∂ a f for any arbitrary function f .In order to select a unique solution some gauge fixations are necessary.Here we shall use the Lorenz gauge condition plus a supplementary condition to fix the second kind gauge, namely This reduces the problem to solving simultaneously For the sake of convenience we will work with the Fourier transform (FT), in terms of which the field equations and gauge fixings respectively read whose general solution is [27] Ẑc (k e ) := where the four-vectors W νc (k) are functions of k, the spatial part of k e ν := (k, ν|k|) , ν = ±.Furthermore, the gauge fixings imply that each of these four-vectors can be written in terms of a polarization vector U µ (k) as Similarly as in the case of Maxwell field, the polarization vectors W ± a (k) are connected with the initial data for the field Z a .Indeed, the spatial FT of the initial data Z a (x, 0) and ∂ t Z a (x, 0) are and where (77) has been included.Then it easily follows that Due to the gauge fixations (73), the solutions (77-78) do not exhaust the entire dynamic space D M .
Indeed, the gauge transformation Z ′ a = Z a + ∂ a f define an equivalence relation among the solutions in D M and each solution (77-78) is a representative of its equivalence class in the quotient space D ph M = D M /gauge , which acts as the physical dynamic space for the alias local theory.
The general solution of the nonlocal set of equations ( 65) is then obtained by solving the implicit functional equation where Φc is the FT of Φ c given by (66), that is As it can be easily checked, the Lorenz gauge condition ∂ a Z a = 0 implies that ∂ a A a = 0, with which we arrive at and its FT is where (120) has been used.
Equation ( 79) can be solved perturbatively by assuming that the more Moyal brackets a term contains, the smaller it is.As Φc is the only term containing Moyal brackets, Âc (k e ) = Ẑc (k e ) + O(1) is the seed on which the perturbative solution is built.As commented before, a more rigorous proof of the existence of one solution Âc for each given Ẑc might be approached on the basis of the inverse function theorem on a Banach space [26].

Hamiltonian formalism on the physical dynamic space
We will next establish a Hamiltonian formalism on D ph ⊂ K, by employing the methodologies introduced in [3] and summarized in Section 3.2.We choose to set up the Hamiltonian formalism on D ph through symplectic techniques rather than Dirac brackets because the former allow a more straightforward implementation of constraints.
According to what is explained and justified in [3], the physical dynamic space is the submanifold of the class of functions (that are differentiable as many times as necessary) such that they fulfill the Euler-Lagrange equations ( 46) taken as constraints.The Hamiltonian formalism is then based on the presymplectic form (27): where the canonical momenta (25) are λ a is defined by ( 4), and the dependence on A b is of functional type.In the language of [3], we refer to this as the momentum constraints p(y) = P (y, A) .Up to this point, we shall use the plural form for canonical momenta (or momentum constraints) to refer to one equation since it represents a continuous infinity of constraints, one for each y ∈ R 4 .Furthermore, we will write P a (y) instead of P a (y, A) to simplify the notation.
The Hamiltonian dynamics is then ruled by the Hamiltonian function (26): where is the Lagrangian.Furthermore, the field A b has to be expressed in terms of a suitable parametrization of D ph , e. g. the parameters U ± (k) in (82).Once this is set, we shall be able to ascertain whether ω is non-degenerate, hence symplectic, or not.
Using (45) and introducing the variable ζ = y − z , the momentum constraints become and, including (135), we have that Developing it a bit further, we arrive at and, combining this with (86) and (87), we obtain that h = R 3 dy H , where is the Hamiltonian density.
For the sake of convenience the Fourier transforms of the potentials and the momenta will be useful, namely dy e −iky P a (y) . (91) In terms of these variables the (pre)symplectic form (84) on the dynamic space is

The momenta
The Fourier transform of the momenta (89) is Then, using the definition (120), we have that and, by the convolution theorem, , where , and similarly .

The symplectic form on the physical dynamic space
The symplectic form ω ∈ Λ 2 (D ph ) results from replacing P a (k) and Âa (−k) in (92) with the constraints (94) and (79).Thus, substituting the lowest orders of these constraints, into the expression (92) and keeping only the lowest order terms, we obtain that where (78) has been included.
The derivation of the symplectic form to the next perturbative order is presented in detail in Appendix B, and the result is Recall that the physical dynamic space D ph is the quotient of D over the full gauge equivalence, first and second kind, which is coordinated with U ± (k) , according to (77-78).
Seemingly the expression (98) should hold at any perturbative order due to the relation (67) and the fact that the alias local theory is just Maxwell theory, whose symplectic form is exact.Equation (98) also points out what the canonical coordinates are to this approximation.Since Z c (x) has to be real, eq. ( 91) Substituting these in (98), we obtain and, introducing the new variables the symplectic form on D ph reduces to It is thus evident that the modes U j (k) and U † l (k) are a set of canonical variables whose non-vanishing elementary Poisson brackets are

The Hamiltonian on the physical dynamic space
As discussed at the beginning of Section 6, the Hamiltonian h defined on the dynamic space arises from implementing the constraints -namely, canonical momenta and field equations-into the Hamiltonian H defined on the kinematic space.
As we have done for the symplectic form, it is convenient to work in Fourier space, therefore, the Taking into account that at the lowest order -see Appendix B- and substituting the lowest orders of the constraints (96-95) into the Hamiltonian (103), the latter becomes (after a bit of algebra) where equation ( 78) has been used.In Appendix B we explicitly derive the next perturbative order of the Hamiltonian and the result is null, that is, In terms of the canonical variables (102) the Hamiltonian reads As commented before for the symplectic form, the vanishing of the O(1) terms in the Hamiltonian probably persists for higher orders and the relation ( 104) is likely exact.Indeed, taking U ± (k) as coordinates of the dynamic space D ph , they are the same as those for the local field Ẑa , which is nothing but a free Maxwell field.In fact, the implicit equation (67) establishes a one-to-one correspondence between the nonlocal field A a and its alias, the free Maxwell field Z a .
Therefore, the nonlocal field Âa is nothing but a fashionable deformation of its local alias, so it might seem appropriated to set the nonlocal Âa aside and stick with the local Ẑa .We will proceed this way in what concerns the Hamiltonian formalism, canonical coordinates and quantization.
However, the nonlocal field comes to fore when the coupling to other fields is considered.For this reason, it is worth writing Âc in terms of the canonical variables U and U † and we have that -see equation (107) Appendix B for details-

Quantization
The quantum commutation relations [28] trivially follow from (102) and the non-vanishing ones are The "Blackboard Bold" type emphasises that U l (k) and U † j (k) are the quantum counterparts of the classical variables U l (k) and U † j (k), and are the adjoint of each other.Additionally, we can introduce the annihilation and creation operators, a l (k) and a † l (k) respectively, as and its adjoint, with The corresponding Hamiltonian operator is where we have used equations (109-110), ω(k) := |k|, and is the number operator.Notice quickly that the Hamiltonian operator is the same as for the electromagnetic field, as we expected [29].
In turn the quantum operator for the nonlocal field in terms of the creation and anihilation operators a and a † is with

Conclusions
In this article, we provide an extensive overview of the nonlocal Lagrangian formalism proposed in [3,8], including the extension of Noether's first theorem.Additionally, we extend Noether's second theorem applied for these Lagrangians, highlighting their practical use in non-commutative U(1) -NCU(1)-gauge theory.
The nonlocal nature of NCU(1) gauge theory is embodied by the Moyal * operator, treatable as an infinite series or integral operator.In this article, we opt for the latter, motivated by the observations made in [1,2], where it is discussed that the functional space of this operator might be wider than the infinite series one.In this context, we have applied the nonlocal Lagrangian formalism and obtained its field equations (46).Next, we have studied how these field equations are transformed through the nonlocal gauge transformation (47).We have observed that it leaves the solution space invariant but not the field equations themselves.They are transformed into equivalent ones.For this reason, we have concluded that the nonlocal gauge transformation cannot be a Noether symmetry, otherwise the field equations would be completely invariant.
Upon close examination, we have observed that the identity (51) holds true even though the nonlocal Lagrangian is not gauge invariant.Remarkably, this identity has been corroborated by the extension of Noether's second theorem for nonlocal Lagrangians, thus affirming the validity of it.It is worth noting that, in this specific case, the use of the developed extension might have been deemed unnecessary, as mentioned earlier, a simple inspection reveals it.However, for more intricate scenarios, this extension can prove exceptionally valuable.
Since the NCU(1) gauge theory does not depend explicitly on the point x b and its Lagrangian density is Poincaré invariant, we have applied Noether's first theorem to obtain the canonical momentum energy tensor and the internal angular momentum current in integral form for this symmetry.Both exhibit a local part and a nonlocal part coming from nonlocality hidden in F ab .
One of the strengths of the nonlocal Lagrangian formalism was to be able to propose a Legendre transform in order to construct a Hamiltonian formalism for nonlocal theories.We have applied this Hamiltonian formalism to the NCU(1) gauge theory.To do so, we have studied the space of solutions, and we have proposed the construction of a perturbative solution through the equation (79) which, for simplicity of calculation, we have kept up to second order; the mathematical foundation of this proposal has been the inverse function theorem in Banach spaces.Once the pertubative solution is obtained, we have introduced it in the momenta constraint (94), and with them, we have calculated the symplectic form and the Hamiltonian.Furthermore, we have obtained the canonical variables and the elementary Poisson brackets, and implemented the canonical quantization.We have observed that up to second order terms the symplectic form and the Hamiltonian coincide with the electromagnetic theory, i.e. the first order of both vanish.This result is not surprising since equation (79) establishes a (pertubative) oneto-one correspondence between the nonlocal field A a and the local field Z a .For this reason, and in the light of this result, we suggest that this theory (which is nothing more than a fashionable deformation of Maxwell's electromagnetic theory) be set aside and work with the local theory.It is important to note that, if this theory were coupled with another field, the latter would not be true, and one would have to work with the nonlocal theory since the coupling would manifest itself in the context of A a and not Z a .
Finally, we have reviewed the Seiberg-Witten map.We have observed that the mapping (96) partially succeeds in connecting the nonlocal and nonabelian gauge theory with a local and abelian theory.We emphasize "partially" because the goal is only achieved at the level of the dynamical space, and not in the kinematic space which is where (at first order of θ ab ) the Seiberg-Witten map connects them.
Currently, the focus in the field of non-locality lies predominantly in the field of nonlocal gravity.It would be very interesting to be able to study this nonlocal formalism, together with Noether's second theorem, in models of nonlocal gravity [30][31][32][33] and to be able to see what conclusions can be drawn from it.Complex but challenging.
dq e iq(x+α) f (q) , can be translated into an integral expression -f (q) is the Fourier transform of f (x)-, and similarly for g(x + β) .Indeed, and, substituting the expressions for the Fourier transforms f (q) and ĝ(k), we arrive at with Defining ka := θ ab k b , the above expression becomes For the sake of simplicity, we shall hereon assume that θ ab is non-degenerate.Therefore, θbc exists such that θ ab θbc = δ a c and, using that Thus, the Moyal product (114) can be written as Similar expressions can be found for the degenerate cases.

The Fourier transform of the Moyal bracket
Given two functions, f (x) and g(x), let f := Ff and ĝ := Fg be their Fourier transforms and define that is, with where the expression (119) for S(u, v) has been included.Then, replacing this into (121), we easily arrive at

A list of useful properties
It is worth listing some properties that will be helpful for the developments throughout the paper, and whose proofs easily follow from the definitions: and fulfills the Jacobi identity 3.-The equality concerning the overall integral is useful to prove three helpful equalities: and 4.-The Leibniz rule for the derivative easily follows from (118-119): 5.
-Let x and z be independent variables, then - 8.-Let ξ = αy + βx , then - - - and, provided that f (y)g(y) → 0 at infinity, the last term on the right hand side vanishes.
Appendix B: The symplectic form and the Hamiltonian up to O(2) The perturbative expansion of (94) demands the previous expansion of (36) and (79).Keeping the two lowest orders only, we have that Combining them and using the FT of (123), one easily arrives at Straightforward computation that includes (77) yields Ẑc , Ẑd which, introduced in (136), yields Using this, the second and third terms in equation ( 94) respectively become and whereas the first term yields where f νµ := |k| 2 − (ν|q| + µ|p|) 2 .Putting them together and keeping only the symmetric part with respect to the exchange (ν, q a , c) ↔ (µ, p a , b) , equation (94) becomes

The Hamiltonian
The first order of the Hamiltonian (103) can be decomposed into where k = (k, σ) and dq dp W νc (q) W µd (p) δ(k − q ν − p µ ) Keeping only the symmetric terms with respect to the change (ν, q, c) ↔ (µ, p, d) and using the Dirac delta function, the right hand side of the latter becomes Therefore, equation (147) can be written as dq dp dk W νc (q) W µb (p) W ρd (k) δ(k + q + p) M cbd νµρ (q, p, k) , with M cbd νµρ (q, p, k) := sin Using (77) and the facts that k d W d = 0, k 2 ρ = 0 and k a ρ + q a ν + p a µ = f ρνµ δ a 4 , we can simplify it further: M cbd νµρ (q, p, k) := sin Moreover, it is worth remarking that M cbd νµρ (q, p, k) = M bcd µνρ (p, q, k) .
Now, if we use the relation C acb ρνµ = 0, we can simplify M cbd νµρ so that M cbd νµρ (q, p, k) = sin The first term is and the second term is To further simplify the right-hand side, we will use (78) together with q a ν + p a µ + k a ρ = −f ρνµ η 4a and k 2 ρ = 0.Then, we finally arrive at current (59) depends on ten independent constant parameters, namely ε b and ω ce , and its local conservation implies the local conservation of both the canonical energy-momentum T b c and the total angular momentum current J b ec separately, i. e. ∂ b T b c = 0 and ∂ b J b ec = 0 , where J b ec := 2x [e T b c] + S b ec (62) splits in an orbital part and an internal part.Now, to simplify the expressions of the canonical energy-momentum tensor and the internal angular momentum current, we can use the identity (133) in Appendix A definition (67) maps the dynamic space D of the nonlocal field A a onto the dynamic space D M of the Maxwell field Z a .This local commutative gauge theory acts as a local alias for our nonlocal non-commutative theory.Let us see what the nonlocal gauge transformation δA a = D a ε becomes in terms of the local field Z a .