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A gauge-symmetrization method for energy-momentum tensors in high-order electromagnetic field theories

For electromagnetic field theories, canonical energy-momentum conservation laws can be derived from the underpinning spacetime translation symmetry according to the Noether procedure. However, the canonical Energy-Momentum Tensors (EMTs) are neither symmetric nor gauge-symmetric (gauge invariant). The Belinfante-Rosenfeld (BR) method is a well-known procedure to symmetrize the EMTs, which also renders them gauge symmetric for first-order field theories. High-order electromagnetic field theories appear in the study of gyrokinetic systems for magnetized plasmas and the Podolsky system for the radiation reaction of classical charged particles. For these high-order field theories, gauge-symmetric EMTs are not necessarily symmetric and vice versa. In the present study, we develop a new gauge-symmetrization method for EMTs in high-order electromagnetic field theories. The Noether procedure is carried out using the Faraday tensor F, instead of the 4-potential A, to derive a canonical EMT T_N. We show that the gauge-dependent part of T_N can be removed using the displacement-potential tensor \mathcal{F}=\mathcal{D}*A/4\pi, where \mathcal{D} is the anti-symmetric electric displacement tensor. This method gauge-symmetrize the EMT without necessarily making it symmetric, which is adequate for applications not involving general relativity. For first-order electromagnetic field theories, such as the standard Maxwell system, \mathcal{F} reduces to the familiar BR super-potential \mathcal{S}, and the method developed can be used as a simpler procedure to calculate \mathcal{S} without employing the angular momentum tensor in 4D spacetime. When the electromagnetic system is coupled to classical charged particles, the gauge-symmetrization method for EMTs is shown to be effective as well.


I. INTRODUCTION
In classical field theories, one can derive canonical Energy-Momentum Tensors (EMTs) T µν N from the underpinning spacetime translation symmetry using the Noether procedure [1]. However, for classical systems of electromagnetic field, the canonical EMTs are neither symmetric with respect to tensor indices nor electromagnetic gauge invariant. Gauge dependence is un-physical, and non-symmetric EMT is not consistent with general relativity. In the present study, we will call a EMT symmetric if it is symmetric with respect to tensor indices, and gauge symmetric if it is gauge invariant. To date, much effort has been focused on symmetrizing the EMTs (with respect to tensor indices), while constructing gauge-symmetric EMTs is oftentimes a challenging task for general systems [2][3][4][5][6][7].
The first method for symmetrizing EMTs was discovered by Belinfante [8,9] and Rosenfeld [10], who added a divergence-free tensor ∂ σ S σµν to obtain a symmetric EMT, i.e., Here, T µν BR is Belinfante-Rosenfeld (BR) EMT, and S σµν is known as BR super-potential that depends on the angular momentum tensor and is anti-symmetric with respect to σ and µ [see Eq. (52)]. General relativity suggests another method to generate symmetric EMTs [11,12], which was modified by Gotay and Marsden, who employed constraints to define symmetric EMTs [13,14]. The relations between these three types of symmetric EMTs have been discussed in the literature [15][16][17].
In many systems, including the standard Maxwell system (6), the symmetrization of T µν N also renders it gauge symmetric. But for general electromagnetic field theories with highorder field derivations, symmetry with respect to tensor indices in general does not imply gauge symmetry and vice versa. High-order electromagnetic field theories appear in the study of gyrokinetic systems [18][19][20] for magnetized plasmas and the Podolsky system [21,22] for the radiation reaction of classical charged particles. In the present study, we propose a new method to gauge-symmetrize the canonical EMTs T µν N in general electromagnetic field theories with high-order field derivations. Our method removes the gauge dependence, but does not necessarily symmetrize the EMTs. In applications that don't involve general relativity, gauge-symmetrized EMTs are adequate.
We first reformulate the equation of motion for the field by the variational principle with respect to the Faraday tensor F µν , instead of the 4-potential A µ as in the standard field theory. The Euler-Lagrange (EL) equation is cast into an explicitly gauge-symmetric form.
The canonical EMT is then separated into a gauge-invariant part and a gauge-dependent part, the later of which contains the anti-symmetric electric displacement tensor D µν . We define a super-potential F σµν ≡ D σµ A ν /4π, called displacement-potential tensor, whose divergence with respect to the first index is divergence free with respect to the second index and removes the gauge dependence in the canonical EMT. It is simpler to calculate the displacement-potential tensor F σµν than the BR super-potential S σµν , and the former only gauge-symmetrizes the canonical EMT without render it symmetric (with respect to tensor indices). For first-order electromagnetic field theories, such as the standard Maxwell system, F σµν reduces to the familiar BR super-potential S σµν , and the method developed here can be used as a simpler procedure to calculate S σµν without employing the angular momentum tensor in 4D spacetime. In addition, when the electromagnetic system is coupled with classical charged particles, we find that the method is effective as well, even though the Lagrangian density is not gauge symmetric in general.
The paper is organized as follows. In Sec. II, we describe the gauge-symmetrization method for the EMT in a general high-order electromagnetic field theory, and highlight the difference in comparison with the BR method using the example of the Podolsky system [21,22]. Section III shows how the gauge-symmetrization method for the EMT works when the electromagnetic system is coupled with classical charged particles.

A. Explicitly gauge-symmetric Euler-Lagrange equation
The Lagrangian density of a general electromagnetic system is written as where A = (ϕ, −A) is the 4-potential and D = ((1/c) ∂ t , ∇) is the derivative operator over spacetime. The EL equation of the Lagrangian density is where the Euler operator E µ A of A µ is defined by . (5) Note that the Lagrangian density L F depends on derivatives of A with respect to the spacetime coordinates up to the (n + 1)-th order. It includes the standard Maxwell system, i.e., as a special case, where is the Faraday tensor. In Eq. (6), L F depends only on first-order derivatives of A. High-order electromagnetic field theories appear in the study of gyrokinetic systems [18][19][20] for magnetized plasmas and radiation reaction for classical charged particles [21,22]. Physics requires that the EL equation (4) is gauge symmetric, i.e, invariant under the gauge transformation In the present study, we assume that L F is explicitly gauge symmetric in the form of From the variational principle, δA = δ L F d 4 x = 0, we have where the boundary term has been dropped, and E µν F denotes the Euler operator for the Faraday tensor F µν defined by In Eq. (9), superscript [µν] represents anti-symmetrization with respect to µ and ν, i.e., Due to the arbitrariness of δA ν in Eq. (9), the equation of motion for the system is where is the electric displacement tensor.
In Sec. III, we will consider electromagnetic systems coupled with charged particles, and the Lagrangian density L will depend on 4-potential A µ i.e.,

Equation (12) then becomes
where is the free 4-current.
For the standard Maxwell system (6) without free 4-current, the electric displacement tensor is the Faraday tensor, i.e., D µν = F µν , and Eq. (12) reduces to Maxwell's equation

B. Infinitesimal criterion of symmetry and conservation laws
A continuous symmetry of the action A is a group of transformation such that where g ǫ constitutes a continuous group of the transformations parameterized by ǫ [23]. The infinitesimal generator of the transformation group is By rewriting the symmetry condition (18) as we can derive the following infinitesimal version of the symmetry condition, where pr (n+1) v is the prolongation of v. The standard prolongation formula for pr (n+1) v can be found in Ref. [23]. In the present study, we rewrite the prolongation formula with respect where and is a characteristic of the Lie algebra.
Combining Eqs. (12) and (21) generates the conservation law corresponding to the symmetry, where The conservation law given by Eq. (25) is not gauge-symmetric in general.

C. Gauge-symmetrization of the canonical EMT
Now we assume the high-order electromagnetic field theory admits the spacetime translation symmetry, i.e., and derive the corresponding energy-momentum conservation law. Because of Eq. (27), the action is invariant under the spacetime translation where X µ 0 is 4D constant vector field. The infinitesimal generator v, characteristic Q ν , and G σρ in Eq. (23) are The Lagrangian density satisfies the infinitesimal criterion because which implies a conservation law. Substituting Eqs. (29)-(31) into to Eq. (25), we obtain the canonical energy-momentum conservation law according the standard Noether procedure, In Eq. (33), T µν N is the canonical EMT derived from the standard Noether procedure. Obviously, T µν N depends on the gauge as expected. In the expression of T µν N given by Eq. (34), the gauge dependence comes from the second term, and the first and third terms are gauge symmetric. Now we show how to gauge-symmetrize T µν N . Note that because electric displacement tensor D σµ is anti-symmetric, the following equations hold, Here, F σµν is a super-potential that is anti-symmetric with respect to the first two indices.
For easy reference, we will call F σµν displacement-potential tensor. The divergence of F σµν defines a divergence-free tensor, i.e., where used is made of Eq. (12). When T µν 0 is added to T µν N , the gauge dependence is removed, i.e., where T µν GS is the gauge-symmetric EMT. It is worthwhile to mention that we derived the gauge-symmetrized EMT T µν GS from the expression of T µν N in Eq. (34), which is calculated from the prolongation with respect to F µν . On the other hand, had we started from Eq. (3) and calculated the EMT from the prolongation with respect to A µ , we would have obtained a canonical EMT in the form of wherê gauge theory [24].

D. Comparison with the BR method
As described above, the method proposed in the present study employs displacementpotential tensor F σµν to gauge-symmetrize the EMT, while the BR method use the superpotential S σµν to symmetrize the EMT. In this subsection, we discuss the difference between the displacement-potential tensor F σµν in Eq. (36) and the BR super-potential S σµν . To calculate S σµν , we need to first derive the 4D angular momentum conservation laws generated by the Lorentz symmetry. Assume that system is invariant under rotational transformation in 4D spacetime where {Λ µσ ǫ } is one-parameter subgroup of the Lorentz group. The infinitesimal generator v, the characteristic Q ρ , and the term G sρ are calculated respectively by Eqs. (19), (24), and where the anti-symmetric tensor Ω µσ = [dΛ µσ ǫ /dǫ] 0 is the Lie algebra element of the Lorentz group. Substituting Eqs. (44)-(46) into Eq. (25), we obtain the angular momentum conservation law in 4D spacetime, which can be rewritten as In above equations, where the superscript [µν] denotes anti-symmetrization with respect to µ and ν.
The anti-symmetric BR super-potentail S σµν is defined from the tensor S σµν in Eq. (48) as [8][9][10] It is clear from Eqs. (49)-(51) that S σµν and F σµν are related as follows, Equation (52) shows that in general F σµν is different from S σµν when ∆ σνµ is non-vanishing.
For a first-order field theory, such as the standard Maxwell system (6), n = 1 and the last three terms vanish such that S σµν = F σµν . In this situation, adding T µν 0 ≡ D σ F σµν to T µν N will render it both symmetric and gauge-symmetric, and the method developed here can be used as a simpler procedure to calculate the BR super-potential S σµν without the necessity to calculate the angular momentum tensor in 4D spacetime.
As an example of high-order electromagnetic field theory, we consider the Podolsky system [21,22], which was proposed to study the radiation reaction of classical charged particles.

The Podolsky Lagrangian density is
or in a manifestly covariant form We substitute the Lagrangian density (55) into Eq. (34) to obtain the canonical EMT where use is made of the following equations, The displacement-potential tensor is and Adding Eq. (63) to Eq. (56), we obtain the gauge-symmetric EMT, It is easy to see that T µν GS for the Podolsky system is not symmetric, i.e., T µν GS = T νµ GS . To calculate the BR EMT T µν BR for the Podolsky system, we evaluate the ∆ µνσ term in Eq. (52). Using Eq. (51), we have Substituting Eqs. (52) and (65) into Eq. (1), we obtain the BR EMT as It is easy to verify that T µν BR for the Podolsky system is both symmetric and gauge-symmetric.

PLED WITH CLASSICAL CHARGED PARTICLES
For self-consistent electromagnetic systems with free currents, the electromagnetic fields are coupled with charged particles. In this section, we apply the theory established in Sec. II to derive gauge-symmetric EMTs for electromagnetic systems coupled with classical charged particles.
For practical applications, such as in the gyrokinetic theory [18][19][20] for magnetized plasmas, reduced theoretical models are often adopted due to the intrinsic complexity of the systems. The equations of motion for the systems are usually gauge invariant, but the Lagrangian densities are not always specified by manifestly covariant forms. For these systems, energy and momentum conservation laws need to be derived separately. We demonstrate how the energy and momentum conservation laws can be transformed into gauge-symmetric forms using the "3+1" form of Eq. (36), i.e.,

A. Weak Euler-Lagrange equation and conservation law
The Lagrangian density of a generic classical electromagnetic field-charge particle system assumes the form of where the subscript a labels particles, X a is its trajectory, L a is its Lagrangian density, and Here, δ(x) is the Dirac δ-function.
In the "3+1" form, the equations of motion for the electromagnetic field are In this study, it is assumed that the Lagrangian density L is linear in terms of ϕ and A, and Eqs. (71) and (72) are thus gauge symmetric. Specifically, we assume that L depends on ϕ and A only through the term −q a δ a (ϕ + A ·Ẋ a /c), i.e., the Lagrangian density can be written as where "GSP (L)" denotes the gauge-symmetric parts of the Lagrangian density L. The right hand side of Eqs. (71) and (72) are the "3+1" form of Eq. (15), the free charge density ρ f and current density j f , respectively. Using Eq. (73), we have The equation of motion for particles is also derived from the variational principle. However, because particles and field reside on different manifolds, the equation of motion for particles will be the weak EL equation [20,[25][26][27] where E Xa is the Euler operator for the trajectory of the a-th particle, To derive a local conservation law from a symmetry, we need the infinitesimal symmetry criterion for the Lagrangian density. A symmetry of the action A ≡ Ldtd 3 x is defined by group transformations such that The corresponding infinitesimal generator of (77) is The infinitesimal criterion of the symmetry condition can be derived using the same procedure in Sec. II B, The prolongation of v now reads where is another characteristic of v induced by particle's trajectory. To obtain the corresponding conservation law, we transform the infinitesimal criterion into where The last two terms on the left-hand-side of Eq. (83) vanish due to Eqs. (71) and (72), but the fourth term does not because of the weak EL equation (75). If the characteristic q a is independent of x, E, and B, the conservation law of the symmetry is established as

B. Gauge-symmetric energy conservation law
We first derive the gauge-symmetric energy conservation law, assuming that the action The infinitesimal generator and characteristic are calculated as And the infinitesimal criterion (80) of the symmetry is The corresponding energy conservation law is thus where The energy density and flux in Eq. (91) are obviously gauge dependent. To gauge-symmetrize the conservation law, we add Eq. (67) to Eq. (91) and obtain, In deriving Eq. (94), we have rewritten the first and second terms of Eq. (67) as We now prove that the energy density and flux in Eq. (94) are gauge symmetric. It suffices to show that the following terms are gauge symmetric. Substituting Eq. (73) into the expression of s 1 , we have Similarly, s 2 is also gauge symmetric,

C. Gauge-symmetric momentum conservation law
We now discuss how to derive a gauge-symmetric momentum conservation law, assuming that the action A ≡ Ldtd 3 x of the electromagnetic field-charged particle system is invariant We emphasize that, different from the situation in standard field theories, this symmetry group simultaneously translates both the spatial coordinate x for the field and particle's position X a [25][26][27]. The infinitesimal criterion of this symmetry is From Eq. (100), the infinitesimal generator and its characteristic are The corresponding momentum conservation law is obtained by substituting Eqs. (102) and (103) into Eq. (86), i.e., wherē (−1) j D µ j+1 · · · D µ i ∇B · D µ 1 · · · D µ j−1 ∂L ∂D µ 1 · · · D µ j−1 D µ D µ j+1 · · · D µ i B . (106)

IV. CONCLUSION
In this study, we developed a gauge-symmetrization method for the energy and momentum conservation laws in general high-order classical electromagnetic field theories, which appear in the study of gyrokinetic systems [18][19][20] for magnetized plasmas and the Podolsky system [21,22] for the radiation reaction of classical charged particles. The method only removes the electromagnetic gauge dependence from the canonical EMT derived from the spacetime translation symmetry, without necessarily symmetrizing the EMT with respect to the tensor indices. This is adequate for applications not involving general relativity.
To achieve this goal, we reformulated the EL equation and infinitesimal criterion in terms of the Faraday tensor F µν . The canonical EMT T µν N is derived using this formalism, and it was found that the gauge dependent part of T µν N can be removed by adding the divergence of the displacement-potential tensor, which is defined as It was shown that the displacement-potential tensor F σµν is related to the well-known BR super-potential S σµν as where ∆ σνµ is defined in Eq. (51). Using the example of the Podolsky system [21,22], we show that ∆ σνµ in general is non-vanishing for high-order field theories. For a first-order field theory, such as the standard Maxwell system (6), ∆ σνµ vanishes such that S σµν = F σµν .
In the case, the method developed can be used as a simpler procedure to calculate the BR super-potential S σµν without the necessity to calculate the angular momentum tensor in 4D spacetime.
Lastly, we applied the method to derive gauge-symmetric EMTs for high-order electromagnetic systems coupled with classical charged particles. Using the "3+1" form of Eq. (36), we obtained the explicitly gauge-symmetric energy and momentum conservation laws in a general setting [see Eqs. (94) and (107)].