Peifeng FAN (范培锋) ,Qiang CHEN (陈强) ,Jianyuan XIAO (肖建元) and Zhi YU (于治)

1 School of Physics and Optoelectronic Engineering,Anhui University,Hefei 230601,People’s Republic of China

2 National Supercomputing Center in Zhengzhou,Zhengzhou University,Zhengzhou 450001,People’s Republic of China

3 School of Nuclear Science and Technology,University of Science and Technology of China,Hefei 230026,People’s Republic of China

4 Institute of Plasma Physics,Chinese Academy of Sciences,Hefei 230031,People’s Republic of China

Abstract In both quantum and classical field systems,conservation laws such as the conservation of energy and momentum are widely regarded as fundamental properties.A broadly accepted approach to deriving conservation laws is built using Noether’s method.However,this procedure is still unclear for relativistic particle-field systems where particles are regarded as classical world lines.In the present study,we establish a general manifestly covariant or geometric field theory for classical relativistic particle-field systems.In contrast to quantum systems,where particles are viewed as quantum fields,classical relativistic particle-field systems present specific challenges.These challenges arise from two sides.The first comes from the mass-shell constraint.To deal with the mass-shell constraint,the Euler-Lagrange-Barut (ELB)equation is used to determine the particle’s world lines in the four-dimensional(4D)Minkowski space.Besides,the infinitesimal criterion,which is a differential equation in formal field theory,is reconstructed by an integro-differential form.The other difficulty is that fields and particles depend on heterogeneous manifolds.To overcome this challenge,we propose using a weak version of the ELB equation that allows us to connect local conservation laws and continuous symmetries in classical relativistic particle-field systems.By applying a weak ELB equation to classical relativistic particle-field systems,we can systematically derive local conservation laws by examining the underlying symmetries of the system.Our proposed approach provides a new perspective on understanding conservation laws in classical relativistic particle-field systems.

Keywords: high-order field theory,weak Euler-Lagrange-Barut equation,infinitesimal criterion of symmetric condition,Noether's theorem,geometric conservation laws

1.Introduction

Classical relativistic particle-field systems,in which a mass of particles interact with the self-generated and high background fields,generally appear in astrophysical objects[1-3],plasma beams[4],and gyrokinetic systems in fusion plasma[3,5,6].For these systems,one of the significant topics of study focuses on the derivation of energy and momentum conservation laws [7-9].

As a widely accepted fundamental principle,conservation laws can be derived by the corresponding symmetry that the Lagrangian (or action) of the system admits.This is the so-called Noether’s theorem [10].This method has been widely used in deriving energy-momentum conservation in quantum field systems by space-time translation symmetry[11].However,for classical relativistic particle-field systems,the derivation procedure is still elusive.For example,for an electromagnetic system coupled with relativistic particles,the energy-momentum conservation was firstly derived by Landau and Lifshitz [12],and just reformulated into a geometric form recently from the space-time translation symmetry [13].

Different from quantum systems,the dynamics of fields and particles lie on heterogeneous manifolds.The fields,such as the electric and magnetic fields,rely on the four-dimensional (4D) space-time domain.However,particles as world lines in Minkowski space are only defined on one-dimensional (1D) parametric space (such as the time-axis),rendering the standard Euler-Lagrange (EL) equation inapplicable[14].This difficulty has been overcome recently by transforming the EL equation into a weak form [14,15],and the corresponding method has been applied in nonrelativistic systems,such as the Klimontovich-Poisson (KP) system,the Klimontovich-Darwin (KD) system,and the gyrokinetic system in plasma physics.

However,for relativistic situations with the a geometric setting,this theory was only reformulated for Maxwell's first order field system coupled with charged particles [13].By splitting space-time,previous studies [16,17] have established a general field theory that is applicable both to relativistic and nonrelativistic classical particle-field systems.In this study,we will geometrically reformulate the field theory of classical particle-field systems established in [16,17].

Geometric formalism is a popular method used to describe particle-field systems in classical relativistic mechanics.To improve this approach,the proper time parameter (instead of the standard time parameter) is often employed.However,this leads to the mass-shell constraint,which must be taken into account when deriving the equation of motion using Hamilton’s principle.In fact,the resulting equation is the Euler-Lagrange-Barut (ELB) equation rather than the standard EL equation [18,19].Furthermore,the Lagrangian density becomes a functional of the particle’s world line,rather than a simple function,when the proper time parameter is utilized.Despite the advantages of this approach,there are still some challenges to overcome.Specifically,the weak form equation of particle motion and the infinitesimal criterion of the symmetry condition are not yet fully understood for general classical relativistic particle-field systems.These issues remain an active area of research in the field and require further investigation.

In this study,we extend the theory developed in [13] to encompass general high-order field systems.Such systems are commonly encountered in the study of gyrokinetic systems for magnetized plasma[20]and the Podolsky system[21,22],which deals with the radiation reaction of classical charged particles.Specifically,we extend the weak ELB equation for Maxwell’s system to accommodate general situations.

In addition,we reformulate the infinitesimal criterion of the symmetry condition in light of the geometric setting.This is important because the Lagrangian density,which is not only a function but also a functional,requires an integrodifferential equation for the infinitesimal criterion.This stands in contrast to the nonrelativistic situation,where the Lagrangian density is a function,and the infinitesimal criterion is therefore a differential equation[23].Using the general weak ELB equation and the infinitesimal criterion developed in this study,the conservation laws can be systematically derived using Noether’s method from underlying symmetries.

Our theory is applicable to a wide range of particle-field systems with nonhybrid variables.To illustrate its effectiveness,we provide a nontrivial example of a high-order case involving the energy-momentum conservation laws for a Podolsky system coupled with relativistic charged particles.Additionally,we derive the energy and momentum conservation laws for zeroth order relativistic gyrokinetic systems using the space-time split form.

The remainder of the paper is organized as follows.The Lagrangian and action of general classical relativistic particlefield systems are given in section 2,as well as the method of establishing the weak ELB equation.In section 3,the geometric infinitesimal criterion of the symmetry is developed as required by the mass-shell constraint.Using the geometric weak ELB equation and geometric infinitesimal criterion,the conservation laws are obtained in section 4.In the last section,the paper derives the energy-momentum conservation laws for high-order electromagnetic systems coupled with charged particles.

2.Classical relativistic particle-field systems and a geometric weak Euler-Lagrange-Barut equation

In this section,we aim to extend the theory previously developed in[13,15]to encompass relativistic situations.We assume a background space-time that is Minkowskian,endowed with a Lorentzian metric.It is important to note that the scope of our current study does not include fundamental plasma dynamics in curved space-time,which have previously been investigated by Elsässer and Popel [24].We begin by considering the geometric action of the particle-field system in order to derive the equations of motion for this system.The geometric action and Lagrangian of a classical relativistic particle-field system are generally written as

where the subscript a labels particles,χ is the space-time position,andψ(χ) is a scalar field,vector,or any kind of tensor field on space-time.Here,pr(n)ψ(χ)is the prolongation of the fieldψ(χ) defined by

which differs from the usual Euler operator in that the last two terms are present.Here,id is the identity operator.To be faithful to history,equation (4) was first obtained by Barut using the Lagrange multiplier method[18].Therefore,we will refer to the linear operatoras the ELB operator,and we will call equation (4) the ELB equation.

The integral of the ath particle’s Lagrangian Lain equation (1) is along with an arbitrary time-like world line(denoted by la),which connected two fixed world points a1and a2at the space-time R4,while the integral of LF(the Lagrangian density for fields) is defined on the space-time domain Ω.Mathematically,particles and fields are stated to exist on heterogeneous manifolds.As a consequence,the action in the form(1)is not applicable when using Noether's method to derive local conservation laws.

To deal with this problem,we can transform equation(1)into

by timing the identity

to the first part of equation (1),whereδa≡δ(χ-χa(τa)) is the Dirac’s delta function.Here,

are Lagrangian densities for the field and the ath particle.Different from the nongeometric situation,the Lagrangian densityL is a functional of the particle’s world line rather than a function of space-time.To differ from the other local variables (such as χ andψ(χ)),we enclosed χawith square brackets (see equation (6)).

We now determine how the action (6)varies in response to the variations of δχaand δψ,

Using Hamilton’s principle,we immediately obtain the equation of motion for fields and particles

Similar to the submanifold EL equation given in [14,15],equation (13) is called the submanifold ELB equation.Using the linear property of the ELB operator,we can easily prove that the submanifold ELB equation (13) is equivalent to the ELB equation (4).

To apply equation(13)to Noether’s method,we need the explicit expression of.For an electromagnetic system,the expression was given in a previous study [13].The general expression ofis derived as follows.We first transform the first and last three terms of equation (13) into the following forms

In deriving equation (15),we use the fact that Lais the function ofbut not of χμ,as well as the relation

where x and y are arbitrary independent variables.The first term on the right-hand side of equation (16) can be rewritten as

where we used the ELB equation (4).We will refer to equation (19) as the weak ELB equation.As a differential equation,the weak ELB equation is equivalent to the submanifold ELB equation (13) and can also be used to determine the motion of particles.Similar to the weak EL equation used in nonrelativistic particle-field systems,the weak ELB equation is also essential in deriving local conservation laws for relativistic situations.

3.Geometric infinitesimal criterion of the symmetry condition

In this section,we discuss the local symmetries of the relativistic particle-field systems.A symmetry of the actionA is a local group of transformation (or a local flow)

such that

and

For the purpose of deriving local conservation laws,an infinitesimal version of the symmetry condition is needed.To do this,we take the derivative of equations(21)and(22)with respect to ϵ at ϵ=0,

Substituting equations(8)and(9)into equation(23),we have

where

Because equation (25) survives for any small integral domains,the integrand must be zero,i.e.

Equation (27) can finally be transformed into

where

is the infinitesimal generator of the group transformation,and where

are characteristics of the infinitesimal generatorv.Equation (28) is the infinitesimal criterion of the symmetry condition(21).To better understand the prolongation formula(31),we refer the interested reader to the detailed derivations presented in [23].Notably,unlike previous references,we encounter an integral along the particle’s world line in the infinitesimal criterion(see equation(28)).This is the result of the fact that the Lagrangian densityL is a functional of the particle’s world lines,rather than a function of space-time variables.As a result,the infinitesimal criterion (3) is not a differential equation but an integro-differential equation that involves this integral.

We can also derive an additional infinitesimal criterion by utilizing equation (24),which yields the result that

In deriving equation (34),we used the equations

4.Conservation laws

The infinitesimal criterion presented in section 3 possesses unique characteristics,particularly the presence of an additional infinitesimal criterion(34),which sets it apart from the standard situation without constraints.As a consequence,the derivation process of the local conservation law and the resulting expressions are notably different.In the following section,we illustrate how the two infinitesimal criteria (28)and (34) combine to determine a conservation law.

To begin,we will convert equation (28) into an equivalent form.The first term of this equation can be written in component form as

The first four terms of the right-hand side of equation(37)can be transformed into

The detailed derivations are shown in appendix C.The last three terms of equation (37)) can be rewritten as

by the standard derivation process [23],where

Similarly,the last two terms in equation (28) can be represented as

where

Combining equations (37)-(39),and (41),equation (28)is then transformed into

where we used

Assuming that ξ and Q are independent of τa,equation (43)becomes

By utilizing the EL equation(14)of the field ψ,the termEψ(L)·Qin equation (45) vanishes.However,the second term in equation (45) remains nonzero due to the weak ELB equation (19).In cases where the characteristicqadoes not involve χ and ψ,this term can be converted to a divergence form,namely

where

Substituting equation (47) into (45),we finally arrive at the geometric conservation law

5.Gauge-symmetric energy-momentum conservation laws for high-order electromagnetic systems coupled with charged particles

We now apply the general theory to a high-order electromagnetic system coupled with charged particles.The Lagrangian Lafor this system can generally be written in a gauge-symmetric form as

Using equation(14),we can obtain the equation of motion for the electromagnetic field,

To derive the gauge-symmetric energy-momentum conservation laws,the derivatives related to the four-potential Aμmust be changed into derivatives with respect to the Faraday tensor Fμν.Using the derivative rule of composite function,the third term in equation (54) is transformed into

Specifically,the second term in equation (54) is

Substituing equations (54)-(56) into (53),we have

and the superscript [μν] in equation (58) denotes the antisymmetrization with respect of μ and ν.Using equation(57),equation (53) can also be transformed into

by defining the electric displacement tensor

Here,dsa=cdτais the line element in the Minkowski space.

Without losing generality,we assume that equation (59)is gauge-symmetric and that the Lagrangian density depends on Aμonly through the term,i.e.

where qais the charge of the ath particle and ‘GSP (L)’denotes the gauge-symmetric parts of the Lagrangian density.For the standard Maxwell system,GSP(L)=,the Lagrangian density is then

by using equations (60) and (61).The general equation of motion(59)is then reduced to the standard Maxwell equation

From equation(64),we can see that the electric displacement tensor is equal to the Faraday tensor,i.e.Dμν=Fμν.In general,for systems involving an electromagnetic field and matter interactions,such as plasma [25] and other polarized media [26,27],the electric displacement tensor is distinct from the Faraday tensor.

The prolongation formula is specifically expressed as

by equation (31) where

and

To get the gauge-symmetric energy-momentum conservation laws,the prolongation formula(67)must be reconstructed by using the derivatives with respect to Fμν.Similar to the transformation process of equations (54)-(57),equation (67)can be also transformed into

by using equations (55) and (56),where

Moreover,the term Qμ(see equation (46)) is transformed into

which can be converted into

using the derivatives with respect to Fμν.Here,Pμin equation (73) is defined by

The expression of equation(73)can also be obtained by using the prolongation formula (70) and the ‘integrated by parts’technique,as described in [23].Substituting equation (74)into equation(49),we finally arrive at the conservation law of the high-order electromagnetic system

We now turn to calculating the energy-momentum conservation law from the space-time translation symmetry.Assuming that the action of the system,given by equation(6),remains unchanged under space-time translations,i.e.

the corresponding infinitesimal generator of the group transformation (76) can be written as

The conservation law (75) becomes

by substituting equations (81) and (82) into equation (75),where

to equation (87) to get the explicitly gauge-symmetric conservation law

where

is the ‘improved’ or gauge-symmetric energy-momentum tensor.Equation (59) is used in deriving equation (90).The completed proof for the gauge-symmetry ofis included in appendix D.

Lastly,as a nontrivial system with high-order field derivatives,we discuss the Podolsky system [21] coupled with relativistic charged particles.In this system,the Lagrangian densities of the fields and particles are expressed as

The subscript ‘P’ is short for Podolsky system.Substituting equations (91) and (92) into equation (59),we obtain the equation of motion for an electromagnetic field

In this study,we assume that the spatial scale is much larger than the electron’s classical radius,and,therefore,we neglect the self-interactions within a single electron.As a consequence,the radiation reaction force is significantly smaller than the background electromagnetic fields,and the energy emitted from the charges can be neglected when compared to the background fields.However,in situations where the space scale is comparable to the electron’s classical radius,the radiation energy emitted from the charges cannot be ignored.

When calculating the radiation reaction force acting on the electron,we encounter a divergence problem as the electromagnetic mass approaches infinity when the electron is confined to an infinitely small volume.To overcome this issue,several theoretical models have been proposed to address this divergence issue by modifying Maxwell’s equations or the radiation reaction force (see [28]).Particularly,Podolsky’s model uses a Lagrangian with high-order field derivatives to cancel the divergence energy arising from a single electron’s self-interactions.Despite ongoing discussions and research,this issue remains elusive,as discussed in recent studies such as[29].We note that,since this study does not consider the radiation effect,the divergence problem is not relevant for our current analysis.

Substituting equation (91) into equation (86),we obtain

The expressions for the tensors Σμνand the tensor Dσμin equation (90) have been previously derived in [17],and we rewrite these two tensors as

Using equations (91)-(96),we can obtain the gauge-symmetric energy-momentum tensor as

6.Conclusions and discussions

This study expands upon previous research[13]to encompass a broader range of scenarios.Specifically,a manifestly covariant or geometric field theory for classical relativistic particle-field systems is established in a more general framework.

In this study,we establish the geometric infinitesimal criterion for the symmetry condition in general transformations,such as space-time translation,space-time rotation,and other transformations,as shown in equation (28).As previously pointed out in[13],the use of a proper time parameter results in the Lagrangian density (8)becoming a function of the field ψ and a functional of the particle’s world line.Consequently,the standard infinitesimal criterion becomes an integro-differential equation rather than a differential equation,as demonstrated in equation (28).Moreover,we derive an additional criterion(34)to ensure the mass-shell condition is satisfied.

We also obtain a weak ELB equation for high-order particle-field systems on the Minkowski space,which is rewritten here as

Combining the weak ELB equation and the infinitesimal criterion of the symmetry conditions,general geometric conservation laws are systematically derived.Using the general theory constructed here,we obtain the energymomentum conservation laws for high-order relativistic electromagnetic systems based on space-time translation symmetry.

It is intriguing to note that,for transformations described by the characteristicas given in equation (20),if they are related to the space-time position χ or the field ψ,the derivative operator Dνcannot be taken outside the integral.This leads to equation(99)being unable to take a divergence form,which in turn implies that,even if the system admits a continuous symmetry,a conservation law cannot be established.So far,no exception symmetry has been discovered that does not result in a conservation law.This raises an open question that we recommend as a promising research direction.Further exploration in this area has the potential to shed more light on the relationship between symmetries and conservation laws,and could lead to exciting new discoveries.

To derive the conservation laws for geometric gyrokinetic systems,there are two possible approaches.One method involves developing a geometric gyrokinetic field theory using canonical variables,where the weak ELB equation remains applicable.Although canonical variables have been discussed for nonrelativistic gyrokinetic systems in previous studies (see [30,31],for example),their application to compute the gyrocenter motion of test particles in tokamaks with a large magnetic perturbation has yet to be established for geometric gyrokinetic systems.The other approach involves developing a different type of weak form equation of motion using hybrid variables.This can be accomplished through the Lagrange multiplier method or by utilizing the arbitrary invariant parameter to deal with the constraints.However,the corresponding methods for determining conservation laws for these approaches have not yet been established.Future work will delve into the possible methods to determine these conservation laws.

Additionally,this study has potential implications for the development of structure-preserving algorithms.In recent years,a series of such algorithms has been developed to simulate relativistic plasma systems over long time periods.These algorithms include volume-preserving algorithms [32]and symplectic algorithms [33,34].Recently,there has been progress in developing Lorentz-invariant volume-preserving algorithms for relativistic dynamics of charged particles [35].Given that our present study is based on a covariant highorder field theory,we believe that it provides a fundamental framework that could be extended to construct structurepreserving algorithms for self-consistent relativistic particlefield systems.Potential methods to achieve this will be investigated in future studies.

Acknowledgments

PF was supported by National Natural Science Foundation of China (No.12005141).QC was supported by National Natural Science Foundation of China (No.11805273).JX was supported by the Collaborative Innovation Program of Hefei Science Center,CAS (No.2021HSCCIP019),National MC Energy R&D Program(No.2018YFE0304100),and National Natural Science Foundation of China (No.11905220).

Appendix A.Geometric energy-momentum conservation laws in curvilinear coordinates

In this appendix,we give the expressions of geometric energy-momentum conservation laws in arbitrary curvilinear coordinates.Typically,the geometric energy-momentum conservation laws can be written as

where Tμνis the EMT,which can be written in a‘3+1’form as

whereRand σ are the momentum density and momentum flux(or stress tensor),and W andSare the energy density and energy flux,respectively.Typically,the tensor Tμνis symmetric,andR=S/c2.In this study,we assume thatR≠S/c2.Substituting equation (A2) into equation (A1),we get the local energy and momentum conservation laws,

Assuming the existence of arbitrary curvilinear coordinates{xi∣i=1,2,3},we can define the coordinate basis (or covariant basis)

The dual basis (or contravariant basis),denoted bygi,is determined by

In the coordinate basis{gi},S,R,and σ can be represented as

Substituting equations (A7) into (A3) and (A4),we obtain

where ∇iis the covariant derivative.When acting on a tensor field,it fulfils the condition

More specifically,if the coordinate{xi}is orthogonal,the Christoffel symbols can be written as

where Hi≡|gi|.Therefore,equations (A8) and (A9) can be rewritten as

Appendix B.Energy and momentum conservation laws for gyrokinetic systems

Relativistic electrons,particularly runaway electrons,play a significant role in the plasma stability of nuclear fusion devices [5,9,36] and have attracted considerable attention due to their deleterious effects on the device during disruption events [37].The orbit of a relativistic electron is often determined by the relativistic guiding-center Lagrangian.Transport equations,including the energy and momentum transport equations,are derived from the conservation laws of energy and momentum.In this appendix,we derive the energy and momentum conservation laws for relativistic gyrokinetic systems using the space-time splitting formalism.The phase space Lagrangian of a relativistic guiding-center motion in the guiding-center coordinate was first derived by Grebogi and Littlejohn [9,36] as

Using the previously developed space-time splitting formalism of the field theory,which is valid for both relativistic and nonrelativistic gyrokinetic systems,the general formulas for energy and momentum conservation laws are expressed here as

Substituting equations(B1)and(B2)into equations(B3)and(B4),we then obtain the energy and momentum conservation laws,

respectively.In deriving equations (B5) and (B6),the following equations are used

The local conservation laws(B5)and(B6)are written in terms of a particle’s phase space coordinates(Xa(t),pa(t)).To express the system in a statistical form using distribution functions of particles and fields,we classify the particles based on their invariants,such as mass and charge.Each particle is assigned a subscript‘a’,which represents a pair of indices,and s and p,indicating the p-th particle of the sspecies,i.e.

For each species,the Klimontovich distribution function is defined to be

For a functionga(x,p),the label a～sp can be replaced just by s,i.e.

In the conservation laws(B5)and(B6),the summations in the form of ∑aga(Xa(t),Ua(t))δacan be expressed in terms of the distribution functionsFs(t,x,u),

Using equation (B12),the conservation laws (B5) and (B6)can be equivalently written in the statistical form in terms of the distribution functionsFs(t,x,p) and fieldsφ(t,x) andA(t,x) as

In the nonrelativistic limit,we have

The energy and momentum conservation are then reduced to

and

Equations(B18)and(B19)agree with the results of Brizard and Tronci [38].

Appendix C.Derivations of equation (38)

In this appendix,we give detailed derivations of equation (38).We can directly transform the first four terms of the right-hand side of equation (37) as follows:

where we use the second infinitesimal criterion (34) in the last step.

Appendix D.Proof of gauge-symmetry of

In this appendix,we prove thatin equation (90) is gaugesymmetric.It is sufficient to show that the first three terms in the right-hand side of equation(90)are gauge-invariant.Substituting equations (62) and (86) into equation (90),these terms are

where equation (61) is used in the last step.Equation (D1)confirms thatis gauge-symmetric.

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