Geometric field theory and weak Euler–Lagrange equation for classical relativistic particle-field systems

Fan, Peifeng; Qin, Hong; Liu, Jian; Xiang, Nong; Yu, Zhi

doi:10.1007/s11467-018-0793-z

Cited by 6 publications

(35 citation statements)

References 31 publications

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“…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…”

Section: B Gauge-symmetric Energy Conservation Lawmentioning

confidence: 96%

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

Fan,

Xiao,

Qin

2021

Preprint

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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 gaugesymmetric (gauge invariant). The Belinfante-Rosenfeld (BR) method is a well-known procedure to symmetrize the EMTs, which also render 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 µν , instead of the 4-potential A µ , to derive a canonical EMT T µν N . We show that the gauge-dependent part of T µν N can be removed using the displacement-potential tensor F σµν ≡ D σµ A ν /4π, where D σµ 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, F σµν reduces to the familiar BR super-potential S σµν , and the method developed can be used as a simpler procedure to calculate S σµν 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.

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Section: B Gauge-symmetric Energy Conservation Lawmentioning

confidence: 96%

“…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]]…”

Section: A Weak Euler-lagrange Equation and Conservation Lawmentioning

confidence: 99%

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

Fan,

Xiao,

Qin

2021

Preprint

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“…We start from the action of particle-field systems and revisit the field theory on heterogeneous manifolds developed in Refs. [70][71][72]. We extend the theory to include high order field derivatives and use noncanonical phase space coordinates (X a , U a ) for particles.…”

Section: A Weak Euler-lagrangian Equationmentioning

confidence: 99%

“…Equations ( 11) and ( 12) are called submanifold Euler-Lagrange equations for X a and U a because they are defined only on the time axis after integrating over the spatial dimensions [70][71][72]. We can easily prove that the submanifold EL equations ( 11) and ( 12) are equivalent to the standard EL equation…”

Section: A Weak Euler-lagrangian Equationmentioning

confidence: 99%

“…Recently, this difficulty is overcome by the development of an alternative field theory for the particle-field system [70][71][72]. This new field theory embraces the fact that different components, i.e., particles and electromagnetic field, reside on heterogeneous manifolds, and a weak Euler-Lagrange equation was derived to replace the standard Euler-Lagrange equation for particles.…”

Section: Introductionmentioning

confidence: 99%

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Discovering exact, gauge-invariant, local energy-momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds

Fan,

Qin,

Xiao

2020

Preprint

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Gyrokinetic theory is arguably the most important tool for numerical studies of transport physics in magnetized plasmas. However, exact local energy-momentum conservation law for the electromagnetic gyrokinetic system has not been found despite continuous effort. Without such a local conservation law, energy-momentum can be instantaneously transported across spacetime, which is unphysical and casts doubt on the validity of numerical simulations based on the gyrokinetic theory. Standard Noether's procedure for deriving conservation laws from corresponding symmetries does not apply to gyrokinetic systems because the gyrocenters and electromagnetic field reside on different manifolds. To overcome this difficulty, we developed a high-order field theory on heterogeneous manifolds for classical particle-field systems and apply it to derive exact local conservation laws, in particular the energy-momentum conservation law, for the electromagnetic gyrokinetic system. A weak Euler-Lagrange equation is established to replace the standard Euler-Lagrange equation for the particles. It is discovered that an induced weak Euler-Lagrange current enters the local conservation laws. And it is the new physics captured by the high-order field theory on heterogeneous manifolds.

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General field theory and weak Euler-Lagrange equation for classical particle-field systems in plasma physics

et al. 2019

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A general field theory for classical particle-field systems is developed. Compared with the standard classical field theory, the distinguish feature of a classical particle-field system is that the particles and fields reside on different manifolds. The fields are defined on the 4D space-time, whereas each particle's trajectory is defined on the 1D time-axis. As a consequence, the standard Noether's procedure for deriving local conservation laws in space-time from symmetries is not applicable without modification. To overcome this difficulty, a weak Euler-Lagrange equation for particles is developed on the 4D space-time, which plays a pivotal role in establishing the connections between symmetries and local conservation laws in space-time. Especially, the non-vanishing Euler derivative in the weak Euler-Lagrangian equation generates a new current in the conservation laws. Several examples from plasma physics are studied as special cases of the general field theory.In particular, the relations between the rotational symmetry and angular momentum conservation for the Klimontovich-Poisson system and the Klimontovich-Darwin system are established.

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Geometric field theory and weak Euler–Lagrange equation for classical relativistic particle-field systems

Cited by 6 publications

References 31 publications

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

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

Discovering exact, gauge-invariant, local energy-momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds

General field theory and weak Euler-Lagrange equation for classical particle-field systems in plasma physics

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