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

Abstract

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.