Energy–momentum tensor from diffeomorphism invariance in classical electrodynamics
摘要
We reexamine the energy–momentum tensor in classical electrodynamics from the perspective of spacetime-dependent translations, i.e., diffeomorphism invariance in flat spacetime. When energy–momentum is identified through local translations rather than constant ones, a unique, symmetric, and gage-invariant energy–momentum tensor emerges that satisfies a genuine off-shell Noether identity without invoking the equations of motion. For the free electromagnetic field, this tensor coincides with the familiar Belinfante–Rosenfeld and Bessel–Hagen expressions, but arises here directly from spacetime-dependent translation symmetry rather than from improvement procedures or compensating gage transformations. In interacting classical electrodynamics, comprising a point charge coupled to the electromagnetic field, diffeomorphism invariance yields well-defined energy–momentum tensors for the field and the particle, while the interaction term itself generates no independent local energy–momentum tensor. Its role is instead entirely encoded in the coupled equations of motion governing energy–momentum exchange, thereby resolving ambiguities in the local definition of the energy–momentum tensor present in canonical and improvement-based approaches.