Relativistic Coupling of Charged Particles to EM Fields

Core Idea

The relativistic Lorentz force dp^μ/dτ = q F^μν u_ν expresses particle motion in manifestly covariant form using 4-momentum and 4-velocity. The action S = -mc²∫dτ - q∫A_μ dx^μ encodes electromagnetic coupling, with canonical momentum p = mv + qA differing from kinetic momentum.

Explainer

The non-relativistic Lorentz force F = q(E + v × B) correctly describes slow charged particles in electromagnetic fields, but it breaks Lorentz symmetry — it mixes components of E and B in a way that depends on the frame. Having studied the electromagnetic field tensor F^μν, you know that E and B are not separate entities but components of a single antisymmetric rank-2 tensor that transforms covariantly under Lorentz boosts. The relativistic equation of motion must be written in terms of this tensor to be frame-independent.

The covariant equation of motion dp^μ/dτ = q F^μν u_ν accomplishes exactly this. Here, p^μ = mγ(c, v) is the four-momentum, u_ν = γ(c, −v) is the covariant four-velocity, τ is the particle's proper time, and F^μν is the field tensor. The μ = 1,2,3 spatial components of this equation reproduce the relativistic generalization of the magnetic and electric forces, while the μ = 0 temporal component gives the relativistic work-energy theorem dp⁰/dτ = γ dE/dt = qγ E·v — power delivered by the electric field. The full equation is a single four-vector equation, manifestly Lorentz covariant, that reduces exactly to the non-relativistic Lorentz force when v ≪ c.

The deeper structure comes from the action principle. The action S = −mc²∫dτ − q∫A_μ dx^μ has two terms: the free relativistic particle term (proportional to proper time, from special relativity) and the coupling term −q∫A_μ dx^μ = −q∫(φ dt − A·dx) that encodes how the particle couples to the electromagnetic potential. Varying this action with respect to the particle's trajectory gives the covariant Lorentz force equation. This action formulation is crucial because it immediately reveals the canonical momentum: differentiating the Lagrangian with respect to velocity gives p_canonical = mγv + qA. This canonical momentum p + qA is the conserved quantity associated with translational symmetry in the presence of a vector potential A, and it differs from the kinetic momentum p = mγv by the term qA. The distinction becomes essential in quantum mechanics, where the canonical momentum is what the momentum operator represents — not the kinetic momentum — leading to the minimal coupling prescription ∇ → ∇ − iqA/ℏ that governs how quantum particles interact with electromagnetic fields.