Fundamentals of Time and Relativity

EM Action

  • The interaction of a particle with electric and magnetic fields is constructed by a linear coupling to the four-vector potential.
  • This minimal coupling to the electromagnetic field in the action reproduces the Lorentz force.

The invariant action describing the interaction of a charged particle with a electromagnetic field is constructed by combing the vector potential A μ with a trajectory segment:

10.7

S em :=−q ∫ 1 2 d x μ A μ (x)=−q ∫ 1 2 dσ x ˙ ⋅A(x)

Like in (10.3), σ is an invariant time parameter, and q is the charge of the particle.

To obtain the variation of S em we proceed as on the previous page. The variation of the vector potential is δ A μ (x)= ∂ ν A μ (x)δ x ν . After a partial integration we derive

10.8

δ S em =− qA⋅δx| 1 2 −q ∫ 1 2 dσ ( ∂ μ A ν − ∂ ν A μ ) x ˙ ν δ x μ

We set σ equal to the proper time τ to arrive at the final result

10.9

δ S em [x(τ)]=− qA(x(τ))⋅δx(τ)| 1 2 −q ∫ 1 2 dτ[ F( x(τ) )⋅u(τ) ] ⋅δx(τ)

in terms of the STA representation (9.15) of the electromagnetic field tensor which, on account of being an anti-symmetric bivector, has six non-vanishing components.

Setting now the variation of the combined action S= S free + S em equal to zero, we obtain the corresponding Euler-Lagrange equation:

10.10

dp(τ) dτ =qF( x(τ) )⋅u(τ)
  • The equation of motion as displayed in (10.10) is the coordinate-free version of the covariant Lorentz equation (9.7).
  • A gauge transformation A μ (x)→ A μ (x)− ∂ μ χ(x) adds a four-gradient to the action (10.7), but has no effect on the equation of motion.