Question

. The action of a charged particle contains a term SdAj" (a) Taking ρ-Σ¡qo3(i-ri(t)) and J-Σ qiv(t)ốja-r,(t)) (where i, ri) verify that J d'xAu.j" leads to the usual terms L -... + Σ¡qi ( 6(t) . Αϊ-o(,)). (b) Explain why J d'aAj" is Lorentz invariant. (c) Verify that d4ΤΑμ]μ is gauge invariant (dropping surface terms associated with integrals of total derivatives) provided that j" is a conserved current

Answer 1

a) given, The action of a particle contains a term, s = ... + 1/c integral d^4 x A_mu j^mu rho = sigma_i q_i s^(3) (vector x -vector x_i (t)) J = sigma_i q_i vector v_i (+) S^3 (vector x -vector x_i (t)) Lagrangian from action, S = 1/c integral d^4 x A_mu j^mu S = integral dt integral d^3 x A_mu j^r /c The Lagrangian is integral d^3 x A_r j^r /c from the definition of action s = integral dt L. L = integral d^3 x /c A_mu j^r L = 1/c integral d^3 x g^mu nu A_r j_2 where, g^mu nu is Min Kowski metric, g^mu nu = matrix (-1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1), A_r = (phi (r), vector A) j_nu = (rho_s, vector J) By solving, [phi (x) vector A] matrix (-1 0 0 0 0 1 0 0 0 0 1 0 0 0