2. Relativistic charged particle Consider the action of a (relativistic) charged particle coupled to a background...

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2. Relativistic charged particle Consider the action of a (relativistic) charged particle coupled to a background gauge field, A. living in Minkowski space. It is governed by the action S[2] =/ds (-me •/do (-mc √-¹²-A₂ (z (0))+¹" (G).) where (a) General covariance in one dimension. Suppose we reparametrize the worldline according to 5+3(s). Use the chain rule to show that this change of coordinates preserves S. (b) Vary with respect to r(s) and find the Lorentz force law. (c) Vary with respect to A,(r) and find the source for Maxwell's equations produced by a trajectory of this particle. 2. Relativistic charged particle Consider the action of a (relativistic) charged particle coupled to a background gauge field, A. living in Minkowski space. It is governed by the action S[2] =/ds (-me •/do (-mc √-¹²-A₂ (z (0))+¹" (G).) where (a) General covariance in one dimension. Suppose we reparametrize the worldline according to 5+3(s). Use the chain rule to show that this change of coordinates preserves S. (b) Vary with respect to r(s) and find the Lorentz force law. (c) Vary with respect to A,(r) and find the source for Maxwell's equations produced by a trajectory of this particle.