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## Homework Statement

Maxwell's Lagrangian for the electromagnetic field is ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}## where ##F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}## and ##A_{\mu}## is the ##4##-vector potential. Show that ##\mathcal{L}## is invariant under gauge transformations ##A_{\mu} \rightarrow A_{\mu} + \partial_{\mu}\xi## where ##\xi=\xi(x)## is a scalar field with arbitrary (differentiable) dependence on ##x##.

Use Noether's theorem, and the spacetime translational invariance of the action, to construct the energy-momentum tensor ##T^{\mu\nu}## for the electromagnetic field. Show that the resulting object is neither symmetric nor gauge invariant.

Consider a new tensor given by ##\Theta^{\mu\nu}=T^{\mu\nu}-F^{\rho\mu}\partial_{\rho}A^{\nu}##. Show that this object also defines four conserved currents. Moreover, show that it is symmetric, gauge invariant and traceless.

Comment: ##T^{\mu\nu}## and ##\Theta^{\mu\nu}## are both equally good definitions of the energy-momentum tensor. However ##\Theta^{\mu\nu}## clearly has the nicer properties. Moreover, if you couple Maxwell's Lagrangian to general relativity then it is ##\Theta^{\mu\nu}## which appears in Einstein's equations.

## Homework Equations

## The Attempt at a Solution

Under gauge transformations ##A_{\mu} \rightarrow A_{\mu} + \partial_{\mu}\xi## where ##\xi=\xi(x)## is a scalar field,

##\delta\mathcal{L} = -\frac{1}{4}\delta(F_{\mu\nu}F^{\mu\nu})##

##\implies \delta\mathcal{L}=-\frac{1}{4}[(\delta F_{\mu\nu})(F^{\mu\nu})+(F_{\mu\nu})(\delta F^{\mu\nu})]##

##\implies \delta\mathcal{L}=-\frac{1}{2}(F_{\mu\nu})(\delta F^{\mu\nu})##

##\implies \delta\mathcal{L}=-\frac{1}{2}(F_{\mu\nu})[\delta(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})]##

##\implies \delta\mathcal{L}=\frac{1}{2}(F_{\mu\nu})[\partial^{\mu}(\delta A^{\nu})-\partial^{\nu}(\delta A^{\mu})]##

##\implies \delta\mathcal{L}=\frac{1}{2}(F_{\mu\nu})[\partial^{\mu}\partial^{\nu}\xi-\partial^{\nu}\partial^{\mu}\xi)]##

##\implies \delta\mathcal{L}=\frac{1}{2}(F_{\mu\nu})[\partial^{\mu}\partial^{\nu}\xi-\partial^{\mu}\partial^{\nu}\xi)]##

##\implies \delta\mathcal{L}=0##.

Therefore, ##\mathcal{L}## is invariant.

P.S.: The problem mentions that ##\xi=\xi(x)## has arbitrary (differentiable) dependence on ##x##. The differentiability of ##\xi=\xi(x)## is used in the lines ##\delta\mathcal{L}=\frac{1}{2}(F_{\mu\nu})[\partial^{\mu}\partial^{\nu}\xi-\partial^{\nu}\partial^{\mu}\xi)]## and ##\delta\mathcal{L}=\frac{1}{2}(F_{\mu\nu})[\partial^{\mu}\partial^{\nu}\xi-\partial^{\mu}\partial^{\nu}\xi)]##.

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Am I correct so far?

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