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## Homework Statement

I want to be able, for an arbitrary Lagrangian density of some field, to derive the energy-momentum tensor using Noether's theorem for translational symmetry.

I want to apply this to a specific instance but I am unsure of the approach.

## Homework Equations

for a field undergoing a translational gauge transformation we also add a rotational gauge transformation to make it gauge invariant so we have the change in the vector field potential as

$$\delta A^\mu = \varepsilon^\mu (-\partial_\nu A^\mu + \partial^\mu A^\nu ) $$

for example the Klein-Gordon field has Lagrangian density

$$L = 1/2 \eta^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - 1/2 m^2 \phi^2 $$

From Noether's theorem the current is

J^\mu = $$ \dfrac{\partial L}{ \partial (\partial_\mu A_\nu)} \delta A_\mu - \epsilon^\mu L$$

for the field

## The Attempt at a Solution

My general procedure is to [/B]

substitute in the Langrangian density and change in the vector field into the expression for the Noether current leading to some expression I then intend on extracting the energy-momentum tensor out of using

$$ J^\mu = \varepsilon^\mu \left( T^{\mu\nu} \right) $$ as per my course notes.

I initially tried to re-arrange the Lagrangian by taking the metric tensor into the vector field to lower its indice to then allow the partial derivative to be resolved in the third equation but this is incorrect. I think this was incorrect because $$\eta^{\mu\nu}$$ cannot lower the index of $$A^\mu$$. I feel that I have the wrong approach.

Can anyone confirm my approach is incorrect and point me in the right direction?

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