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A gauge symmetry is a symmetry (often local) which doesn't affect the physical quantities or the Lagrangian.

A translation of the whole Universe 3 meters to the left won't affect anything physically.

So does this mean translation is a gauge symmetry?

If so, how do we fix a gauge in this symmetry?

I imagine a scalar field under a translation in the direction $a^\mu$ has the transformation $\delta \phi(x) = a^\mu\partial_\mu \phi(x)$. Can we fix a gauge in this case?

But if we fixed a gauge does this mean the gauge-fixed theory is no-longer translation invariant?

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Teleparallel gravity (a theory equivalent to general relativity) can be described as a "gauge theory for the translation group". See e.g. [1]. There are some differences from the usual gauge theory, which the paper describes.

[1] V. C. de Andrade and J. G. Pereira. Gravitational Lorentz force and the description of the gravitational interaction. Phys. Rev. D, 56:4689–4695, Oct 1997. https://arxiv.org/abs/gr-qc/9703059

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