Mixing internal and spacetime transformations: some examples and counterexamples

Understand

This note addresses the question whether in a gauge theory coupled to gravity internal and spacetime transformation can be mixed.

• It is shown that if the VEV of the gauge field is flat, the symmetry group is always a product of internal and spacetime symmetries.
• On the other hand, if the VEV of the gauge field is not flat it is impossible to properly define the notion of a ``spacetime'' transformation; as a consequence, if the symmetry group is nontrivial, mixing generically occurs.