Mixing internal and spacetime transformations: some examples and counterexamples

TL;DR

This paper investigates whether internal and spacetime symmetries can be mixed in gauge theories coupled to gravity, showing conditions under which such mixing occurs or is prevented based on the gauge field's vacuum expectation value.

Contribution

It clarifies the conditions under which internal and spacetime symmetries can be combined or must remain separate in gauge theories with gravity.

Findings

If the gauge field VEV is flat, the symmetry group is a product of internal and spacetime symmetries.

Non-flat gauge field VEVs prevent a proper definition of spacetime transformations, leading to possible symmetry mixing.

Mixing of symmetries occurs when the symmetry group is nontrivial and the gauge field VEV is not flat.

Abstract

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.