# The relation between gauge symmetry and global internal symmetry

I'm a little confused about the relation between the gauge symmetry and global internal symmetry of a field theory. I'd appreciate any clarification on this. My question can be phrased as the following:

Is it correct to say that for a global internal symmetry (abelian or non-abelian) like $$SU(2)$$ or in general a symmetry group $$G$$, the gauge transformations and the transformations from group $$G$$ are not related to each others but for a local internal symmetry the gauge transformations become related to the action of $$G$$ (by enforcing the invariance of the theory as demonstrated in the usual procedure in textbooks) and the gauge fields become the connections on the manifold associated to $$G$$?

• I believe the OP is asking what the relation is between a global internal symmetry and a local internal symmetry. But this question is not very clearly written. I suggest the OP rephrases this. Dec 3, 2020 at 12:55
• Hi there, many thanks for your comment. I edited the question and hopefully it's more clear now. Just to clarify, my question is not about the relation between global and local internal symmetry. It's about the relation between gauge symmetry and internal symmetry. I'd appreciate any thoughts on this. Dec 4, 2020 at 1:07
• @AliN. Perhaps you could define for us what you mean by "global internal symmetry," "local internal symmetry," and "gauge symmetry." In the standard terminology, "local symmetry" and "gauge symmetry" are synonyms.
– d_b
Dec 4, 2020 at 1:23
• @ d_b, thanks for the comment. My confusion arises exactly from your point. I thought when we say we have a gauge theory we have a theory with a local symmetry and the gauge fields are the connections of the Lie group but then I came across the following paper.There, we have a color vector charge $\vec{q}$ with a global SO(3) symmetry in color space and gauge symmetry for the gauge field. The definition of strength tensor and the transformations are the same as E&M only now we have a SO(3) symmetry instead of U(1). By internal symmetry I mean a symmetry group G (not Lorentz or Poincare). Dec 4, 2020 at 4:05
• " By internal symmetry I mean a symmetry group G (not Lorentz or Poincare)". You might be supersized to learn that for gravy the local Lorentz symmetry is an "internal symmetry"(does not involve coordinate transformation!), while the diffeomorphism is an "external symmetry" (involves coordinate transformation). See answer here: physics.stackexchange.com/questions/502982/… Dec 4, 2020 at 16:11

I believe you may be slightly confused about terminology used. It is really pretty simple.

An internal symmetry is indeed a symmetry of the theory that does not impact the spacetime coordinates. It can be a global symmetry or a local symmetry.

Examples of a global symmetry are $$U(1)$$ for the EM charge, $$SU(3)$$ flavour or $$SU(2)$$ isospin.

When the internal symmetry is local it is a gauge theory. The gauge fields are then a connections of the symmetry. Examples are $$U(1)$$ for the EM force, or $$SU(2)\times U(1)$$ for the electroweak force or $$SU(3)$$-color for QCD. Note that gravity has a local symmetry as well: diffeomorphism invariance. It is a gauge theory, but not an internal symmetry.

1. traditionally gauge theories are taught in the following way: one starts from a global internal symmetry, usually $$U(1)$$, and then asks oneself the question: what happens if we make this local. One then see that one has to introduce a connection etc. There is a prior no reason to do it that way; it is just educational convenience/laziness

2. a local internal symmetry is, strictly speaking, not a symmetry. It is a redundancy in the description of the system that allows to simplify it. It introduces d.o.f. that then need to be gauged away by fixing the gauge.

3. the example of gravity that is a gauge theory but not an internal symmetry

4. the fact that some people use gauge theory and Yang-Mills theory interchangeably. They are not the same. YM are just special examples of gauge theories.

• Thanks for the answer. It clarified some stuff. Dec 6, 2020 at 1:54