What does it mean to say "internal symmetry"? Let me try to express the way I see it, so you can have it as a starting point.

There are spacetime symmetries, which are global since any Lorentz transformation, at any point in spacetime, will be invariant. On the other hand there are also internal symmetries (which I understand as local), which are only invariant in a certain region of the spacetime.

  1. Does this make sense?
  2. Could anyone give examples of internal symmetries?

PS: I'm currently studying Classical Field Theory. That's where I see the terminology.


An internal symmetry is a transformation acting only on the fields, therefore not transforming spacetime points, and leaving the lagrangian or the physical results invariant. Example of internal symmetries are gauge symmetries. These are local symmetries, which means the transformations are in general spacetime dependent in the sense they are, in general, different for each spacetime point. Example: The $U(1)$ gauge symmetry of Maxwell theory and the $SU(3)$ color symmetry of quantum chromodynamics.

A brief remark in your quote: spacetime symmetries can be local, for example the coordinates transformation of General Relativity.

  • $\begingroup$ follow up question: Can I then say a local transformation is one that acts on the object (like a field), while a global transformation acts on the space where is object is "placed" (like the spacetime)? $\endgroup$
    – Patrick
    Jun 19 '16 at 15:30
  • $\begingroup$ Hmm.. say we have some vector (field). Performing a transformation upon it, say a (rotation) IS equivalent to performing the opposite transformation upon the space (time) points, either one of which should leave a reasonable lagrngian invariant. Therefore, I'm not sure I find this answer regarding distinction between Internal and external symmetries satisfactory. $\endgroup$
    – R. Rankin
    Sep 22 '19 at 21:36

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