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This is more a terminology question. I have seen that some people differentiate between the two types of symmetry: internal symmetry and gauge symmetry (of a field theory).

Is there a difference (in both local and global case), and if there is what is it?

Edit: For example in the case of Dirac equation interacting with electromagnetic field (plus the gauge fixing term) which is invariant under the set of local transformations: \begin{equation} \psi \rightarrow e^{i\theta(x)}\psi\\ A_{\mu}\rightarrow A_{\mu} + \partial_{\mu}\theta(x) \end{equation}

which is internal symmetry (if any) and which is gauge symmetry?

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An internal symmetry only involves transformations on the fields of a theory, and must act the same independent of the point in spacetime. For example, consider a Lagrangian,

$$\mathcal{L} = \partial_\mu \psi^\star \partial^\mu \psi - V(|\psi|^2)$$

for some potential $V$, and complex field $\psi$. The theory has an internal symmetry, namely one which rotates the field, i.e.

$$\psi \to \psi'=e^{i\alpha}\psi$$

where infinitesimally we would have $\delta \psi = i\alpha \psi$. The corresponding conserved current is,

$$j^\mu = i(\partial^\mu \psi^\star)\psi - i\psi^\star(\partial^\mu \psi)$$

which after quantization adopts the interpretation of charge or particle number.


A gauge transformation, on the other hand, is one which is dependent on the point in spacetime wherein one operates, and it may act on spacetime itself, or the fields. An example is the $U(1)$ gauge symmetry of electrodynamics, described by,

$$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

which is invariant under $A_\mu \to A\mu + \partial_\mu \lambda(x)$, for any function $\lambda(x)$. (To see this clearly, note $F=dA$, and hence the change $A \to A + d\lambda$ has no effect as $d^2\lambda=0$.)

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  • $\begingroup$ i think this answer is off-target, slightly. gauge symmetries are a type of internal symmetry, contrariwise to your answer, and you don't make the distinction between internal and spacetime symmetries that clear to me. $\endgroup$ – innisfree Oct 20 '14 at 15:26
  • $\begingroup$ @innisfree: Yes, perhaps I should labor that point more. $\endgroup$ – JamalS Oct 20 '14 at 16:06
  • $\begingroup$ It seems like what you're calling internal symmetries can just be called a global gauge symmetry and your latter a local gauge symmetry. $\endgroup$ – R. Rankin Oct 17 '18 at 4:30
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I think I have came up with the answer, which I hope is correct.

Although, neither internal nor the gauge symmetry operations affect the space-time coordinates, there is a big difference:

Internal symmetry is an actual symmetry of the system (field): two physically distinct field configurations (or in QM, two physically distinct states in Hilbert space) are related via an internal symmetry operation.

However, the gauge "symmetry" is not an actual symmetry of the physical system. Two states which are related via a gauge transformation, are physically equivalent. In other words, "we use different labels to label the same state"(1). For this reason, some physicists don't call it a symmetry (For example Prof. Wen calls it "gauge structure").

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