[Physics] Is internal symmetry the same as gauge symmetry

field-theorygauge-symmetrysymmetryterminology

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?

Best Answer

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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