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If I had a Lagrangian of the form

$$ \mathcal{L} = (\partial_{\mu}\phi,\partial^\mu \phi)-U[(\phi,\phi)]$$

where $\phi = (\phi_1,\phi_2,\ldots,\phi_n)$ is an $n$-dimensional vector of complex fields, and $(u,v)=v^\dagger u$ is the usual inner product, how do I know what the symmetry group is? At first glance I would say $SU(n)$ as it will preseve the inner products, but surely an $n$-dimensional unitary representation of ANY group would preserve the inner products too?

Qmechanic
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Hermitian_hermit
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  • Under your definition, every Lagrangian has every group as a symmetry group -- you just have to pick the representation that does absolutely nothing. (After all, that certainly won't change the Lagrangian!) – knzhou May 25 '18 at 14:21
  • Implicitly we're restricting to faithful representations, i.e. representations where nontrivial group elements act by doing nothing don't count. – knzhou May 25 '18 at 14:22
  • I'm not sure I understand the question. Is your point the following?: The Lagrangian $\mathcal L$ is invariant under any $n$-dimensional unitary representation of any Lie group $G$. How do we determine $G$? If this is your question, note that $\mathcal L$ is invariant under any unitary transformation. OTOH, a unitary representation of arbitrary $G$ is not in general surjective on $\mathrm U(n)$, and therefore "you are missing symmetries". This pretty much singles out $G=\mathrm U(n)$ as the correct group, because it includes all symmetries and misses none. – AccidentalFourierTransform May 25 '18 at 14:39
  • One usually demands that the field(s) stay canonically normalized. So the symmetries are more or less by default U(n) unless there are additional noncompact symmetries. If in your case U=0, say, there will be all sorts of global shift symmetries which leave the Lagrangean invariant as well. Otherwise, the symmetry will be U(n). –  May 25 '18 at 14:58
  • Related: https://physics.stackexchange.com/q/206355/2451 and links therein. – Qmechanic May 25 '18 at 17:05
  • You don't state what U(phi,phi) is. Is this notation for inner product of a U(n) valued vector? The way your question is worded you seem to be asking, if you stumble across a Lagrangian in the woods how do you know what symmetries are present. For example, I would hope that it is symmetric under space-time coordinate transformations too. In QFT and PP we usually make an invariant L with a group in mind. Proving that your choice is good requires applying the group action and showing invariance. –  May 25 '18 at 18:46

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