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Here's an excerpt from Schwartz's QFT text, section 8.2.2, that's always totally confused me.

A natural guess for the Lagrangian of a spin 1 field is $$\mathcal{L} = - \frac12 \partial_\nu A_\mu \partial_\nu A_\mu + \frac12 m^2 A_\mu A_\mu.\tag{8.17}$$ Then the equations of motion are $$(\Box + m^2) A_\mu = 0.\tag{8.18}$$ In fact, this Lagrangian is not the Lagrangian for a massive spin-1 field, but the Lagrangian for four massive scalar fields, $A_0$ through $A_3$. You might wonder how we know if $A_\mu$ transforms as a vector or four scalars. As a very general statement, we do not get to impose symmetries on a theory. We just pick the Lagrangian, and let the theory go.

I find this passage incredibly confusing. First, how is it apparent that there are four massive scalar fields, when it looks like the equation of motion works either way?

Second, Schwartz seems to be saying that the Lagrangian determines how the fields inside it transform, which seems exactly backward to me. Don't we first postulate that fields exists with certain transformation properties, then write down scalar Lagrangians for them? That's exactly how we construct the Dirac spinor, for instance -- we write down how it transforms, and only afterward write down the Dirac Lagrangian.

Qmechanic
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knzhou
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  • Possible duplicates: https://physics.stackexchange.com/q/149018/2451 and links therein. – Qmechanic Oct 06 '17 at 23:23
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    Comment 2 - Your last paragraph of course presents the standard modern view of effective field theories and how research in quantum field theory is done. You pick your favorite symmetries and then write down the most general action that preserve those symmetries. Schwartz is pointing out a slightly different point of view where a theory is defined by its Lagrangian and everything else (including symmetries) follows from it. – Prahar Oct 06 '17 at 23:29
  • @Prahar Okay, but then how does Schwartz see that there are four scalers above? To me the Lagrangian appears totally agnostic about that. – knzhou Oct 07 '17 at 08:10

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