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I am currently familiarizing myself with QFT and have a question about this article. My understanding is that the Lagrangian density in (2) couples my gauge fields to the Higgs field. And with variation of (2) I should be able to get the Yang Mills equations (3). But I haven't really understood the variation principle in QFT yet. According to which quantities do I have to vary in order to obtain the Yang Mills equations? Could someone explain to me,perhaps in analogy to Hamiltonian mechanics, how to apply the variational principle in QFT? There should be no difference to the mechanics, but somehow all the tensor indices cause confusion.

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    $\begingroup$ See section 1.1.3 of David Tong's QFT lecture notes. There he shows how to obtain Maxwell's equations from the Lagrangian. The Yang-Mills case is very similar. $\endgroup$
    – Bairrao
    Commented Jun 19 at 19:34
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    $\begingroup$ It just means to use Euler-Lagrange Equations. In the language of Variations you're varying the action with respect to the fields in your theory. This alone is not a QFT-centric thing, the procedure is the same whether you're in classical or quantum field theory. The difference is just that in QFT, your operator valued fields will obey these EoMs (if you're doing canonical quantization), or your fields only respect them on average (if you're doing path integrals). $\endgroup$
    – FranDahab
    Commented Jun 19 at 19:40

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