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The strategy is described in the book "Quantum Field Theory" by Itzykson and Zuber, Sec. 6-2, and partucularly sec. 6-2-2 (pag. 289 of my edition). The derivation can be generalised to fields of any statistic. Useful references include the lecture notes by K. Narain (from the Diploma in High Energy Physics, lectured at ICTP - Trieste), although I'm ...

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I found a reliable source that addresses my question (Appendix 4.1 - M. Gasperini, Theory of Gravitational Interactions; DOI 10.1007/978-88-470-2691-9), so I will present the resolution to my confusion here for posterity's sake. The issue is not with using the structure equations on-shell (indeed, this is purely a classical theory) and there is no error in ...

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I guess I should answer this :) In the context of the Lagrangian formalism the equations of motion (EOM) refer to Euler-Lagrange (EL) equations. The phrase "equality modulo EOM" means that a left-hand side is equal to a right-hand side of an equation if one uses the EOM. It is often important to keep track of when one uses EOM. Sometimes one is not ...

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This part of yours is incorrect: \begin{align} S &\sim \int\varepsilon_{abcd}\left(e^a\wedge e^b\wedge D\omega^{cd}\right) \\ &= \int\varepsilon_{abcd}\left(-De^a\wedge e^b\wedge\omega^{cd} + e^a\wedge De^b\wedge\omega^{cd}\right) \end{align} The correct derivation is: \begin{align} S &\sim \int\varepsilon_{abcd}\left(e^a\wedge e^b\... 5 Here is a second way to see the correct result for taking the functional derivative of the spacetime derivative of the field, which I hope will be helpful. Recall that the definition of the functional derivative is \frac{\delta\phi(y)}{\delta\phi(x)}=\delta(y-x) .$$You know that Dirac deltas are distributions. That is, you should always think of them ... 5 Once nice way to calculate functional derivatives is to use the concept of the Gateaux derivative as follows:$$\frac{d}{d\epsilon}S[\phi+\epsilon \eta]\bigg|_{\epsilon=0} = \int d^4x\frac{\delta S}{\delta \phi} \eta$$In your case,$$S[\phi+\epsilon \eta]= \int d^4x \ \bigg\{\frac{1}{2}\big((\partial \phi)^2 + 2\epsilon (\partial_\mu\phi)(\partial^\mu\eta) +...

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The safest way to compute the functional derivative is to use the following prescription: $$S[\phi + \delta \phi] = S[\phi] + \int {\rm d}^4 x \frac{\delta S}{\delta \phi}\delta \phi + O(\delta \phi^2)$$ In other words, add a small perturbation to the field, and manipulate the action so it has the form of an integral times the ...

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