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So I was learning about the Faddeev-Popov method and my instructor briefly told to use it on the following action:

$$S = \int d^dx\left( \frac12 \partial_{\mu}\phi_1 \partial^{\mu}\phi_1+ \frac12 \partial_{\mu}\phi_2 \partial^{\mu}\phi_2 + m \chi (\phi_1^2+\phi_2^2)\right),$$

where $\phi_1,\phi_2,\chi$ are scalar fields. (Note: No kinetic term for $\chi$ field). Now I can see this is like a linear sigma model so there is a rotation symmetry $\phi_i \rightarrow R_{ij} \phi_j$. But I am not able to see how to use the method. All examples I have seen are for gauge fields. There is one for a 2D Gaussian integral, but except for that I haven't seen it being applied anywhere else. Any hint will be useful!

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  • $\begingroup$ Is it important for your question that the constraint is precisely $\phi_1^2+\phi_2^2= 0$ or would your underlying question still stand if the constraint is instead, say, $\phi_1^2+\phi_2^2= 1$? $\endgroup$
    – Qmechanic
    Apr 2, 2021 at 16:11

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Well you have seen the Faddeev–Popov method for gauge fields because it is designed to help with the path integral in a gauge theory; namely with summing over equivalent gauge configurations. If you don't have a gauge symmetry you don't have a problem to try to solve with Faddeev–Popov to begin with.

Now, in your case you have a global $\mathrm{SO}(2)$ symmetry which you can gauge. The simplest way would be to make the two real scalars $\phi_1$ and $\phi_2$ into a single complex scalar and the $\mathrm{SO}(2)$ symmetry becomes a $\mathrm{U}(1)$ symmetry (multiplying the scalar by a phase). Next you covariantise the derivatives introducing a gauge connection and it is standard $\mathrm{U}(1)$ Faddeev–Popov from there on.

Actually, however, once you're at this point you're not required to gauge fix the connections (i.e. gauge fields). You can easily fix the matter fields instead, e.g. by introducing the Faddeev–Popov determinant as $$ 1 = \Delta_\text{FP}[\phi]\sum_{\bullet}\int \mathrm{D}\alpha\ \delta\left[\phi-\hat{\phi}_\bullet^\alpha \right],$$ where here $\hat{\phi}$ is a reference matter field, the superscript $\alpha$ is the gauge parameter of an infinitesimal gauge transformation, e.g. in this case $\hat{\phi}^\alpha=\hat{\phi}+i\alpha\,\hat{\phi}$ and the $\bullet$ and the sum over these denotes any gauge invariant data that you might have (for example if $\phi$ was compact you'd have to sum over winding sectors, or maybe depending on your manifold you'd have to sum over holonomies, moduli etc.).

I don't know if that's what you had in mind, hopefully I didn't derail your question :)

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