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In Rubakov's "Classical Theory of Gauge Fields", in the chapter on Higgs mechanism, he mentions that you can switch to the polar form of the complex scalar field only for non-zero vacuum value of the scalar field and only for small perturbations about the vacuum. He also mentions the same in the previous chapter on spontaneous symmetry breaking where he essentially says that the the complex scalar field can be written in the polar form only when the symmetry is broken.

I don't understand why this is the case. Since we can write any complex number as either $(a+ib)$ or $re^{i\theta}$, why can't we always do the same thing for the field here?

Polar Decomposition of a Complex Scalar Field

The question attached above addresses what the polar decomposition of the field means but my question is why is the decomposition only possible under specific conditions?

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    $\begingroup$ as @user26872 says, I think the issue is that the uncertainty of the phase diverges when the amplitude goes to zero $\endgroup$ – lurscher May 22 at 18:08
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Note that $\theta$ is not well-defined at the origin.

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