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0

The factors $e^{i\xi\cdot A}$ and $e^{iu\cdot V}$ are chosen this way so there is a unique label $(u,\xi)$ for each group element $g=e^{i\xi\cdot A}e^{iu\cdot V}$. To see that $\xi$ is a label for $G/H$, it's enough to check that $g$ and $gh$ are associated with the same value $\xi$ when $h\in H$. This is easy to see: set $h=e^{iu_h\cdot V}$, so that ...

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Here are two facts - A vacuum expectation value of a quantum field is equal to the minimum of the effective potential (taken from the 1PI effective action). The effective potential takes the general form $$V_{\text{eff}}(\phi) = V_{\text{classical}} (\phi) + \text{quantum corrections}$$ In perturbation theory, where quantum corrections are assumed to be ...

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Let's introduce a bit more notation because I think you're confusing yourself with the $\to$ notation: Let $\theta : \mathbb{R}^4\to\mathbb{R}$ be any function. Then the gauge transformed fields are \begin{align} \phi^\theta & := \mathrm{e}^{-\mathrm{i}\theta}\phi \\ A^\theta& := A - \frac{1}{q}\mathrm{d}\theta \end{align} and a gauge ...

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