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3

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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One has to distinguish between fields and particles. Fields are a mathematical construct , similar to a coordinate system, defined at all (x,y,z,t) points . Quantum mechanical fields are at the same time operators with expectation values. Particles are excitations on the fields and their interactions are measurable in the laboratory. If no electron exists, ...

5

There is a single Higgs field that fills all of space and always has. Similarly there is a single electron field filling all of space. And an up quark field, and a photon field and a $W^+$ field and a Z field and a gluon field and a $W^-$ field and some neutrino fields and fields for down quarks and top and bottom quarks and charm and strange quarks and ...

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If you will allow me to make your question slightly more precise, I think you are asking the following: In spontaneous symmetry breaking (SSB), generally speaking, we say that the system has a range (either continuous or discrete) of possible degenerate values, and as a result it picks at random one of these configurations, resulting in a state without the ...

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