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The vacuum state is the state with the minimum energy, which implies no excitations, which I assume is the same as a state with no particles. Then I am confused about a static electric coulomb field. The electric field in empty space has an energy density larger than in a vacuum state with no electric fields generated by nearby electrically charged particles, right? (It is a state with no particles as there are no electromagnetic waves or photons due to the field, right?

Question: If a field has a value different than zero in some region of space, will its state in that region still be a vacuum? And such a field which has a value larger than for the vacuum (not just the value of the field, but the energy density too), is not the same as an excited field? so it is not necessarily associated with real particles of that field?

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An electric field with a value different than zero in some region of space can be considered as a coherent state of photons Coherent state vs. classical world. This is not the vacuum state. We can associate a non-dynamical background gauge field with this state: $A^{b.g.}_{\mu}$.

Then you can have quantum fluctuations on top of this configuration. Such fluctuations are usually studied in path integral form. You need to take $A^{b.g.}_{\mu}+\mathcal{A}_{\mu}$ as the total gauge field. Here $\mathcal{A}_{\mu}$ is the dynamical part. You can then perform standard perturbation theory techniques to investigate the effect of quantum fluctuations and find the effective action of the classical field, namely; $\Gamma[A^{b.g.}_{\mu}]$.

See Peskin-Schroeder section 16.6 for details.

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