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What is the expectation value of the number operator when the vacuum has a VEV?

The number operator N applied to a field whose vacuum has zero VEV gives $N|0>=0$. What if we apply it to the Higgs field? The background of this question is that in popular scientific accounts, ...
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“Space” in General Relativity and “vacuum” in Standard Model, is it the same thing?

And expansion of space is equal to expansion of vacuum?