I’ve been thinking about vacuum states in QFT, especially in the context of the Higgs mechanism. When the vacuum expectation value (VEV) of a field is zero, the vacuum is often described as the simplest possible state—the state with no particles, where the annihilation operator acts on it and gives zero.
But can I think of this vacuum as something like a coherent state of the field operator, where the eigenvalue is set to zero? That seems intuitive, but I’m not sure if it’s exactly right.
Now, when the VEV is nonzero, like with the Higgs field, the vacuum changes. The field gets a classical shift—something like $\phi(x) = v + \hat{\phi}'(x)$—where $v $ is the VEV, and $\hat{\phi}'(x)$ are the quantum fluctuations. So the vacuum we’re talking about now doesn’t match the simple “no particles” idea from before. What does this new vacuum actually mean in terms of states? Can I think of it in terms of a coherent state as suggested here? I have also read in the book Quantum Gases by Nick Proukakis that, "if a field has an expectation value in the ground state, the ground state cannot be an eigenstate of the number operator."
Here’s another thing: Field eigenstates in QFT are defined as states where the field operator acts on them and gives back a classical value. How do those states relate to this shifted vacuum? And how does all of this connect to the classical expectation value of a field? Can this situation be interpreted as quantizing the field around a different value $\phi=v$ and seen as analogous to applying a displacement operator, as in quantum mechanics?
I came across this post that suggests that specific types of expectation values provides a link between quantum fields and classical fields. So where exactly do these definitions converge? How do the vacuum state, the classical expectation value, and field eigenstates come together into a single, unified meaning?
