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There seem to be two ways of defning what a vacuum is in QFT:

  1. It is state $|0\rangle$ such that $a_k|0\rangle = 0$ for all anihilation operators $a_k$, with creation operators $a_k^{\dagger}$. Thus, it is defined in Fock space.

  2. It is state $|0\rangle$ such that derivative $V_{eff}'(\phi_c) = 0$ for effective potential $V_{eff}$ with $\langle 0|\hat{\phi}|0\rangle = \phi_c$.

Two definitions do not seem to be completely compatible at first glance, but it seems that at least second definition follows from first definition when Fock space can be defined. What exactly is happening?

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The first definition you give assumes you can define the creation and annihilation operators, and these can be seen (in the classical theory) as the coefficients of the expansion of the fields around an equilibrium field configuration. This equilibrium configuration is what corresponds to your second definition.

Another way to phrase this is as follows: when you have a classical field theory, you first look for the classical configurations which minimize the effective potential. Then when you quantize, you take this as the vacuum. In the Fock space construction, you can construct excited states by acting with creation operators, and this corresponds classically to oscillations in phase space around the equilibrium position.

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