Definition of vacuum in field theory; Connection between the classical definition and the connection to QFT

I am a bit confused by what is defined to be a vacuum in field theory.

Classically a vaccum state is defined to be the state where the field sits at some minima of the potential $\frac{\partial V}{\partial \phi}=0$.

My main confusion is that when we go to quantum framework the vacuum state is defined to be a unique state in the Fock space with no-particles $|0\rangle$? I would be grateful if anyone can clarify the correspondence between these various definitions.

• Well, it's when the potential term in the Lagrangian $V(\phi)$ is at a minimum, i.e., when $\partial V/\partial\phi = 0$... – Alex Nelson Feb 12 '14 at 17:15
• How does it correspond to the notion in QFT? – user40469 Feb 12 '14 at 17:19
• IIRC, the vacuum expectation value $\langle\phi\rangle$ relates to the classical notion when taking the classical limit; but it has been too long since I have examined this in detail, so I don't want to mislead you posting an answer... – Alex Nelson Feb 12 '14 at 17:32
• @user40469 The classical limit can be dealt with in the functional integral formulation using the stationary phase approximation. Try reading a bit about it; it won't answer all of your questions, but it does address at least some of them. – joshphysics Feb 12 '14 at 18:21

In classical field theory, the system will indeed be in the minima of the potential, i.e., the point at which $\partial V/\partial \phi_i=0$ for all fields $\phi_i$ for all fields $\phi_i$ (more precisely one should include the fermionic fields here as well but fermions don't exist in a classical world).
• Are you saying $<\phi> \to \text{solution of }\frac{\partial V}{\partial \phi}$ in the $\bar h \to 0$ limit – user40469 Feb 12 '14 at 17:45
• I think that is correct, though I never thought much about the $\hbar \rightarrow 0$ in QFT. – JeffDror Feb 12 '14 at 17:53
• The vacuum is not the classical minimum, since there are fluctuations that change the vacuum state. The vacuum is the minimum of the effective action $\Gamma[\varphi]$, where $\varphi=\langle \phi\rangle$. – Adam Feb 12 '14 at 18:13