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The relation between classical and quantum vacua

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First let me clarify what I mean by vacuum.

Suppose we are concerned with a theory of fields $\phi ^i$ defined on a stationary globally hyperbolic spacetime $M$ (I want the spacetime to be stationary so that I have a canonical choice of time-derivative and I want the spacetime to have a Cauchy surface so that I can speak of the Lagrangian) by an action functional $S(\phi ^i)$. For $\phi ^i$ stationary (i.e. $\dot{\phi}^i=0$), we define the potential by $V(\phi ^i):=-L(\phi ^i)|_{\dot{\phi}^i=0}$, where $L$ is the Lagrangian of $S$.

A classical vacuum (the definition of quantum vacuum is a part of the question) of this theory is a solution $\phi _0^i$ to the equations of motion $\tfrac{\delta S}{\delta \phi ^i}=0$ such that (1) $\phi _0^i$ is stationary and (2) $\phi _0^i$ is a local minimum of $V(\phi ^i)$ (by this, I mean to implicitly assume that $V(\phi ^i)<\infty$).

In what way do these vacuum solutions of the classical equations of motion correspond to quantum vacuums? For that matter, what is a quantum vacuum? In particular, I am interested in theories with interesting space of vacua, for example, how $SU(3)$ instantons relate to the QCD vacuum.

This post imported from StackExchange Physics at 2014-06-11 21:25 (UCT), posted by SE-user Jonathan Gleason
asked Jun 10, 2014 in Theoretical Physics by Jonathan Gleason (255 points) [ no revision ]
I don't think that even your classical vacuum is defined correctly: $\phi_0$ is the vacuum expectation value of some operator, whereas one use the term vacuum for the quantum state $|vac\rangle$ (such that $\phi_0=\langle vac|\hat \phi|vac\rangle$).

This post imported from StackExchange Physics at 2014-06-11 21:25 (UCT), posted by SE-user Adam
One way I like to think about the quantum vacuum (although not strictly speaking accurate) is that the quantum effects manifest themselves as non-linearaties in the classical vacuum as a complicated medium. If you write out the field equations resulting from, say, the Euler-Heisenberg Lagrangian, you find they can be re-cast into the form of Maxwell's equations but with non-trivial polarisation $\mathbf{P}$ and magnetisation fields $\mathbf{M}$.

This post imported from StackExchange Physics at 2014-06-11 21:25 (UCT), posted by SE-user Arthur Suvorov

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