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