# The relation between classical and quantum vacua

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.

• 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 $\phi_0=\langle vac|\hat \phi|vac\rangle$). – Adam Jun 10 '14 at 22:04
• 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 $\mathbf{M}$. – Arthur Suvorov Jun 10 '14 at 22:58

## 1 Answer

I'm not sure why you are asking, because you seem to mention the answer already. This problem has been studied thoroughly in the late 70's by Belavin et al and 't Hooft.

As far as I understand, the quantum vacuum is the lowest energy eigenstate of a Hamiltonian. It turns out that the classical solutions to the equations of motion (of a particle, or a field) are very good tools to make approximations as to the corresponding energy eigenvalue to this eigenstate. If the topology of the solution is non-trivial (as happens indeed with SU(2) or SU(3) gauge theory in 4-spacetime) then the quantum vacuum becomes complicated, and is described by instantons.

The situation is a bit similar to Bloch's theorem: it turns out that each classical vacuum solution is like a minimum in a sinusoidal potential and the true quantum vacuum, which must be an eigenstate of the translation operator, is thus a Bloch mode, a Fourier transform of all these vacua. Thus the various possible vacuum states are indexed by a continuous parameter, the $\theta$ angle, and we are said to live in a universe with a particular $\theta$ angle (this is the source for the famous "strong CP violation" problem). In essence, you could say that the only effect the instantons have on the QCD vacuum, is thus, to add a term in the Lagrangian which violates CP, and has some arbitrary strength (or one which is determined by Peccei-Quinn theory).

See: - S. Coleman "The Usese of Instantons" chapter 2 and 3 mainly