OK.
In quantum field theory one introduces a so-called vacuum state |vac>. This is not trivial mathematically, but the main idea is that
- the vacuum state is the state with lowest energy
- the vacuum state is empty, so the are no particles present
1) Now one can calculate the energy expectation value (in a finite volume) and one can calculate the expectation value for the particle numbers (particles species s, e.g. electrons and positrons, photons, quarks, gluons). One expectes something like
##\langle H \rangle_\text{vac} = \langle\text{vac} | H | \text{vac}\rangle = 0##
##\langle N_s \rangle_\text{vac} = \langle\text{vac} | N_s | \text{vac}\rangle = 0##
where H and N are so-called operators which define energy and particle number.
If you do that for QED (for s = electrons, positrons and photons) you find zero (as expected).
But for QCD the two definitions do not coincide!
So one has a vacuum state with lowest energy 0, but for which the expectation value of N does not vanish. So the two equations become
##\langle H \rangle_\text{vac} = \langle\text{vac} | H | \text{vac}\rangle = 0##
##\langle N_s \rangle_\text{vac} = \langle\text{vac} | N_s | \text{vac}\rangle \neq 0##
The so-called quark condensate which I introduced in the previous post is something that measure the quark content of the vacuum. So the non-vanishing of this condensate means that the vacuum (defined as the state with lowest energy) is not empty.
2) There is a related phenomenon, namely the excitatons of the vacuum. In quantum field theory these quantized excitatons are interpreted as particles.
In QED one can find states with arbitrary small energy ε>0 (ε can be any positive number). This is rather simple b/c the energy of a photon is just E=hf, so a single photon with small frequency f (long wave length λ) defines such a state
##\langle f| H | f \rangle = hf##
Again in QCD the situation is different. There is no such state with arbitrary small but non-zero energy ε. Adding a single excitatation (a quark, a gluon) results in an unphysical state which is forbidden due to symmetry reasons and due to (infinite) energy. So in QCD there is a mass-gap, which means that one must add a rather large energy (a few hundred MeV) to find the next state above the vacuum state.
3) In order not t confuse you too much I due not (yet) discuss exceptions to 2) which are closely related to 1)
