# Difference between QED & QCD Vacuum

#### dev70

Hi pf, i have been wondering what differentiates QED Vacuum from QCD Vacuum? How would u explain its implications? I mean, how can u define pure vacuum in 2 ways?

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#### Bill_K

Hi pf, i have been wondering what differentiates QED Vacuum from QCD Vacuum?
The difference is, QED vacuum contains no electrons, while QCD vacuum contains no quarks. #### tom.stoer

There are several differences. The first one is that in low-energy QCD there is a phase with broken chiral symmetry indicated by order parameter (the so-called quark condensate) with non-vanishing expectation value

$\langle \bar{q} q \rangle > 0$

whereas in QED the electron-positron expectation value vanishes

$\langle \bar{\psi} \psi \rangle = 0$

#### dev70

ok..fine..then would you please explain the QED vacuum and the creation of virtual particles and more about vacuum fluctuations?

#### tom.stoer

What is your background in physics and your knowledge in quantum field theory?

#### dev70

well.nothing much..just 12th grade..nd beginner of quantum physics...

#### tom.stoer

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)

• jakob1111 and dev70

#### dev70

this is awesome..though i wont get all of it now...but still, i will. thanks a lot...its wonderful..

#### Bill_K

$\langle N_s \rangle_\text{vac} = \langle\text{vac} | N_s | \text{vac}\rangle \neq 0$
Ok then, if it is not zero, what is its value?

#### tom.stoer

Ok then, if it is not zero, what is its value?
I was cheating a little bit b/c one does not determine <N> but the (flavor-specific) quark condensate; afaik the (ren.-scheme dep.) values are in the range of 300 MeV3; afaik in two-flavor QCD the value can be related to the pion mass and decay constant via the Gell-Mann–Oakes–Renner relation, e.g. in current algebra and chiral perturbation theory; there should be lattice gauge calculations as well.

I have to find some references.

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