# A QCD as a classical field theory?

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1. Aug 5, 2017

### Sleuth

Hi everyone,

I have a question that, when came to me, sounded a bit silly to me as well, but then I realized, I myself maybe don't understand the logic behind this 100%, so why not discussing with you about it.

So my question is the following. Usually we are used to do quantum field theory starting from a classical lagrangian with classical fields (say electromagnetism), and then do our usual quantization proceedure, one way or another and get say qed.

Now when we look at electromagnetism, we can also study the classical field theory, and this already provides us with a lot of interesting information, for example the existence of electromagnetic waves. Similarly with gravity, that we still cannot quantize, we study the classical field theory and we get a lot of information on the dynamics including the existence of gravitational waves.

Now my silly question is, can we extract anything reasonable from qcd as a classical field theory? Or say yang mills in general. For example, would we find gluonic waves?

Clearly I understand very well that the difference between qcd and qed ist that we see phenomena of qcd only on very small scales (due to confinement) and somehow the theory and the phenomena we want to understand are naturally quantum phenomena. Moreover something like asymptotic freedom I guess makes sense only at the quantum level. Still, I was wondering if we can really exclude a priori that anything interesting can be extracted from classical qcd ( I should probably call it just cd, instead of qcd). Or said in other terms, suppose I didn't know about color confinement, asymptotic freedom and so on, and I just wanted to investigate classical yang mills theory for a given group SU(N). Would I be able to see some interesting physics compared to what I know from em and gravity, already at the classical level?

Sleuth

2. Aug 5, 2017

### tom.stoer

We can.

The classical theory is just the classical limit of the quantum field theory, i.e. the zeroth order in certain approximations.

The main differences to QED are that 1) QCD has self-coupling gluons and 2) is strongly coupled at low energies. That means at low energies a free-field approximations is mathematically useless, and free plain waves do not exist in nature.

However we do know classical solutions = zeroth order approximations to QCD which are relevant in some calculations, e.g. instantons.

3. Aug 5, 2017

### king vitamin

Like pure classical electrodynamics, pure classical Yang-Mills theory is conformally invariant. You have massless classical gluon waves as your excitations, even if the coupling is very large.

Unlike pure QED, quantum Yang-Mills theory does not have conformal invariance. There is a "conformal anomaly" meaning quantum effects break the classical conformal symmetry, and there is a generation of an energy scale. The coupling is renormalized and quantum effects overwhelm the classical physics at low energy, so the classical theory isn't a good starting point for understanding IR physics.

4. Aug 6, 2017

### tom.stoer

Just for clarification: no plane waves b/c they do not solve the equations of motion. So the question is whether it really make sense to call the solution waves.

5. Aug 6, 2017

### king vitamin

That's a good point, which reminds me that the same question can be posed for gravitational waves in classical gravity in the strong-curvature limit. There's a section on the gravitational wave problem towards the end of MTW mentioning that these waves are not actually periodic, and that the plane wave is just a (useful) abstraction in a nonlinear theory. It seems they choose to adopt a terminology where you keep calling them waves at strong coupling.

6. Aug 6, 2017

### Sleuth

Thanks for the answers. So just to see if I understand your answers, the difference has to do mainly with the largeness of the coupling...?

Self coupling exists also in gravity, but we still talk about gravitational waves, at least in the linear regime (where the coupling is small and self coupling is not so important...). While of course if the coupling is too large, there is no linear approximation. On the other hand, in principle, given a Yang Mills theory (classical) there is no running and therefore no low- or high-energy couplings, there is only one coupling, which must be fixed from the experiment.

So one could imagine a maybe very academic exercise of studying classical Yang Mills theories with small couplings, as one would do for classical electromagnetism. There would still be a self coupling here, with the coupling constant would be small and one could do an expansion similar to gravity I guess...?

Cheers

7. Aug 6, 2017

### tom.stoer

Yes, you are right.

At vanishing (small) coupling g the classical vector potential Aa fulfills (approximately) a plane wave equation with zero (very small) coupling of different components a (a is the Lie-algebra index). This is a good starting point in the UV / asymptotic-freedom regime for a perturbative definition of QCD at small g, using free plane-wave states. It's useless in the IR / strong coupling regime with g of order one.