# When can we say we fully understand QCD?

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What constitutes a sufficient "understanding" of QCD? Take an analogy, if an undergrad understands Newton's three laws well, the formula for Newtonian gravity and know how to solve for conic sectional orbits, then I won't cringe if this undergrad claims to have understood Newtonian gravity.

Now, I'd like to ask what makes people say we haven't understood low energy QCD yet: if it means we must have a systematic nonperturbative computational scheme to calculate everything of interest, isn't lattice QCD (in principle) enough? If it means we must have a rigorous mathematical foundation for it, then in the Newtonian analogy, wouldn't that imply no physicist understood Newtonian gravity before mathematical analysis was discovered, which is an absurd statement?

Still, my impression is that people talk about QCD as if some fundamentally new enlightenment is needed, what's the reason for that? What do we really hope to accomplish so that a complete "understanding" can be claimed? How do we know we are not just pushing the technical boundary further and further?

asked Mar 11, 2015
edited Mar 12, 2015

It is true that we know the formula.

Like we know the formular for newton's law of mechanics, But does that mean you can compute the precise orbits of 9 planets accurate to 1000 years? You would simply put it in a computer, if you had to do that kind of a calculation.

I don't know of anyone who seriously thinks he can solve a 9 body problem analytically. There is probably very little hope, unless ofcourse someone comes along and actually does solve it.

It seems to me that there is some hope with QCD to say atleast something more analytically. Yang mills mass gap conjucture for instance is one way to motivate research in that direction. I think the problem is strictly of a mathematical nature.

However there are other related reseach ideas, especially in understanding the supersymmetric versions of QCD  to provide deeper insight into the structure of physics itself.

@Prathyush,

Like we know the formular for newton's law of mechanics, But does that mean you can compute the precise orbits of 9 plants accurate to 1000 years? You would simply put it in a computer, if you had to do that kind of a calculation.

What do you want to convey by this paragraph? Not knowing how to do a calculation that precise doesn't mean we don't understand Newtonian gravity.

It seems to me that there is some hope with QCD to say atleast something more analytically. Yang mills mass gap conjucture for instance is one way to motivate research in that direction. I think the problem is strictly of a mathematical nature.

But again in the Newtonian analogy, could mass gap problem be like the problem of stability of the solar system? It's also strictly of mathematical nature, and heavy machinery of analytic nature is probably crucial, but not understanding it seems to pose no big threat to the claim that we understand Newtonian gravity. I admit I'm partly struggling with the philosophical underpinning of "understanding". I need to go to bed after writing this comment, don't hold your breath for my next reply:-)

Yes, there are 2 senses we use the word understanding.

One is to obtain the correct action principle. We which we do have for QCD.

The second is use the action to make predictions and confirm Experiments.

We have some predictive power, when it comes to observed phenomenon.(Gell-mann quark models and so on) However in most cases there are no exact answers, like mass of a proton and so on.

The second problem is much like stability of the solar system. You can do it on a lattice, but you are confined to accuracy of the simulations, and can't say anything beyond that scope.

When I said there is some hope in QCD, I mean in simplifying the expressions analytically. When in the case of the 9 body problem it is seemingly impossible.

As I said the problem is strictly mathematical.

@JiaYiyang Take a look at this for instance, Even though we know something about QCD a lot of stuff need explanation.

http://guava.physics.uiuc.edu/~nigel/courses/569/Essays_Fall2011/Files/Garcia.pdf

The mass gap is a statement about the smallest mass bound state of the theory, corresponding in the solar analogy to the Kepler problem, which can be solved exactly and hence is easy to understand. Whereas the stability of the solar system is the analogue of the question of whether a particular set of particles with given properties (sun and planets) form a bound state. Clearly understanding the latter is far more demanding than understanding the former.

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The understanding of an ordinary differential equation has nothing to do with being able to successfully execute a Runge-Kutta method. The latter only gives numerical values for an individual trajectory (or if called multiple times, for many).

But understanding means to know where its fixed points are, how it behaves for large times, how sensitive the solutions are to a change of initial conditions, etc.. Not single numbers or curves but the general pattern of arbitrary solutions.

We are far for such an understanding of QCD except in the very high energy region. Thus people say that QCD is not understood because in the infrared domain (confinement, mass gap, bound states) we have very little grasp on how to obtain properties of arbitrary solutions at a level that conveys more than individual numerical numbers. Lattice QCD is just a black box that spits out a (fairly inaccurate) number for every numerical question we ask, it give not understanding in any sense.

One can say we understand QCD if we can derive from its action the low energy Hamiltonian and the bound state content (mesons, baryons, and perhaps glueballs) to an extent that we can match the meson and baryon data from the Particle Data Group to its particle spectrum.

We wouldn't understand Newtonian gravity if we coudn't do a qualitative analysis of the 3-body problem. There is no analytic solution but still we understand (and can approximate) essentially everything about its behavior, independent of the detailed parameters of the problem.

Lattice calculations in the 3-body problem would correspond to discretizing the dynamics using second-order divided differences, which very poorly resolves the dynamics of a 3-body problem, so a very fine lattice would be required to give good results over a significant time span, and then it would just be for a single system - nothing general.

Perturbation theory around a 2-body problem gives very useful analytic approximations not just for a single system but for all problems in the class, and one can deduce a lot fronm its sutdy, whereas even a better discretization method (like modern symplectic integrators) give just a single trajectory, or if repeated a bunch of trajectories, from which one cannot deduce much about the qualitative behavior.

Understanding always means to be able to derive qualitative understanding, not just numbers. (And even the numbers obtained from lattice QCD are not impressive. I haven't seen even a single attempt to compute the full baryon and meson spectrum from QCD. Given that QCD needs no numerical input to define it, the accuracy for basic predictions such as the proton mass (by lattice QCD or by Schwinger-Dyson equations) is perhaps 5 percent, and it will not grow much even if the speed of computers and algorithms increase by a factor of $10^6$ (which is not realistic). No, we need a much better understanding!

We understand QED much better than QCD, because there are many good approximation schemes which give qualitative information about almost everything of interest. But even QED is not completely understood as we don't have a logically satisfying setting for the theory, and things like the nonperturbative existence of the Landau pole are unsettled.

answered Mar 11, 2015 by (13,969 points)
edited Mar 15, 2015

Thanks and +1.

But understanding means to know where its fixed points are, how it behaves for large times, how sensitive the solutions are to a change of initial conditions, etc..

But can't these be resolved with large and fine enough lattice and powerful enough computer? So that more effort toward algorithms and hardware would suffice?

Lattice QCD is just a black box that spits out a (fairly inaccurate) number for every numerical question we ask, it give not understanding in any sense.

This is a statement I intuitively subscribe but can't quite articulate the reason. Perturbative calculations seems in no way fundamentally better than lattice simulations, except sometimes it's possible to go through  them by pen and paper. Could it be only a delusion to say we only understand what we are studying when we are employing a method that can be done on paper(suppose we can achieve the same accuracy with computer)?

One can say we understand QCD if we can derive from its action the low energy Hamiltonian and the bound state content (mesons, baryons, and perhaps glueballs) to an extent that we can match the meson and baryon data from the Particle Data Group to its particle spectrum.

But the best results from lattice are already quite impressive, aren't they?

@JiaYiyang Nothing beats an analytic solution, in terms of efficiency, simplicity and versatility. They are equally harder to comeby.

We are yet to fully understand how to deal with non perturbative theories analytically.

Its not just QCD that is interesting, there are a lot of examples of non perturbative systems that are being studied right now.

Which is why it is such an interesting problem, and open.

@Pathyush,

I agree with everything you said, but they all seem to be beside the point. The question is why these unsolved problems are considered very central to our understanding of QCD. Again, we are quite entitled to say we understand Newtonian gravity even if we can't prove or disprove the stability of solar system, what would be wrong with this analogy?

It depends of what you mean by understanding of QCD?

Words only have a meaning in context.

We do understand QCD in the sense of Newtonian gravity, we have the correct laws.

But we don't know the "Phase structure of quarks and gluons" and when we say we don't "Understand QCD", we mean questions like this, How does one apply the formalism correctly to undestand the observed phenomenon involving QCD?.

The analogy breaks because the word understanding is used with 2 different meanings, One is the physical law itself and second is understanding of physical phenomenon.

For instance we say we understand QED, because all important experimental observations can be explained to a very high accuracy.

@JiaYiyang: See my expanded answer.

@ArnoldNeumaier: Thanks for extending your answer! +1!

Hi Arnold, thanks for all the valuable input. I'm aware it's usually too exacting to envision what's the next big thing(or if there is any) before a big thing actually emerges, but I'm currently facing a choice problem between going into more formal QFT/string or going into more down-to-earth QCD study for a PhD, so I can't help thinking about these "big-picture" issues, which sometimes can simply be ill-posed(but I certainly hope it hasn't been the case for this question). Let me chew on the issue for some more time.

@JiaYiyang: I think there are many unsolved issues in QCD (and some even in QED). It is a pity that many of the best minds go to a more speculative side of theoretical physics such as string theory rather than work on conventional QFT. I believe that standard QFT remains valid even at the Planck scale and below, and that the real progress in fundamental physics will come from getting a stronger nonperturbative analytic grasp on QFT - e.g., through finding a valid Hilbert space setting - rather than from changing the foundations. (Of course, this doesn't necessarily affect the choice of a Ph.D. topic, as this must more be something tractable rather than something aimed at the physics of the future.)

A lot of intuition to understanding standard non perturbative QFT arises from string theory. Most of the string theorist are primarily experts in QFT. Although a lot of work is being done in understanding string theory on its own merit a lot of work is being done in understanding QFT in general using tools from string theory. I think that at this time only using string tools we can get a better grasp into QFT. Tools like non-perturbative dualities, AdS/CFT and so on.

When Enrico Fermi was asked about "what QFT is", he answered briefly that "it is a formalism of occupation numbers". It was long ago and occupation numbers were thought to be understood well as those for known (theoretical or experimental) states. Since then the notion of a state in QFT "evolved" and QFT "understanding" wiggles even today.

@conformal_gk: Nothing in AdS/CFT as applied to QCD uses the notion of a string. Of course string theorists develop a lot of QFT along the side, but the tools they develop for the latter are QFT tools, not string theory tools. Only the motivation for having developed them comes from string theory.

@ArnoldNeumaier: D-branes are one of the most important objects in string theory. The whole AdS/CFT and its extensions are based in completely geometrical constructions using D-branes. Indeed, QCD is still far from being understood via a holographic dual description, but this, nonetheless, is a very promising area of research and one needs to understand to some extent string theory in order to understand the various constructions. In any case most of the current string theory research  that is not focused on phenomenology it is focused on understanding some sort of dual field theories.

@conformal_gk: Possibly, but I know very little about strings and still had no difficulties understanding AdS/CFT as applied to QCD from a QFT perspective. (Not just the papers mentioned in the link but all related stuff as well.) I think it is phrased in stringy language only because of its historical origin, not because tools outside of QFT are needed. Anyway, this is likely to be controversial, and I can understand that you have a different view on this.

@ArnoldNeumaier I will not pretend I know a lot about the models of holographic QCD but in principle you do calculations on the bulk where SUGRA is present. In that sense was my comment above. I agree, you do not need to understand a whole big deal of string theory to apply holography in some models and this is why a lot of CMT people have shifted in holography. To create new ones that would be better and better approximations to various kinds of dual field theories though, you need to understand very well string theory and the various geometric constructions.

@ArnoldNeumaier, @conformal_gk, @VladimirKalitvianski, @Prathyush, thank you all and you've been helpful, I'd like to reply but I've unfortunately run out of thought......

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We will understand completely QCD when we are able to compute any quantity we want at any energy scale we want. The main object of interest are of course $n$-point correlation functions in the IR below $\Lambda_{\text{QCD}}$. We also  need to understand QCD instantons, QCD phase transitions like the transition to the quark-gluon plasma. Also, an open problem is the mathematical proof of the existence of the Yang-Mills (e.g. QCD) mass gap. The problem with the lattice is that it is Euclidean and that you solve numerically. Understanding the theory and being able to predict requires in some sense analytical and exact results. Lattice as well as other approaches give you intuition but this does not mean you have solved the theory.

answered Mar 11, 2015 by (3,625 points)

QCD instantons? I havent heard about them. Can you comment on this topic, even references would be appriciated.

I don't know, just google search it. It is very standard for example http://arxiv.org/abs/hep-ph/9610451

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