# Yang–Mills and Mass Gap

The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap" as we know: the quantum particles have positive masses, even though the classical waves travel at the speed of light.

Now, my question is, this property has been discovered experimentally and computationally; but how can it be understood from a theoretical point of view?

• – Qmechanic Feb 13 '17 at 20:41

A true theoretical understanding of the Yang-Mills mass gap is a major open problem in physics and mathematics; in fact, it is one of the seven Millennium prize problems, stated as follows:

Prove that for any compact simple gauge group $G$, a non-trivial Yang-Mills theory exists on $\mathbb R^4$ and has a mass gap $\Delta >0$.

Whilst the problem is stated in some generality for any simple compact gauge group $G$, for the case of $SU(3)$, the strong interaction, a proof has not been shown yet.

For an understanding of the problem itself, see the official description. The mass gap itself can be understood that the Hamiltonian $H$ has no spectrum in $(0,\Delta)$. A key consequence is that,

$$|\langle \Omega, \mathcal O (\vec x) \mathcal O (\vec y) \Omega\rangle |\leq e^{-C|\vec x - \vec y|}$$

for some $C < \Delta$. Thus the mass gap can be understood not only physically in terms of the content of the field theory, but also in terms of the behaviour of correlations which have geometrical implications, namely the extension of the theory to other four-manifolds.

For an elaboration of the status of the problem and further insights, see here.

• I looked at your last reference Jamal, and I didn't find this at all insightful: "Its starting point is the 'AdS/CFT correspondence' of Maldacena, according to which this version of Yang-Mills theory can be reformulated as a string theory in anti de Sitter (AdS) space". Given that the paper was written in 2004 and it's now 2017, I don't think anybody else did either. PS: you didn't answer the question at all, but +1 for a useful reference. – John Duffield Feb 13 '17 at 21:29
• @JohnDuffield "At all"? The question asks for a "theoretical point of view" of the mass gap which is extremely broad. Since a true mathematical understanding is not known as outlined in my answer, I thought the best option was to offer a brief description of what a mass gap is mathematically, and refer to references in the two links. – JamalS Feb 13 '17 at 22:04
• No, because the question asked how the property can it be understood from a theoretical point of view. Of course, if the OP begs to differ, I stand corrected. And besides, don't worry about it. At least you offered something. The Wikipedia Yang–Mills existence and mass gap article makes for interesting reading. – John Duffield Feb 13 '17 at 22:17
• All of this still does not answer my question, it does help indeed understand better. As @JohnDuffield mentioned, my question was how the property can be understood from a theoretical point of view. – Edwardo9 Feb 13 '17 at 23:38

The successful use of Yang-Mills theory to describe the strong interactions of elementary particles...

Who says it's successful? The nuclear force remains one of the unsolved problems in physics: "What is the nature of the nuclear force that binds protons and neutrons into stable nuclei and rare isotopes? What is the origin of simple patterns in complex nuclei?" Forces between quarks and gluons doesn't make up for that. Especially when the gluons in ordinary hadrons are virtual, and we've never ever seen a free quark.

depends on a subtle quantum mechanical property called the "mass gap" as we know: the quantum particles have positive masses, even though the classical waves travel at the speed of light.

IMHO quantum particles such as the electron have a positive mass whilst electromagnetic waves are massless because of E=mc². As Einstein said, "the mass of a body is a measure of its energy-content". When you trap a massless photon in a mirror-box, you increase the mass of that system. When you open the box, it's a radiating body that loses mass. See https://arxiv.org/abs/1508.06478 by van der Mark and (not the Nobel) 't Hooft. Think of photon momentum as resistance to change of motion for a wave moving linearly at c. Think of the extra mass of the mirror-box as resistance to change of motion for a wave going round and round at c. As for the mass gap, see Wikipedia:

"the mass gap is the difference in energy between the lowest energy state, the vacuum, and the next lowest energy state. The energy of the vacuum is zero by definition and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest".

It's not a good idea to make assumptions in physics. An electron is a spinor. We can diffract electrons. The wave nature of matter is not in doubt. But an electron isn't going past you at c like a photon. Check out Hans Ohanian’s 1984 paper what is spin? Pair production works the way that it does and the electron has the mass it has for a good reason. However that isn't part of the standard model.

Now, my question is, this property has been discovered experimentally and computationally; but how can it be understood from a theoretical point of view?

By looking outside the standard model. In particular I'd say it's important to appreciate that space has particular properties associated with E=hf electromagnetic waves. And that in atomic orbitals electrons exist as standing waves. So the mass gap concerns the way in pair production you form an electromagnetic wave into a spin ½ standing-wave spinor called the electron. The standard model doesn't tell you anything about this, so you end up in something of a catch-22 situation. To understand something in the standard model, you have to understand something that isn't in the standard model, and is arguably in conflict with it.