From: goldenfieldquaternions@gmail.com   
      
   On Wednesday, June 21, 2017 at 8:54:47 PM UTC-5, Jos Bergervoet wrote:   
   > On 6/18/2017 6:40 PM, Lawrence Crowell wrote:   
   > > On Sunday, June 18, 2017 at 8:17:13 AM UTC-5, James Goetz wrote:   
   > >> Gluons bond quarks into baryons (i.e., protons and neutrons). For   
   > >> example, two up quarks and one down quark form a proton while one up   
   > >> quark and two down quarks form a neutron. Is there one gluon per one   
   > >> baryon or two gluons per one baryon or what is the ratio of gluons to   
   > >> baryons?   
   > >   
   > > If you or I could definitively answer this question a Nobel prize would   
   > > be next. In a perturbation series there are various orders. The lowest   
   > > order has each quark in a baryon connected by a gluon. So that is one   
   > > gluon per quark, which are sometimes called baryons, or for say the   
   > > proton there are 3 gluons per proton. However, to higher order there ar=   
   e   
   > > loops and higher hbar corrections and these are amplitudes with more   
   > > gluons. The higher order diagrams are a tangle or mesh of gluons. Also   
   > > at higher orders quark-antiquark pairs come into play.   
   > >   
   > > QCD has a vast solution space, and perturbative QCD lets us look at som=   
   e   
   > > tiny slices or are like small paths in it. The solution space in its   
   > > entirety is a vast unknown.   
   >   
   > But wouldn't it be possible, despite not having the full solution,   
   > to have a number based on the field shape and strength?   
   >   
   > For instance take the similar question about the number of photons   
   > in a Hydrogen atom. We can say that the field in coordinate space   
   > has an extent of the atomic size, so in momentum space the modes   
   > of the photon fields we need are of energy in the order of 1 keV   
   > (the inverse of the Bohr radius or van der Waals radius). And the   
   > energy in the photon field is in the order of the binding energy,   
   > i.e. about 10eV, so the number of photons is:   
   >     = 10eV / 1keV = 0.01   
   > Or to put it differently: the photon field will be a superposition   
   > consisting mainly of the vacuum state (99%) with about 1% admixture   
   > of 1-photon states (an integral over the required Fourier components   
   > of course). And most likely it also contains in the order of about   
   > 0.01% of 2-photon state terms, and so on (if the 'harmonic state'   
   > for the field is any guideline, that is..)   
   >   
   > Now we can play this game for a baryon (especially if we use a   
   > simple description of bound quarks, for instance the old Isgur-Karl   
   > model, but that's an exercise for the reader). Since quarks are   
   > quite light to begin with, half of the total energy might be in the   
   > gluon field now. And the radius of the baryon is of course 1 fm,   
   > so requiring a field shape with momentum space modes of 200 MeV,   
   > on average. Therefore:   
   >    = 0.5GeV / 200MeV = 2.5   
   > So a baryon contains about 2.5 gluon quanta, it seems.   
   >   
   > One might wonder (in both cases) whether the Fourier components for   
   > the shape of the fields do contain an infra-red tail with infinitely   
   > many soft photons (or gluons) so that simply dividing field energy   
   > by energy of the average photon would not be correct. After all   
   > this is exactly the case for scattering problems, where we do indeed   
   > have infinitely many photons, and most of them extremely low energy.   
   > But for these bound states there is no such effect as far as I can   
   > see (the tails in k-space do not give divergent integrals). The   
   > above estimates should be more or less right.   
   >   
   > --   
   > Jos   
      
   In QCD the u and d quarks are about 10MeV at most and the remaining   
   mass of the proton is due to the QCD field of gauge bosons or gluons.   
   The majority of the mass of a proton is due to the mass-gap induced   
   by confinement. This has some bizarre consequences, and one of those   
   is that the number of virtual gauge bosons running around the   
   confined QCD vacuum is divergent. In other words it is infinite.   
   In effect by the anti-sceening physics of chromo-charges the IR   
   contribution to physics dominates and all order diagrams contribute   
   equally. In orders of radiation corrections g^n if g is unity then   
   the most high order diagram contributes. As a result QCD is great   
   at calculating scattering amplitudes where the coupling constant g   
   ~ 1/E, for E the energy of scatter or transverse momentum, but it   
   is no good at calculating the mass of a proton.   
      
   So what do we do? We use computers. The space is gridded up into   
   cells and the gauge potentials run around the edgelinks of the   
   cells. The difference in gauge potentials across nodes determines   
   the chromo-electric field, just as E = -&A/&t - grad A (& = partial).   
   The areas bounded by edgelinks carry chromo-magnetic fields. In   
   doing this I prefer to set up a dual lattice in 3-d where the   
   chromo-electric field is along the dual edgelinks. Think of Faraday's   
   principles. I prefer this because it puts the chromo-electric and   
   magnetic fields on a dual and more equal footing. This system in   
   euclidean 4-d is set to go to equilibrium or a Monte-Carlo calculation   
   determines the most probable distribution of fields.  These are run   
   into the stress-energy tensor to find energy.   
      
   On supercomputers a fair amount of progress has been made. The   
   masses of low energy hadrons has been computed to fair accuracy.   
   There is of course the unsatisfying sense that we still do not   
   understand this. The Claymath problem on mass gap is the open problem   
   to address this.   
      
   Is there a way to think about this analytically? Maybe. The S-dual   
   of QCD is where the chromo-electric field is very weak. This is the   
   gist of the Mantenen-Olive duality   
      
   qg = n\hbar,   
      
   where for a gauge charge q large the dual charge g is small. I think   
   the graviton is maybe an entanglement of colorless pair of these   
   s-dual gluons. By this if we extend QCD to SU(4) then conformal   
   gravity is SU(2,2), and this has a duality with this extended QCD.   
   The advantage is that a weak field theory is not hard to understand.   
   The problem is that it is gravitation now, and that is not hard to   
   understand either. In fact the UV limit of quantum gravity is lots   
   of Planck oscillators collapsed into a black hole. This is in a way   
   dual to the IR limit in QCD. So we have converted a hard problem   
   "here" into a hard problem "there."   
      
   Cheers LC   
      
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