From: goldenfieldquaternions@gmail.com
guess I don't entirely disagree with what is written here. Lattice
gauge calculations provide an IR cut-off by the lattice scale. The last
I read lattice QCD does get the masses of nucleons and light mesons
within a few MeV.
The only part I would question is the idea there can be an infinite
number of gluons as k --> 0. I suppose the problem is just infinity,
which in physics we really prefer to have removed. To be somewhat
speculative if we must invoke something as a real IR cutoff then maybe
the cosmological constant /\ ~ 10{-52}m^{-2} will work. This gives an
energy density rho = 2.4x10^{-8}j/m^3 or 5.8x10^{-45}ev^4. If nature
somehow provides an cutoff this might be it. This should keep the number
of gluons from being infinite.
LC
On Thursday, July 6, 2017 at 2:12:23 AM UTC-5, Jos Bergervoet wrote:
> As already indicated above, I don't think it's bizarre, I just
> wouldn't expect it here. You can only have an infinitude of soft
> quanta if their field has a long-range tail (like 1/r or 1/r^2
> for instance). Now the transition form confined QCD vacuum to
> the surrounding ordinary vacuum does not seem to have this tail.
> The strong interaction range outside a baryon is determined by
> the pion mass, approximately, so it falls off exponentially.
>
> Put differently, if you sum up infinitely many long-wavelength
> quanta (in occupation number space, or Fock space) you will be
> constructing a long-range tail of the gluon field around the
> baryon. Does such a tail really exist?!
>
> > In effect by the anti-sceening physics of chromo-charges the IR
> > contribution to physics dominates and all order diagrams contribute
> > equally.
>
> That is to be expected since we talk about a bound state. Even
> for a simple hydrogen atom you cannot solve that case with
> perturbation theory in finite order. The question I was trying
> to answer is just what the expectation value of total photon
> number in Fock space would be, regardless how we have solved
> the problem.
>
> > 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.
>
> You mean "perturbative QCD" is not good at that. So you need
> another computational tool. Still that does not prove (nor
> disprove) that the bound state must have an infinitude of
> gluons (in the sense of the expectation value ).
>
> > 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 a lattice of finite extent you cannot have soft gluons at
> all, the shortest possible wavelength is restricted. So in
> that case I would expect that you would certainly *not* find
> infinitely many of them!
>
> > On supercomputers a fair amount of progress has been made. The
> > masses of low energy hadrons has been computed to fair accuracy.
>
> Yes, I heard they are within 0.5% correct. But that isn't even
> accurate enough to show the difference between proton and neutron
> mass.. Nevertheless, I would expect that after the calculation one
> can see the statistics of the gluon field in Fock space, is that
> correct? I'm still trying to zoom in on the value of (purely
> to help OP with his question, of course!) and since I don't expect
> it to be infinite, a reasonably sized lattice would suffice to
> indicate the approximate value. (The proof it can't be infinite
> would then have to be added in another way.)
>
> > 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.
>
> Still I am afraid that computing a bound state cannot become "simple"
> just by using any such trick..
>
> > 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."
>
> But is it hard? (To find out whether for gluons is finite or not,
> I mean?) I still do not see what is wrong with my original reasoning:
> without long-range tail, no infinitude of soft quanta is possible.
>
> --
> Jos
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