QuoteOriginally posted by: scholarQuoteFor the pion, I can visualize it as two quarks connected by a gluon flux tube: o ================= oMy question: what is the visualization for the lightest particle in the pure Yang-Mills theory of the Mass Gap prize??It could be some topologically non-trivial (and hence stable) solution of the gluon field, like a soliton or something like that, properly quantized. This would be similar to viewing the proton as a soliton of the pion field, as was originally suggested by Skyrme, and then developed by others (including Witten).I found this discussion over at Physics Forums useful.In particular, these comments reproduced from Polyakov's book on the possible role (or not) of instantons were interesting:QuoteSo : here are quotations from "Gauge Fields and strings". Even the negative results are interesting, remember that it is an excellent book to read.(from intro §6)We have seen in the previous chapters that in Abelian systems the problem of charge confinement is solved by instantons.In Non-Abelian theories instantons are also present. However, due to the large perturbative fluctuations, dicussed in Chapter 2, it is difficult to judge whether they play a decisive role in forming a mass gap and a confining regime. In such theories we had a kind of instanton liquid which is difficult to treat. It is possible that due to some hidden symmetries, present in these systems, instantons may form a useful set of variables for an exact description of the system, but this has not yet been shown.At the same time, due to the fact that instantons carry non-trivial topology (they describe configurations of the fields which cannot be "disentangled"), some manifestations of instantons cannot be mixed up with perturbative fluctuations.[...](from end of §6.2)As happened in the case of n-fields, the instanton contribution has an infrared divergence. This implies that in the multi-instanton picture, individual instantons tend to grow and to overlap. The vaive dilute gas approximation is certainly inapplicable then, and we should expect somethig like dissociation of dipole-like instantons to their elementary constituents, as happened in the case of n-fields. However, even one loop computations on the multi-instanton background have not yet been performed, and nothing similar to the Coulomb plasma of the previous section has been discovered. This is connected partly with the fact that multi-instanton solutions have not yet been explicitely parameterized up to now. I expect many interesting surprises await us, even on the one loop level, in this hard problem.[...](from end of §6.2)So, our conclusion is that on the present level of understanding of instanton dynamics, we cannot obtain any exact dynamical statements concerning Non-Abelian gauge theory. In the case of n-fields the situation is slightly better, since we were able to demonstrate the apearance of the mass-gap on a qualitative level. Even in this case one would like to have much deeper understanding of the situation. There are reasons to believe that some considerable progress will be achieved in the near future. In the case of gauge fields we have to pray for luck.At the same time, the existence of fields with topological charge has a deep qualitative influence on the dynamical structure of the theory. [...](from end of §6.3)(...) exchange of a massless fermion pair leads to long-range forces between instantons and anti-instantons. The result of this may have several alternative consequences. The first one is that since (6.87) implies quenching of large fluctuations in the presence of massless fermions, the system looses the confining property and we would end up with massless gauge fields together with fermions. This option seems highly improbable to me on the basis of some analogies and some model considerations. However, I am not aware of any strict statements permitting us to reject it.The second possibility, which in my opinion is realized in the theory, is the following. Due to the strong binding force between fermions the chiral symmetry gets spontaneously broken and as a result the fermions acquire mass. After that has happened, the long range force between instantons and anti-instantons disappears, being screened by the fermionic mass term in the effective lagrangian. The only remaining effect of anomalous non-conservation will consist of giving a mass to the corresponding Goldstone boson.There is also another improbable option, namely that instantons get confined but some type of large fluctuations, not suppressed by fermions, disorder the system.
Last edited by Alan
on September 21st, 2008, 10:00 pm, edited 1 time in total.