Dear Dmitry, a very nice homework. Truth to be told, I am not sure about the asewnr. But it is true that Instanton's asewnr, currently shared by 14% of the readers, is the only acceptable asewnr in democracy where every solution can always be correct. Saying that one solution is more correct than others is a case of discrimination.Also, I didn't quite understand whether the homework was physical or mathematical. Are you asking which of them is more correct in the real world ie. the full QCD? Otherwise, mathematically you seem to be solving three different problems, pretty well? But when you say that the quarks are heavy and nonrelativistic, doesn't it eliminate the relativistic solution 2? Also, if you define the quarks to be the heavy degrees of freedom in BO and you define the flux tube to be the light degrees of freedom, doesn't it mean that you say that the lightest energy gaps must come from the string's vibration? That would make solution 3 sound pretty good except that isn't the flux tube always a relativistic string? For instanton: the scaling in (2) with the cubed root can be seen by dimensional analysis, without knowing Airy functions. You solve a quantum Hamiltonian, H = p^2 / M + sigma X. Now, to find the scalings, choose a new variable y=Qx and its dual p_y=x/Q (scaled in the opposite way) in such a way that H = K (py^2 + y). By mapping these two things with two coefficients, you have K/Q^2 = 1/M, KQ = sigma. Dividing the latter equation by the former, you have Q^3 = sigma M, Q=(sigma M)^(1/3). And K = (sigma^2 / M)^{1/3}. This K is Dmitry's gap of H because the remaining factor, py^2 + y, has spacing of order one by dimensional analysis: there are no parameters here.Best wishesLubos
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Dear Dmitry, a very nice homework. Truth to be told, I am not sure about the asewnr. But it is true that Instanton's asewnr, currently shared by 14% of the readers, is the only acceptable asewnr in democracy where every solution can always be correct. Saying that one solution is more correct than others is a case of discrimination.Also, I didn't quite understand whether the homework was physical or mathematical. Are you asking which of them is more correct in the real world ie. the full QCD? Otherwise, mathematically you seem to be solving three different problems, pretty well? But when you say that the quarks are heavy and nonrelativistic, doesn't it eliminate the relativistic solution 2? Also, if you define the quarks to be the heavy degrees of freedom in BO and you define the flux tube to be the light degrees of freedom, doesn't it mean that you say that the lightest energy gaps must come from the string's vibration? That would make solution 3 sound pretty good except that isn't the flux tube always a relativistic string? For instanton: the scaling in (2) with the cubed root can be seen by dimensional analysis, without knowing Airy functions. You solve a quantum Hamiltonian, H = p^2 / M + sigma X. Now, to find the scalings, choose a new variable y=Qx and its dual p_y=x/Q (scaled in the opposite way) in such a way that H = K (py^2 + y). By mapping these two things with two coefficients, you have K/Q^2 = 1/M, KQ = sigma. Dividing the latter equation by the former, you have Q^3 = sigma M, Q=(sigma M)^(1/3). And K = (sigma^2 / M)^{1/3}. This K is Dmitry's gap of H because the remaining factor, py^2 + y, has spacing of order one by dimensional analysis: there are no parameters here.Best wishesLubos
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