marlon said:
selfAdjoint is completely right, humanino. This is how it is, point final.
Cristal clear explanation...
regards
marlon
Of course, since selfAdjoint is a superMentor, and one of the best available in this forum.
Thanks for answer sA. I am not flattering. :shy:
Let me elaborate a little. Instantons are selfAdjoint (too !) classical solutions of the pure YM dynamics, and thus minimize the action as :
S\geq \frac{8\pi^2}{g\2}Q_T
with the topological charge (Pontryagin index) related to Chern-Simmons number as Q_T = N_{CS}(+\infty)-N_{CS}(-\infty) = \int d^4x \partial_\mu K_\mu
where \frac{1}{32\pi^2}F \tilde F = \partial_\mu K_\mu is the topological term that can be added to the usual QCD lagrangian.
The energy of the field is a periodic function of the topological charge Q_T, and oscillator-like in the other directions. This leads to the interpretation of instantons as tunneling effect between different vacua (similar to solitons).
The tunneling amplitude is given by
{\cal A} \sim e^{-S} = e^{-\frac{8\pi^2}{g^2} }= e^{-\frac{2\pi}{\alpha_s} }
which makes it clear that instantons are non-perturbative in the coupling constant.
Let me come to the point : chiral symetry breaking by instantons.
It is obvious from the well-known fact that the quark condensate acquire a nonzero value in the presence of instantons :
\langle \bar{q}_i q_i\rangle \approx -(250 MeV )^3
To see this, one has to calculate the partition function of QCD, separate the pure-gluon contribution, and in the remaining part, interpret the fermionic functional integral as a determinant :
<br />
{\cal Z} = \int DA_{\mu} D\Psi D\Psi^{\dagger}<br />
\exp[-\frac{1}{4g^2}\int F^2 + \sum_f\int \Psi_f^{\dagger}(\imath \gamma_\mu \nabla^\mu+\imath m_f) \Psi_f] <br />
=\int DA_{\mu} <br />
\exp[-\frac{1}{4g^2}\int F^2] \prod_f \det(\imath \gamma_\mu \nabla^\mu+\imath m_f) <br />
= \overline{det(\imath \gamma_\mu \nabla^\mu+\imath m_f) }<br />
with the average taken over the instanton gas. I am beginning to think that only those already knowing the Banks-Casher relation are following
The classical problem with this determinant is that it is formally not hermitean because of the \imath m term. Here, by acting on a solution with \gamma_5, one obtains another eigenvector of the Dirac operator, with opposite eigenvalue (classic trick in chiral stuff) :
det(\imath \gamma_\mu \nabla^\mu+\imath m_f) = \sqrt{ \prod_n (\lambda_n^2 + m^2)} <br />
=\exp[\frac{1}{2}\sum_n (\lambda_n^2 + m^2)]<br />
=\exp[\frac{1}{2}\int_{-\infty}^{+\infty} d\lambda \overline{\nu(\lambda)} \ln(\lambda_n^2 + m^2)]<br />
with the spectral density of the Dirac operator \nu(\lambda)} averaged over the instanton ensemble.
A few more manipulations lead to the celebrated Banks-Casher relation :
\langle \bar{q}_i q_i\rangle = -\frac{1}{V}<br />
\frac{\partial}{\partial m}<br />
\left[ \frac{1}{2}\int_{-\infty}^{+\infty} d\lambda \overline{\nu(\lambda)} \ln(\lambda^2 + m^2) \right] _{m \rightarrow 0}<br />
= -\frac{1}{V}<br />
\left[ \int_{-\infty}^{+\infty} <br />
d\lambda \overline{\nu(\lambda)} \frac{m}{\lambda^2 + m^2}\right] _{m \rightarrow 0}<br />
And finally :
\langle \bar{q}_i q_i\rangle = -\frac{1}{V} sign(m)\pi \overline{\nu(0)}
I hope that was not too long, or at least will motivate those not already familiar who could be interested. I made it technical because I am not able to sum up with concepts in a clear manner those tools I recently discovered in the literature.
The Banks-Casher relation relates the quark condensate to the spectral density of the Dirac operator at the origin.
I would like to know if other people think it is (as I am convincing myself) an appealing direction to compute the mass gap ?
: the instantons indeed spontaneaously break chiral invariance by giving the quark condensate a non vanishing value 
The Planck epoch presides this. And then, there was light? 
