- #1
QuantumCosmo
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Hi,
I was wondering about the [tex]G\tilde{G}[/tex] term that can be added to the QCD Lagrangian.
It seems clear that the analogous QED term can be left out because it is (just as the [tex]G\tilde{G}[/tex] term) a total divergence and thus has no influence on the physics.
But in the case of a non-abelian gauge theory that doesn´t seem to be the case. So
a) why is only the QCD case important? The weak interactions would exhibit that as well, wouldn´t they?
b) I don´t really understand why the term doesn´t drop out just as in QED. When integrating over the [tex]G\tilde{G}[/tex] term, one gets the winding number n - which is simply a constant. Sure, it depends on the type of instanton, so n=n([tex]A_\mu[/tex]) still depends on the gauge fields and therefore does the action S, but if one fixes the intial vacuum at t=-\infty (lets say it has winding number n') and the final vacuum at t=\infty (winding number n''), then all possible gauge fields connecting those two vacua will have winding number n=n'' - n' and so the corresponding term in the action will not depend on [tex]A_\mu[/tex] at all.
Thus, variating the action with respect to [tex]A_\mu[/tex] will give the same result wether or not I leave the term out.
I understand that the term is important because it gives us the winding number and therefore there is physics in it, but I don´t understand why it has to stand in the Lagrangian for that...
(Maybe it is not possible to fix intial and final vacuum? But why?)
I hope somebody here can help me with that :)
Thanks in advance,
Quantum
I was wondering about the [tex]G\tilde{G}[/tex] term that can be added to the QCD Lagrangian.
It seems clear that the analogous QED term can be left out because it is (just as the [tex]G\tilde{G}[/tex] term) a total divergence and thus has no influence on the physics.
But in the case of a non-abelian gauge theory that doesn´t seem to be the case. So
a) why is only the QCD case important? The weak interactions would exhibit that as well, wouldn´t they?
b) I don´t really understand why the term doesn´t drop out just as in QED. When integrating over the [tex]G\tilde{G}[/tex] term, one gets the winding number n - which is simply a constant. Sure, it depends on the type of instanton, so n=n([tex]A_\mu[/tex]) still depends on the gauge fields and therefore does the action S, but if one fixes the intial vacuum at t=-\infty (lets say it has winding number n') and the final vacuum at t=\infty (winding number n''), then all possible gauge fields connecting those two vacua will have winding number n=n'' - n' and so the corresponding term in the action will not depend on [tex]A_\mu[/tex] at all.
Thus, variating the action with respect to [tex]A_\mu[/tex] will give the same result wether or not I leave the term out.
I understand that the term is important because it gives us the winding number and therefore there is physics in it, but I don´t understand why it has to stand in the Lagrangian for that...
(Maybe it is not possible to fix intial and final vacuum? But why?)
I hope somebody here can help me with that :)
Thanks in advance,
Quantum
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