I just want to make sure that I am doing some things correctly. I'll be using(adsbygoogle = window.adsbygoogle || []).push({});

http://www.physics.umd.edu/courses/Phys741/xji/chapter5.pdf

from about 5.64 on.

The kinetic term :

[tex]

\frac{f^2}{4} Tr[D_{\mu} \Sigma D^{\mu} \Sigma^{\dagger}]

[/tex]

Now if I want to expand this out, as [itex] \Sigma =e^{i \vec{\pi}^a \cdot \tau^{a}/f}[/itex] (5.64)

Now, first, am I missing something. The pion fields have a vector sign, AND an index, AND are dotted into the generators. Now I assume "a" is the index of the color group, and the pion "vector" is really just a 1-D vector, and so the vector sign is redundant.

Then as the tau is a matrix, and the pion now a scalar, the cdot between them is wrong and it should be implied that it is a normal multiplication and not a dot product.

So it should either be [itex]\vec{\pi} \cdot \tau[/itex] or [itex]\pi^a \tau^a[/itex] with a sum implied. Right?

Now on to the next question:

Expanding it out:

Exp[x] = 1 + i x -1/2 x^2

So knowing that there are matrices and vectors involved, how does it expand:

[tex]

e^{i \pi^a \tau^a/f} = 1 + \frac{i}{f} \pi^a \tau^a - \frac{1}{2 f^2} (\pi^a \tau^a) (\pi^a \tau^a)

[/tex]

or

[tex]

e^{i \pi^a \tau^a/f} = 1 + \frac{i}{f} \pi^a \tau^a - \frac{1}{2 f^2} (\pi^a \tau^a) (\pi^b \tau^b)

[/tex]

or

[tex]

e^{i \pi^a \tau^a/f} = 1 + \frac{i}{f} \pi^a \tau^a - \frac{1}{4 f^2} [(\pi^a \tau^a) (\pi^b \tau^b) +(\pi^b \tau^b) (\pi^a \tau^a) ]

[/tex]

etc. Do i need to add new indices each time I expand it, and how do I keep track of order.

The order will matter as when I take the conjugate transpose for the second term in the kinetic part itll reverse the order of the generators.

This is probably what I'm most stuck on. My lagrangian is more complicated than just a kinetic term and so I want to be really careful about the ordering of the generators.

Question 3:

The trace that is implied is a trace over COLOR (the generators)? Or a trace over dirac structure? Or both? The strong interaction term that appears between heavy mesons and the chiral fields includes a similar trace.

Question 4:

When taking the hermitian conjugate of the fine structure constants does [itex](f^{abc})^{\dagger} = f^{cba} = -f^{abc}[/itex]. Do I change the order of indices thusly, as if it were a matrix?

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# Chiral Perturbation Theory : Some quick questions

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