# Kinetic lagrangian for non-abelian gauge theories

1. Aug 6, 2008

### patrice

hello everyone!

i've got a very special question on the gauge invariance (gauge group: SU(N), non-abelian) of the kinetic term in the lagrangian. is it invariant for any representation? i only know, that for the fundamental rep. this is true.

to be more specific:

the usual kinetic term is $$-\frac{1}{4} F_{\mu \nu}^{a}F^{\mu \nu a}$$

arbitrary rep.: $$F_{\mu \nu} =F_{\mu \nu}^a T^a$$

the field tensor transforms like this: $$F_{\mu \nu}'=U F_{\mu \nu} U^{-1}$$, where $$U(x)=\exp(i \theta^a (x) T^a)$$. (in an arbitrary representation with generators $$T^a$$)

in the fundamental representation of SU(N), i.e. $$T^a=\frac{\lambda^a}{2}$$, the lagrangian is invariant under gauge transformations, which can easily be proved.

and now for the adjoint representation of SU(N): $$(T^a)_{ij}=-if^{aij}$$
(note: the generators are here $$(N^2 -1)\times (N^2-1)$$-matrices.) does anyone know, or is anyone smart enough :-) to proof, that this also holds for this representation?

thank you, i'm looking forward to any answer!

patrice

2. Aug 6, 2008

3. Aug 7, 2008

### patrice

hi sam!

thank you for your answer! it was a great help for me. i remember my professor saying that the gauge fields transform in the adjoint representation. up to now it's not yet clear to me, how the gauge transformation "prescription" $$A_{\mu}'=UA_\mu U^{-1}+\frac{i}{g} U (\partial _\mu U^{-1})$$ is connected with the generators of the adjoint representation of Lie group SU(N), namely the structure constants of the corrensponding Lie Algebra.

Does $$A_\mu= A_\mu ^a T^a$$ not mean (which one follows from the prescription above), that $$A_\mu$$ is Lie Algebra valued for the representation of our choice for the matter multiplett $$\psi$$, even the fundamental rep?

by the way: before i read your post i wanted to say, that the problem is solved, because $$F_{\mu \nu}^a$$ is independent from the representation and so the Lagrangian is too. but now i'm unsure...

i found a maybe good article on adjoint endomorphisms at wikipedia which i will try to understand later on. but now for the moment my head is full :) i'm going to tell you, if i could get wise from that arcticle..

best regards

patrice

4. Aug 7, 2008

### samalkhaiat

5. Aug 15, 2008

### patrice

hello

sorry for not posting that long. yes, indeed this wikipedia article is a little short and cannot help me with my problem.
neither could the textbooks that i tried: peskin & schroeder (introduction to quantum field theory), ryder (qft) and bjorken drell (relativistic quantum field theory)

do you know or does anyone know a textbook which gives a more detailed discussion on representations of gauge potentials? that would be really nice.

thanks a lot

patrice

6. Aug 15, 2008

### Avodyne

In an arbitrary rep, the kinetic term is ${\rm Tr}\,F^{\mu\nu}F_{\mu\nu}$ (up to an overall constant).

Try Srednicki's book (google to find a free draft version online).

7. Aug 21, 2008

### patrice

thank you, avodyne! i'll try this book. it really seems to be well structured and quite instructive.

patrice

8. Aug 21, 2008

### schieghoven

Hi, you seem to be mostly on the right track....

Gauge theory posits a symmetry described by the transformation rule $$\psi \rightarrow U \psi$$, where U belongs to a gauge group G, and $$\psi$$ belongs to a vector space which is referred to as the fundamental representation of G. The covariant derivative $$D_{\mu} \psi$$ transforms in the same way (in other words, it also belongs to the fundamental representation), and this can be used to show gauge invariance of the kinetic term for $$\psi$$ in the Lagrangian, as you seemed to refer to in your original post. The adjoint representation is the vector space of quantities which transform as $$Z \rightarrow UZU^{-1}$$; the connection $$A_\mu$$ is of this type for *global* gauge transformations, while the field strength $$F_{\mu\nu}$$ is an even better example, since it transforms this way even under local gauge transformations. $$A_\mu$$ belongs to this space as a consequence of its definition in the covariant derivative. For your question, yes, both $$A_\mu$$ and $$F_{\mu\nu}$$ take values in the Lie algebra of G.

I'm not entirely sure, but perhaps your question in your original post asks what changes if we substitute G for one of its other representations? If so, the answer is nothing, because G can be whatever group you like; the fundamental representation is still the space which holds \psi, and A and F will be in the adjoint representation of that space.

Good luck with the studies!
Dave