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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 [tex]-\frac{1}{4} F_{\mu \nu}^{a}F^{\mu \nu a}[/tex]

arbitrary rep.: [tex]F_{\mu \nu} =F_{\mu \nu}^a T^a[/tex]

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

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

and now for the adjoint representation of SU(N): [tex](T^a)_{ij}=-if^{aij}[/tex]

(note: the generators are here [tex](N^2 -1)\times (N^2-1)[/tex]-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

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# Kinetic lagrangian for non-abelian gauge theories

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