Wednesday, September 9, 2015

conventions - Index of representation of $SU(N)$ fundamental and adjoint


Im getting crazy trying to derive this simple expression.


Say $f^{abc}$ are structure constants of a Lie algebra of $SU(N)$ with $[T^a, T^b]=i f^{abc}T^c$. Then chosing normalization such that



$$\sum_{c,d}f^{acd}f^{bcd}=N\delta^{ab}$$


Show that for fundamental rep of $SU(N)$: $$tr(T^aT^b)=\frac{1}{2}\delta^{ab}$$


And for adjoint rep: $$tr(T^aT^b)=N\delta^{ab}$$


Let me know if you have any suggestions, all much appreciated.


EDIT: Adjoint representation case is simply a matter of definition $(T_{adj}^a)^{bc}=-if^{abc}$


$$tr(T^aT^b)=-f^{amn}f^{bnm}=f^{amn}f^{bmn}=N\delta^{ab}$$




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