Thursday 7 June 2018

group theory - Fundamental Representation of $SU(3)$ is a complex representation


Let in a $D(R)$ dimensional representation of $SU(N)$ the generators, $T^a$s follow the following commutation rule:
$\qquad \qquad \qquad [T^a_R, T^b_R]=if^{abc}T^c_R$.


Now if $-(T^a_R)^* = T^a_R $, the representation $R$ is real. Again if we can find a unitary matrix, $V(\neq I)$ such that


$ \qquad \qquad \qquad -(T^a_R)^*=V^{-1} T^a_R V \quad \forall a $


then the representation $R$ is pseudoreal.


If a representation is neither real nor pseudoreal, the representation $R$ is complex.


Claim: One way to show that a representation is complex is to show that at least one generator matrix $T^a_R$ has eigenvalues that do not come in plus-minus pairs.


Now let us consider $SU(3)$ group. The generators in the fundamental representation are given by


$T^a =\lambda^a/2; \quad a=1,...8$,

where $\lambda^a$s are the Gell-Mann matrices. We see that $T^8$ has eigenvalues $(1/\sqrt{12}, 1/\sqrt{12}, -1/\sqrt{3} )$.


My doubt is:


According to the claim, is the fundamental representation of $SU(3)$ a complex representation?




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...