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I'm reading a rather elementary particle physics text, Modern Particle Physics by Thomson. He is staying away from the heavy group theoretic stuff. He derives the transformation law for an SU(2) antiquark doublet and shows it is possible to construct a $\bar 2$ that transforms exactly like the $2$. Then a few pages later, he says there is no way to make a $\bar{3}$ transform like a $3$ in SU(3). The explanation is that it just can't be done. That's obviously not very satisfying, but I image the true explanation is not suited for an undergrad text.

So, what is the real reason we can't create an antiquark triplet that transforms like the quark triplet, i.e. if $q$ is the $3$ and transforms like $$q\longrightarrow q'=\exp(\tfrac{1}{2}i\vec\alpha\cdot\vec\lambda)q$$ then why can't we find a $\bar 3$ $\bar q$ such that $$\bar q\longrightarrow \bar q'=\exp(\tfrac{1}{2}i\vec\alpha\cdot\vec\lambda)\bar q$$

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The transformation of a quark field under a group require you have to choose a representation of that group. It happen that the fundamental representation and the anti fundamental (bar) of $SU(N)$ with $N>2$ are inequivalent in the sense that there no non singular matrix independent of the representation chosen that allow us to make a change of basis and passing to one representation to another. I mean, if $\lambda$ is a matrix of the fundamental and $\bar{\lambda}$ is of anti fundamental there is no matrix $M$ that make : $M\lambda M^{-1}=\bar{\lambda}$. For $SU(2)$ it happen that there is that matrix $M$ and is the Levicivita tensor in $2$ dimension.So there's no way to create an antiquark triplet that transforms like the quark triplet because there is no way (in $SU(3)$ for passing from the fundamental representation to anti fundamental.

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  • $\begingroup$ Are you aware of a more rigorous proof of this fact? If yes, I'll give you the check. $\endgroup$ – Ryan Unger Feb 8 '15 at 11:51
  • $\begingroup$ Mhm, i think it will more useful for you read from this paper from my group theory class dl.dropboxusercontent.com/u/9571828/mathphys.pdf because i don't understand very well what kind of proof you talking about but i'm sure that you'll find what you want in that lectures at the section "Tensor representation" $\endgroup$ – Andrea89 Feb 8 '15 at 12:06
  • $\begingroup$ I'll give you the check. Lol, your instructor thinks people learn polynomials in elementary school? $\endgroup$ – Ryan Unger Feb 8 '15 at 12:15
  • $\begingroup$ why? (thanks for the check) $\endgroup$ – Andrea89 Feb 8 '15 at 12:22
  • $\begingroup$ When he says the $p(x)$ form a ring, he says we learned how to $+$ and $\cdot$ them in elementary school. Or maybe he's just saying we learned general addition and multiplication in elementary school. Anyway, thanks for the PDF. I've been meaning to learn Young Tableaux for a while now. $\endgroup$ – Ryan Unger Feb 8 '15 at 12:30
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This might not be the answer you expect, but if antiquarks were in the 3 representation, mesons made of a quark and an antiquark would have a global color. In addition, we would have found "baryon" states made of $qq\bar{q}$ or $q\bar{q}\bar{q}$. That would be in contradiction with the experiments.

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