# Why are 3 colors used in QCD?

The mapping of strong charge to RGB left me believing that there are only 3 conserved quantities in QCD. I recently came to the understanding that there are in fact 8 conserved quantities, as explained in the answers to the following question: Why is color conserved in QCD?

My question is this. If we used 8 colors, or other analogies, to describe QCD, would this be equally representative? Is the choice of the number 3 significant, beyond an intuitive trick of simplification?

The language of "colors" maps more-or-less straightforwardly to the actual group theory that controls the strong force:

The gauge symmetry that is associated to the strong force is the unitary group in three (complex) dimensions, $\mathrm{SU}(3)$. The analogon for the other forces would be $\mathrm{U}(1)$ for electromagnetism and $\mathrm{SU}(2)$ for the weak force. The lowest-dimensional "fundamental" representation is three-dimensional, it's just the natural action of $\mathrm{SU}(3)$ on $\mathbb{C}^3$. We pick three basis vectors in $\mathbb{C}^3$ and call them (arbitrarily) "red, blue, green". This is the representation in which the quarks transform, so for quarks, we speak of red/blue/green quarks.

Now, the number of conserved quantities is unrelated to the dimension of hte fundamental representation - the number of conserved quantities is the dimension of the group as a Lie group, which is 8. Not coincidentally, this is also the number of gluons - the force carriers always transform in the so-called adjoint representation, which is the natural action of the group on its own Lie algebra. Rather by definition, the dimension of the Lie algebra is that of the group.

Finally, let's discuss how the naming scheme of "blue-antired" etc. for gluons arises: If we call $V_\text{f}$ the fundamental representation, $V_\text{ad}$ the adjoint and $V_1$ the trivial representation, then we have $$V_\text{f}\otimes V^\ast_\text{f} = V_1\oplus V_\text{ad}$$ where $V^\ast$ is the dual representation. It's sort-of traditional to see the dual/conjugate representation as the antipode to the original representation, so the three basis vectors of $V_\text{f}^\ast$ would be called antired/antiblue/antigreen. So the nine basis vectors of the l.h.s. would be called red-antired,blue-antired,green-antired,red-antigreen, etc. Reorganizing those into eight vectors that span $V_\text{ad}$ gives the usual color nomenclature for gluons.

• It is interesting to read the history, how the necessity of extra charges was imposed by the existence of Delta++ amd Omega minus and who where the first to realize extra charges were necessary en.wikipedia.org/wiki/Quantum_chromodynamics#History – anna v Oct 20 '16 at 13:57
• I know very little of Lie algebra, but I surmise that this means the 3-complex-dimensional symmetry group SU(3) is homomorphic to the 8-dimentional Lie algebra su(3)? Alternatively, would it be correct to say that there are 8 conserved quantities, but 3 independent ways of doing symmetry transformations among them? – Ketil Tunheim Oct 20 '16 at 14:41
• @KetilTunheim No, none of that is true. The group $\mathrm{SU}(3)$ is 8-dimensional, which is why there are 8 conserved quantities. That it is 8-dimensional means there are 8 independent ways of doing symmetry transformations, always. What is 3-dimensional (in the complex sense) is the representation space on which the group acts. $\mathrm{SU}(3)$ are 3-by-3 matrices, and all 3-by-3 matrices form a nine-dimensional vector space, although they act on the three-dimensional vectors. – ACuriousMind Oct 20 '16 at 14:45

The choice of the number $3$ has to do with the underlying symmetry of the strong interaction. The gauge group of quantum chromodynamics is $SU(3)$.

With 3 colors, one would naively expect 9 gluons, since gluons carry one color charge and one anti-color charge. But the group structure only allows for 8 gluons.

We could have chosen some other names... for example, we could have used complementary colors to describe anti-color charge, so instead of anti-red, anti-green and anti-blue, we could have had cyan, magenta and yellow. That's 6 colors. But it wouldn't matter, those are just names, arbitrary labels.

It's the $SU(3)$ and the representation theory of its Lie algebra $\mathfrak{su}(3)$ that matter.