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There is a gluon that is $$\frac{1}{\sqrt{3}} (red \cdot\overline{red} + blue\cdot\overline{blue} - 2\cdot green \cdot\overline{green})$$ This confuses me because I do not understand how adding and subtracting and dividing these colors would work. I know that in matrix form it is $$ A = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}$$ This still confuses me, please help me. This is the 8th Gell-Mann matrix.

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What you are writing as red is actually a quantum mechanical state, which is a vector in a Hilbert space. What you are writing as a dot is something called a tensor product; it produces a vector in another Hilbert space. You can add and subtract the vectors in Hilbert spaces, and you can multiply them by scalars like $2$ or $1/\sqrt 3$, just like in any vector space.

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