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Benedict Irwin edited Ternary Ladders.tex
almost 10 years ago
Commit id: 02f2999372754608d1a32abb9fe06cd7e65f9fdb
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So in Triplet operations exists operations which have an eigenstate of $|\alpha>$ and counting triplets and also binary operations in the same way quarks combine. \begin{equation}
aaa^+|\alpha>=\beta^{\alpha}_{a^+}\beta^{\alpha+\frac{2}{3}}_a\beta^{\alpha+\frac{1}{3}}_a|\alpha> \\
a^+a^+a|\alpha>=\beta^{\alpha}_{a}\beta^{\alpha-\frac{1}{3}}_{a^+}\beta^{\alpha+\frac{1}{3}}_{a^+}|\alpha+1>
\\
aaa|\alpha>=\beta^{\alpha}_{a}\beta^{\alpha-\frac{1}{3}}_{a}\beta^{\alpha-\frac{2}{3}}_{a}|\alpha-1>
\end{equation}
For doublet (meson) operations take an operator and it's anti-operator\begin{equation}
\hat{a}^+a^+|\alpha>=\beta^{\alpha}_{a^+}\beta^{\alpha +\frac{2}{3}}_{\hat{a}^+}|\alpha> \\
a^+\hat{a}^+|\alpha>=\beta^{\alpha}_{\hat{a}^+}\beta^{\alpha -\frac{2}{3}}_{a^+}|\alpha>
\\
aaa|\alpha>=\beta^{\alpha}_{a}\beta^{\alpha-\frac{1}{3}}_{a}\beta^{\alpha-\frac{2}{3}}_{a}|\alpha-1>
\end{equation}