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  • Ternary Ladder Operators

    Abstract

    We develop a triplet operator system which encompasses the structure of quark combinations. Ladder operators are created. The constants \(\beta\) are currently being found.

    Ternary Ladders

    Based on quarks

    Define: \[a|\alpha>=\beta^{\alpha}_a|\alpha-\frac{1}{3}> \\ a^+|\alpha>=\beta^{\alpha}_{a^+}|\alpha+\frac{2}{3}> \\ \hat{a}|\alpha>=\beta^{\alpha}_{\hat{a}}|\alpha+\frac{1}{3}>\\ \hat{a}^+|\alpha>=\beta^{\alpha}_{\hat{a}^+}|\alpha-\frac{2}{3}>\]

    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. \[aaa^+|\alpha>=\beta^{\alpha}_{a^+}\beta^{\alpha+\frac{2}{3}}_a\beta^{\alpha+\frac{1}{3}}_a|\alpha> \\ a^+a^+a|\alpha>=\beta^{\alpha}_{a}\beta^{\alpha-\frac{1}{3}}_{a^+}\beta^{\alpha+\frac{1}{3}}_{a^+}|\alpha+1> \\ aaa|\alpha>=\beta^{\alpha}_{a}\beta^{\alpha-\frac{1}{3}}_{a}\beta^{\alpha-\frac{2}{3}}_{a}|\alpha-1> \\ a^+a^+a^+|\alpha>=\beta^{\alpha}_{a^+}\beta^{\alpha+\frac{2}{3}}_{a^+}\beta^{\alpha+\frac{4}{3}}_{a^+}|\alpha+2>\\\]

    So considering the \(\beta\) terms. The only time \(\beta^{\alpha-\frac{2}{3}}_a\) might arise is if an a has operated on a state that is already filled by \(\alpha-\frac{2}{3}\). So, it should seem appropriate that if \(\alpha=0\) this coeficcient drops to zero to prevent negative integer fillings. This proposes a trial function of \[\beta^{\alpha}_a=\sqrt[3]{\frac{2}{3}+\alpha}\]

    Then the operation \[aaa|\alpha>=\sqrt[3]{\frac{2}{3}+\alpha}\sqrt[3]{\frac{2}{3}+\alpha-\frac{1}{3}}\sqrt[3]{\frac{2}{3}+\alpha-\frac{2}{3}}|\alpha-1>\]

    and for \(\alpha=0\) the coefficient drops to zero.

    One could forsee that if the is a base number operator \(N=aaa^+\) then combinations of these transitions could be such that \[(aaa)(a^+a^+a)|\alpha> = a(aaa^+)a^+a|\alpha>\]

    For doublet (meson) operations take an operator and it’s anti-operator\[\hat{a}^+a^+|\alpha>=\beta^{\alpha}_{a^+}\beta^{\alpha +\frac{2}{3}}_{\hat{a}^+}|\alpha> \\ a^+\hat{a}^+|\alpha>=\beta^{\alpha}_{\hat{a}^+}\beta^{\alpha -\frac{2}{3}}_{a^+}|\alpha>\]

    Question? Does \[\beta^{\alpha}_{a^+}\beta^{\alpha +\frac{2}{3}}_{\hat{a}^+}=\beta^{\alpha}_{\hat{a}^+}\beta^{\alpha -\frac{2}{3}}_{a^+}?\]

    Another Method

    Another method to try is 3 (or 6 really) operators, for example \[a^r_{+}|\alpha>=\sqrt[3]{\alpha}|\alpha+\frac{2}{3}> \\ a^r_{-}|\alpha>=\sqrt[3]{\alpha}|\alpha-\frac{1}{3}> \\ a^g_{+}|\alpha>=\sqrt[3]{\alpha-\frac{1}{3}}|\alpha+\frac{2}{3}> \\ a^g_{-}|\alpha>=\sqrt[3]{\alpha-\frac{1}{3}}|\alpha-\frac{1}{3}> \\ a^b_{+}|\alpha>=\sqrt[3]{\alpha+\frac{1}{3}}|\alpha+\frac{2}{3}> \\ a^b_{-}|\alpha>=\sqrt[3]{\alpha+\frac{1}{3}}|\alpha-\frac{1}{3}> \\\]

    Then is was found that the combination \[a^r_+a^g_-a^b_+|\alpha>=(\alpha+\frac{1}{3})|\alpha+1>\]

    Stable Number Operator

    An eigenstate for a ternay triplet was found. Define: \[r^+|\alpha>=\sqrt[3]{\alpha}|\alpha+\frac{2}{3}> \\ r|\alpha>=\sqrt[3]{\alpha}|\alpha-\frac{1}{3}> \\ g^+|n>=\sqrt[3]{n+\frac{2}{3}}|n+\frac{2}{3}> \\ g|\alpha>=\sqrt[3]{\alpha-\frac{2}{3}}|\alpha-\frac{1}{3}>\\ b^+|n>=\sqrt[3]{n+\frac{1}{3}}|n+\frac{2}{3}> \\ b|\alpha>=\sqrt[3]{\alpha-\frac{1}{3}}|\alpha-\frac{1}{3}>\]

    Then the operation \[a^b_-a^g_-a^r_+|\alpha>=a^b_-a^g_-\sqrt[3]{\alpha}|\alpha+\frac{2}{3}> \\ =a^b_-\sqrt[3]{\alpha}\sqrt[3]{\alpha}|\alpha+\frac{1}{3}> \\ a^b_-a^g_-a^r_+|\alpha>=\alpha|\alpha>\]

    and then acts like a number operator

    Then there are operations: \[a^ra^ba^g|\alpha>=(\alpha-\frac{2}{3})|\alpha-1> \\ ... \\ bgr|\alpha>=\]

    Hamiltonian basic version:

    \[H=Wbgr^+\\\]

    TERNARY COMMUTATOR? [a,b,c]=abc-acb-bac+bca+cab-cba \[\begin{array}{| c | c |} \hline combin. & coeff \\ \hline bgr^+ & \alpha \\ br^+g & \sqrt[3]{\alpha}\sqrt[3]{\alpha-\frac{1}{3}}\sqrt[3]{\alpha-\frac{2}{3}} \\ gbr^+ & \sqrt[3]{\alpha}\sqrt[3]{\alpha+\frac{1}{3}}\sqrt[3]{\alpha-\frac{1}{3}} \\ gr^+b & \alpha -\frac{1}{3} \\ r^+bg & \alpha -\frac{2}{3} \\ r^+gb & \sqrt[3]{\alpha-1}\sqrt[3]{\alpha-\frac{1}{3}}\sqrt[3]{\alpha-\frac{2}{3}} \\ \hline \end{array}\]

    Thus \[[b,g,r^+]=\alpha - \sqrt[3]{\alpha}\sqrt[3]{\alpha-\frac{1}{3}}\sqrt[3]{\alpha-\frac{2}{3}} \\ - \sqrt[3]{\alpha}\sqrt[3]{\alpha+\frac{1}{3}}\sqrt[3]{\alpha-\frac{1}{3}} + (\alpha -\frac{1}{3}) \\ + (\alpha -\frac{2}{3}) - \sqrt[3]{\alpha-1}\sqrt[3]{\alpha-\frac{1}{3}}\sqrt[3]{\alpha-\frac{2}{3}}\]

    Out of all possible operators involving 3 of the primitive operators above there are 10 solutions with “nice” coefficients. 1 behaves as an annihilation operator, 3 behave as creation operators and the others are like number operators and thus eigen operators of the states. \[\begin{array}{| c | c |} \hline r^+bg|n> & (n - \frac{2}{3})|n> \\ r^+bb^+|n> & (n + \frac{1}{3})|n+1> \\ rbg|n> & (n-\frac{2}{3})|n-1> \\ rbb^+|n> & (n+\frac{1}{3})|n> \\ gr^+g^+|n> & (n+\frac{2}{3})|n+1> \\ gr^+b|n> & (n-\frac{1}{3})|n> \\ b^+rg^+|n> & (n+\frac{2}{3})|n+1> \\ b^+rb|n> & (n-\frac{1}{3})|n> \\ bgr^+|n> & n|n> \\ bb^+r|n> & n|n> \\ \hline \end{array}\]