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blue2script
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Hi all!
I am currently preparing for an oral exam in quantum field theory and particle physics and I have some problems with the SU(3)-Hadron Multipletts and the relation to the Gell-Mann-Nishijima equation: First, for SU(2) Multipletts you take your Casimir-Operator J, some commuting operator [tex]J_3[/tex] and index your states by [tex]\left|j,m\right\rangle[/tex]. Then, in principle, you get all states of a multiplett by starting with some highest [tex]J_3[/tex]-state and take the [tex]J_-[/tex] operator to generate all states. You know that all states with [tex]j < m[/tex] must be zero.
Now, the treatment of the SU(3) multipletts is a lot different and I have only a vague idea why that is the case. For the full SU(3) flavour symmetry you index the states by means of the two diagonal generators [tex]T_3 = F_3, Y = 2/\sqrt 3 F_8[/tex] where [tex]T_3[/tex] is the third component of the isospin and Y is the hypercharge. Than you introduce the ladder operators [tex]T_\pm, U_\pm, V_\pm[/tex]. What I don't understand is: You label the states by [tex]t_3, y[/tex]. Why? Sure, both generators commute, but where are the eigenvalues of the two Casimir-Operators of the SU(3)? In text-books it is argued that the ladder operators above change [tex]t_3, y[/tex] and so you construct your nice diagrams in this plane. But why for example do I get a triangle decuplett? How do I know which states give zero under the action of one of the ladder operators?
Why don't I just take the Casimir-Operators of the SU(3) (which I can't find in any book...), take two generators, calculate the eigenvalues and dependences of the eigenvalues of the four commuting generators and construct my decuplett? Especially: What is the implication of the Gell-Mann-Nishijima equation? What does it mean that the eigenvalues of [tex]T_3, Y[/tex] give the charge of the particle?
Thank you very much for answering my questions! Hope I clarified them good enough!
Blue2script
I am currently preparing for an oral exam in quantum field theory and particle physics and I have some problems with the SU(3)-Hadron Multipletts and the relation to the Gell-Mann-Nishijima equation: First, for SU(2) Multipletts you take your Casimir-Operator J, some commuting operator [tex]J_3[/tex] and index your states by [tex]\left|j,m\right\rangle[/tex]. Then, in principle, you get all states of a multiplett by starting with some highest [tex]J_3[/tex]-state and take the [tex]J_-[/tex] operator to generate all states. You know that all states with [tex]j < m[/tex] must be zero.
Now, the treatment of the SU(3) multipletts is a lot different and I have only a vague idea why that is the case. For the full SU(3) flavour symmetry you index the states by means of the two diagonal generators [tex]T_3 = F_3, Y = 2/\sqrt 3 F_8[/tex] where [tex]T_3[/tex] is the third component of the isospin and Y is the hypercharge. Than you introduce the ladder operators [tex]T_\pm, U_\pm, V_\pm[/tex]. What I don't understand is: You label the states by [tex]t_3, y[/tex]. Why? Sure, both generators commute, but where are the eigenvalues of the two Casimir-Operators of the SU(3)? In text-books it is argued that the ladder operators above change [tex]t_3, y[/tex] and so you construct your nice diagrams in this plane. But why for example do I get a triangle decuplett? How do I know which states give zero under the action of one of the ladder operators?
Why don't I just take the Casimir-Operators of the SU(3) (which I can't find in any book...), take two generators, calculate the eigenvalues and dependences of the eigenvalues of the four commuting generators and construct my decuplett? Especially: What is the implication of the Gell-Mann-Nishijima equation? What does it mean that the eigenvalues of [tex]T_3, Y[/tex] give the charge of the particle?
Thank you very much for answering my questions! Hope I clarified them good enough!
Blue2script