- #1
Slereah
- 7
- 0
I am currently trying to find how to derive the decomposition for two particles via the tensor notation, for instance for the product of two particles of spin 1/2 :
[tex]\frac{1}{2} \otimes \frac{1}{2} = 1 \oplus 0 [/tex]
Giving the components of spin 1 and 0. So to do it, I write down the product of two spinors and write it as its symmetric and antisymmetric part :
[tex]\psi_a \chi_b = \frac{1}{2} (\psi_{[a} {\chi_{b}}_] + \psi_{\{a} {\chi_{b}}_\})[/tex]
The antisymmetric part is the scalar part, being simply
[tex]\psi_{[a} {\chi_{b}}_] = \left( \!\!\begin{array}{cc}
0&-1\\
1&0
\end{array}\! \right) (\psi^+ \chi^- - \psi^- \chi^+) [/tex]
And the symmetric part is something of the form
[tex] \psi_{\{a} {\chi_{b}}_\} = \left( \!\!\begin{array}{cc}
2\psi_+ \chi_+&\psi_+ \chi_- + \psi_- \chi_+\\
\psi_+ \chi_- + \psi_- \chi_+&2\psi_- \chi_-
\end{array}\! \right) [/tex]
Which looks like it contains all the right components, but then I try to compare it to the actual vector quantity :
[tex]V^{ab} = \varepsilon^{ca} V_c^b = \varepsilon^{ca} (\sigma^\mu)_c^b V_\mu = \left( \!\!\begin{array}{cc}
- V_x - i V_y &V_z\\
V_z&V_x - i V_y
\end{array}\! \right) [/tex]
There may be a sign wrong in here somewhere because of the spinor metric ([tex]\varepsilon[\tex], the levi-civita tensor), but even up to a sign, the symmetric part does not really lend itself to being put in vector form, and it would seem to mix the states (+,+) with (-,-) as well. I tried using directly the product of a contravariant and covariant spinor, but it did not help. So where does the problem stem from?
[tex]\frac{1}{2} \otimes \frac{1}{2} = 1 \oplus 0 [/tex]
Giving the components of spin 1 and 0. So to do it, I write down the product of two spinors and write it as its symmetric and antisymmetric part :
[tex]\psi_a \chi_b = \frac{1}{2} (\psi_{[a} {\chi_{b}}_] + \psi_{\{a} {\chi_{b}}_\})[/tex]
The antisymmetric part is the scalar part, being simply
[tex]\psi_{[a} {\chi_{b}}_] = \left( \!\!\begin{array}{cc}
0&-1\\
1&0
\end{array}\! \right) (\psi^+ \chi^- - \psi^- \chi^+) [/tex]
And the symmetric part is something of the form
[tex] \psi_{\{a} {\chi_{b}}_\} = \left( \!\!\begin{array}{cc}
2\psi_+ \chi_+&\psi_+ \chi_- + \psi_- \chi_+\\
\psi_+ \chi_- + \psi_- \chi_+&2\psi_- \chi_-
\end{array}\! \right) [/tex]
Which looks like it contains all the right components, but then I try to compare it to the actual vector quantity :
[tex]V^{ab} = \varepsilon^{ca} V_c^b = \varepsilon^{ca} (\sigma^\mu)_c^b V_\mu = \left( \!\!\begin{array}{cc}
- V_x - i V_y &V_z\\
V_z&V_x - i V_y
\end{array}\! \right) [/tex]
There may be a sign wrong in here somewhere because of the spinor metric ([tex]\varepsilon[\tex], the levi-civita tensor), but even up to a sign, the symmetric part does not really lend itself to being put in vector form, and it would seem to mix the states (+,+) with (-,-) as well. I tried using directly the product of a contravariant and covariant spinor, but it did not help. So where does the problem stem from?