Where Does the Problem Stem from in Decomposing Spinor Products?

[Slereah]

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 :

\frac{1}{2} \otimes \frac{1}{2} = 1 \oplus 0

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 :

\psi_a \chi_b = \frac{1}{2} (\psi_{[a} {\chi_{b}}_] + \psi_{\{a} {\chi_{b}}_\})

The antisymmetric part is the scalar part, being simply

\psi_{[a} {\chi_{b}}_] = \left( \!\!\begin{array}{cc}<br /> 0&amp;-1\\<br /> 1&amp;0<br /> \end{array}\! \right) (\psi^+ \chi^- - \psi^- \chi^+)

And the symmetric part is something of the form

\psi_{\{a} {\chi_{b}}_\} = \left( \!\!\begin{array}{cc}<br /> 2\psi_+ \chi_+&amp;\psi_+ \chi_- + \psi_- \chi_+\\<br /> \psi_+ \chi_- + \psi_- \chi_+&amp;2\psi_- \chi_-<br /> \end{array}\! \right)

Which looks like it contains all the right components, but then I try to compare it to the actual vector quantity :

V^{ab} = \varepsilon^{ca} V_c^b = \varepsilon^{ca} (\sigma^\mu)_c^b V_\mu = \left( \!\!\begin{array}{cc}<br /> - V_x - i V_y &amp;V_z\\<br /> V_z&amp;V_x - i V_y<br /> \end{array}\! \right)

There may be a sign wrong in here somewhere because of the spinor metric (\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?

 

[sgd37]

you can extract the vector by using

V^{ \mu } = (\varepsilon \sigma^{ \mu })^{ab} V_{ab}

and shouldn't one of your i's have a positive coefficient

 

[Slereah]

Extracting the vector isn't the hard part, but merely comparing the two gives me something like

\psi_+ \chi_+ \propto - V_x - i V_y

and

\psi_- \chi_- \propto V_x - i V_y

But the vector should be something like

(\psi_+ \chi_+, \psi_- \chi_-,\frac{1}{\sqrt{2}}(\psi_+ \chi_- + \psi_- \chi_+)

Plus maybe some factors.

One of the i's does have a + when using the transformation

\sigma^\mu V_\mu

But this disappears after index raising with the spinor metric epsilon (I also now realize that this should be index lowering and not raising, but the difference between the two is just a sign, which does not solve the problem)

 

[sgd37]

I calculated there to be a +i but i guess you are using different conventions . Anyway if you did have the +i you can reduce to the form you want

using 2 \psi_{+} \chi_{+} = -V_x - i V_y and 2 \psi_{-} \chi_{-} = V_x + i V_y you can get \psi_{+} \chi_{+} = - \psi_{-} \chi_{-}

from which you can write 2 \psi_{+} \chi_{+} = \psi_{+} \chi_{+} - \psi_{-} \chi_{-} which is in the form you want up to some minus signs