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

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

And the symmetric part is something of the form

$$\psi_{\{a} {\chi_{b}}_\} = \left( \!\!\begin{array}{cc} 2\psi_+ \chi_+&\psi_+ \chi_- + \psi_- \chi_+\\ \psi_+ \chi_- + \psi_- \chi_+&2\psi_- \chi_- \end{array}\! \right)$$

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

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

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?

 

[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

 

[paranormal]

I would first like to commend you for your efforts in trying to derive the decomposition of spinor products using tensor notation. This is a complex and challenging topic, and it is clear that you have put a lot of thought and effort into it.

Now, to address your specific question, the problem may stem from the fact that the decomposition of spinor products is not a simple vector quantity. In fact, it is a tensor quantity known as the spinor metric, which is a rank-2 tensor that acts on spinors to produce vectors.

The symmetric part that you have derived is indeed the spinor metric, and it contains all the necessary components for constructing vectors. However, as you have noticed, it may not seem to lend itself easily to being put in vector form. This is because the spinor metric is a tensor quantity, and it cannot be directly represented as a vector.

To understand this better, it may be helpful to think about how tensors behave under coordinate transformations. While vectors and spinors transform in a simple way under coordinate transformations, tensors can exhibit more complex transformation rules. This is why the symmetric part of the spinor product, which is a tensor, may not seem to easily correspond to a vector quantity.

In summary, the problem may not necessarily stem from any mistakes in your calculations, but rather from the fact that the decomposition of spinor products is a tensor quantity, and it cannot be directly represented as a vector. I would suggest exploring the transformation rules of tensors and their relation to spinors to gain a deeper understanding of this topic. Keep up the good work!