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James1238765
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- TL;DR Summary
- What are the derivation steps for the tensor product leading to Dirac spinors?
Could anyone help with some of the later parts of the derivation for Dirac spinors, please?
I understand that an arbitrary vector ##\vec v##
$$ \begin{bmatrix}
x \\
y \\
z
\end{bmatrix} $$
can be defined as an equivalent matrix V with the components
$$ \begin{bmatrix}
z & x - iy \\
x + iy & -z
\end{bmatrix} $$
Now let ##R^-## be defined as
$$ \begin{bmatrix}
e^{-i\frac{\theta}{2}} & 0 \\
0 & e^{i\frac{\theta}{2}}
\end{bmatrix} $$
and let ##R^+## be defined as
$$ \begin{bmatrix}
e^{i\frac{\theta}{2}} & 0 \\
0 & e^{-i\frac{\theta}{2}}
\end{bmatrix} $$
Next, consider the matrix product ##R^-VR^+##:
$$
\begin{bmatrix}
e^{-i\frac{\theta}{2}} & 0 \\
0 & e^{i\frac{\theta}{2}}
\end{bmatrix}
\begin{bmatrix}
z & x - iy \\
x + iy & -z
\end{bmatrix}
\begin{bmatrix}
e^{i\frac{\theta}{2}} & 0 \\
0 & e^{-i\frac{\theta}{2}}
\end{bmatrix}
$$
which simplifies component-wise to
$$ \begin{bmatrix}
z & x\cos\theta - y\sin\theta - i(x\sin\theta + y\cos\theta) \\
x\cos\theta - y\sin\theta + i(x\sin\theta + y\cos\theta) & -z
\end{bmatrix} $$
Substituting ##\theta## ("rotating") by 180 ##^\circ## , we obtain as per normal
$$ \begin{bmatrix}
z & -(x-iy) \\
-(x+iy) & -z
\end{bmatrix} $$
and substituting ##\theta## by 360 ##^\circ##, we obtain as per normal the original state
$$ \begin{bmatrix}
z & x - iy \\
x + iy & -z
\end{bmatrix} $$
Now, the matrix V is said to be decomposable into 2 tensor products.
$$ \begin{bmatrix}
z & x - iy \\
x + iy & -z
\end{bmatrix} =
\begin{bmatrix}
\Psi_1 \\
\Psi_2
\end{bmatrix} \otimes
\begin{bmatrix}
-\Psi_2 & \Psi_1
\end{bmatrix} $$
I am lost at this step: why are the ##\Psi_1, \Psi_2, \Psi_3, \Psi_4## left symbolic? I have tried to put concrete variables into ##\Psi_1, \Psi_2, \Psi_3, \Psi_4## but could not find what the correct arrangements should be.
Could anyone write down the explicit variables what ##\Psi_1, \Psi_2, \Psi_3, \Psi_4## should be set to, please?
I understand that an arbitrary vector ##\vec v##
$$ \begin{bmatrix}
x \\
y \\
z
\end{bmatrix} $$
can be defined as an equivalent matrix V with the components
$$ \begin{bmatrix}
z & x - iy \\
x + iy & -z
\end{bmatrix} $$
Now let ##R^-## be defined as
$$ \begin{bmatrix}
e^{-i\frac{\theta}{2}} & 0 \\
0 & e^{i\frac{\theta}{2}}
\end{bmatrix} $$
and let ##R^+## be defined as
$$ \begin{bmatrix}
e^{i\frac{\theta}{2}} & 0 \\
0 & e^{-i\frac{\theta}{2}}
\end{bmatrix} $$
Next, consider the matrix product ##R^-VR^+##:
$$
\begin{bmatrix}
e^{-i\frac{\theta}{2}} & 0 \\
0 & e^{i\frac{\theta}{2}}
\end{bmatrix}
\begin{bmatrix}
z & x - iy \\
x + iy & -z
\end{bmatrix}
\begin{bmatrix}
e^{i\frac{\theta}{2}} & 0 \\
0 & e^{-i\frac{\theta}{2}}
\end{bmatrix}
$$
which simplifies component-wise to
$$ \begin{bmatrix}
z & x\cos\theta - y\sin\theta - i(x\sin\theta + y\cos\theta) \\
x\cos\theta - y\sin\theta + i(x\sin\theta + y\cos\theta) & -z
\end{bmatrix} $$
Substituting ##\theta## ("rotating") by 180 ##^\circ## , we obtain as per normal
$$ \begin{bmatrix}
z & -(x-iy) \\
-(x+iy) & -z
\end{bmatrix} $$
and substituting ##\theta## by 360 ##^\circ##, we obtain as per normal the original state
$$ \begin{bmatrix}
z & x - iy \\
x + iy & -z
\end{bmatrix} $$
Now, the matrix V is said to be decomposable into 2 tensor products.
$$ \begin{bmatrix}
z & x - iy \\
x + iy & -z
\end{bmatrix} =
\begin{bmatrix}
\Psi_1 \\
\Psi_2
\end{bmatrix} \otimes
\begin{bmatrix}
-\Psi_2 & \Psi_1
\end{bmatrix} $$
I am lost at this step: why are the ##\Psi_1, \Psi_2, \Psi_3, \Psi_4## left symbolic? I have tried to put concrete variables into ##\Psi_1, \Psi_2, \Psi_3, \Psi_4## but could not find what the correct arrangements should be.
Could anyone write down the explicit variables what ##\Psi_1, \Psi_2, \Psi_3, \Psi_4## should be set to, please?
Last edited: