Recently I've read a little about the description of particles with spin in the book Quantum Mechanics by Cohen-Tannoudji. Although I yet didn't fully study the subject, I've read one interesting part in which the author considers the description of such particle being given by the state space $\mathcal{E}=\mathcal{E}_{S}\otimes \mathcal{E}_{\mathbf{r}}$ being $\mathcal{E}_S$ the spin state space generated by $\{|+\rangle, |-\rangle\}$ and $\mathcal{E}_{\mathbf{r}}$ the usual state space of a particle without spin.
In that setting we can consider the representation $\{|-\rangle\otimes| \mathbf{r}\rangle, |+\rangle\otimes |\mathbf{r}\rangle\}$. With this, denoting $|\mathbf{r}, -\rangle = |-\rangle\otimes| \mathbf{r}\rangle$ and $|\mathbf{r}, +\rangle =|+\rangle\otimes |\mathbf{r}\rangle$ we find that if $|\varphi\rangle \in \mathcal{E}$ we can express it as
$$|\varphi\rangle = \int_{\mathbb{R}^3}(\varphi_-(\mathbf{r})|\mathbf{r}, -\rangle + \varphi_+(\mathbf{r})|\mathbf{r},+\rangle )d^3\mathbf{r}.$$
The author then states the following:
In order to characterize completely the state of an electron, it is therefore necessary to specify two functions of the space variables $x,y,z$:
$$\varphi_{+}(\mathbf{r})=\langle \mathbf{r}, + | \varphi \rangle$$ $$\varphi_{-}(\mathbf{r})=\langle \mathbf{r}, - | \varphi \rangle$$
These two functions are often written in the form of a two-component spinor, which we shall write $[\varphi](\mathbf{r})$:
$$[\varphi](\mathbf{r}) = \begin{pmatrix}\varphi_{+}(\mathbf{r}) \\ \varphi_{-}(\mathbf{r})\end{pmatrix}$$
The main point is that it seems here that a spinor is just an element of $L^2(\mathbb{R}^3)\times L^2(\mathbb{R}^3)$.
Now, some time ago, before reading this I tried to learn what a spinor is, and looking on the internet I've found the Wikipedia page on which many things are said. I couldn't locate at the time any direct definition on the page.
On the page many things are said, considering Clifford algebras, spin groups and many other things. It highly depends on Clifford Algebras (which although I know the definition I haven't had the time yet to fully study) and IMHO it doesn't relate immediately to the idea of spin found in quantum mechanics.
On the other hand the idea of spinor introduced by Cohen is a thousand times clearer, simpler and much more connected to the idea of spin. I believe the relationship to rotations could be made even clearer seeing as the spin operators are generators of rotations in $\mathbb{R}^3$ and that the elements $|+\rangle$ and $|-\rangle$ are its eigenvectors.
My question here is: what is the relation between these two points of view about spinors? What is the relation between Cohen's idea of just picking those functions together and the quite complex algebraic construction? How can we connect those two viewpoints?
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