I am trying to understand the matrices and vectors presented in this section
https://en.wikipedia.org/wiki/Spin_(physics)#Spin_projection_quantum_number_and_multiplicity
I am looking for a reference where these objects are defined in a sufficiently precise manner that I can derive the matrices and vectors from their definition. All I could find were long and quite vague descriptions of physical backgrounds which ask me to understand things intuitively but lack necessary precision. For example, the Wikipedia article talks of operators but does not provide their domain of definition. Also the $|\alpha, \beta >$ notation is used without ever defining these notations and the vector spaces behind them in a precise manner.
I also find abstract mathematics as in "are elements of a unitary representation of $SU(2)$". I know what a unitary representation is but this does not provide me with a sufficiently clear definition from which I can derive the matrices in the article.
Ideally, I am looking for a starting point which defines these objects in a mathematically precise way and from which I can dash off calculating these matrices.
Added for clarification:
I obviously get the task of interpreting this wrong but I do not know where exactly.
In order that a commutator relation $[\sigma_x, \sigma_y] = 2i\sigma_z$ makes sense,
I need to know the space where these operators live. So let's pick a space.
In the spin 1/2 case I can get 2 different experimental results.
Thus, I am using projective space ${\mathbb P}({\mathbb C}^2)$
for the particle state.
I consider Hermitean operators of the type
${\mathbb C}^2 \to {\mathbb C}^2$. The four Hermitean
operators $\sigma_x, \sigma_y, \sigma_z, \sigma_0$ form a (real) basis of the
4-dimensional space of Hermitean operators. If I only consider trace zero operators this breaks
down to $\sigma_x, \sigma_y, \sigma_z$. I can use a normed real vector $\vec{a} = (a_x, a_y, a_z)^t$
to define an observable $a_x\sigma_x + a_y\sigma_y + a_z \sigma_z$. Measuring the
observable provides two possible results, spin-up or spin-down, which I interpret as
measuring the spin in direction $\vec{a}$.
In the spin 1 case I can get 3 different experimental results. Thus, I am using
projective space ${\mathbb P} ({\mathbb C}^3)$ for the particle state.
I consider Hermitean operators
${\mathbb C}^3 \to {\mathbb C}^3$. The space of Hermitean operators of this
signature has dimension 9, reducing them to trace zero leaves 8 dimensions.
I thus expect a basis consisting of 8 Hermitian operators.
How should I now arrive at only three operators $\sigma_x, \sigma_y, \sigma_z$
since I need 8?
How should they obtain an interpretation in 3 dimensional real space? If
the characteristic thing I need is the commutator relation, I can satisfy that
with the following embeddings as well:
$
\sigma_x =
\begin{pmatrix}
0 &1 &0 \\
1 &0 &0 \\
0 &0 &0 \\
\end{pmatrix}$,
$\sigma_y =
\begin{pmatrix}
0 &-i &0 \\
i &0 &0 \\
0 &0 &0 \\
\end{pmatrix}$ and
$
\sigma_z =
\begin{pmatrix}
1 &0 &0 \\
0 &-1 &0 \\
0 &0 &0 \\
\end{pmatrix}
$
Further clarification:
What I am looking for as well is a complete definition of what spin is. In my mind this needs a definition of the domains as in $A\colon H \to H$ (and what is $H$). If a starting point is "spin obeys commutation relations such-and-such" then I expect to find an existence and uniqueness theorem somewhere. Currently I fail in these attempts…
Amendment:
The article mentions operators $S_x$, $S_y$, $S_z$ and calls them
spin operators.
They seem to make
sense on every $n$-dimensional complex vector space. The article lists them for $n=2, 3, 4, 6$.
In the case $n=2$, three operators $S_x, S_y, S_z$ are given on ${\mathbb C}^2$.
Probably they are interesting because we want to study an observable
$a_x \cdot S_x + a_y \cdot S_y + a_z \cdot S_z$ which we can
compose as real linear combinations of these operators.
That we chose these operators as Pauli matrices is a matter of convention and convenience.
In the cases $n=3, 4, 6$ the space of observables is much larger.
However, we still only consider three operators $S_x, S_y, S_z$.
Probably we again are interested in studying the observables of the form
$a_x \cdot S_x + a_y \cdot S_y + a_z \cdot S_z$.
The particular choice of the $S_x, S_y, S_z$ probably again is a matter of convention
and convenience, and they simply are obtained if one follows a particular Kronecker
or tensor product type of construction. Fine.
However there must be a particular condition by which we single-out these operators.
We are not considering all traceless Hermitean operators (as we did in the $n=2$ case)
but only a very specific subspace of traceless Hermitean operators.
One part of the conditions, so it seems, is that the operators must have full rank,
but this is not enough. A further condition might be connected with the
commutator relation. However: The structure constants of a Lie-algebra are basis dependent
and the specific choice of the $S_x, S_y, S_z$ seems a bit arbitrary – they just generate
that class of operators. So I do not expect this to translate 1:1. Moreover, I am interested in a base independent condition.
It is this condition and its physical significance which I am looking for.
Comment added only to the suggestion to use representations of SU(2) as operators: It has been suggested to define the operators as values of group elements under a representation of $SU(2)$ in a suitable $GL ({\mathbb C}^n)$. I see several problems here. 1) This definition would be dependent on the specific choice of a representation. When $\omega\colon SU(2) \to GL(V)$ is a representation also $A\cdot \omega \cdot A^{-1}$ is a representation and so we get way too many operators again (certainly more than the 3 degrees of freedom in the SU(2)). So we would again need some way of connecting this to the observable. 2) The values of a representation are unitary operators, how do we ensure Hermeticity? 3) The suggestion to start with generators of the Lie algebra then would depend on the specific choice of the generating elements, which probably also is not unique.
Best Answer
I am not really sure about the scope of this question and the type of answer OP is looking for but computing those higher spin matrices/representations successively is relatively straight forward:
Lets assume for now that we have understood the spin $\frac{1}{2}$ case: so we know that the spin $\frac{1}{2}$ operators are $$ \begin{align} S_x = \frac{\hbar}{2} \sigma_x= \frac{\hbar}{2}\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}, \quad S_y = \frac{\hbar}{2} \sigma_y= \frac{\hbar}{2}\begin{pmatrix} 0 & -\mathrm{i}\\ \mathrm{i} & 0 \end{pmatrix}, \quad S_z = \frac{\hbar}{2} \sigma_z= \frac{\hbar}{2}\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}. \end{align} $$ We can verify that those sattisfy the defining commutator relation $\left[ S_j, S_k\right] = i \hbar \varepsilon_{jkl} S_l$ with $i,j,k\in\{x,y,z\}$. Furthermore we can compute the Casimir operator $S^2$: $$ S^2=S_x^2+S_y^2+S_z^2 =\hbar^2\begin{pmatrix} \frac{3}{4} & 0\\ 0 & \frac{3}{4} \end{pmatrix}. $$ The eigenvalues of $S^2$ characterize the present representation and are $\hbar^2\frac{3}{4}=\hbar^2s(s+1)$ with $s=\frac{1}{2}$. A spin state can be completely characterized by specifying $s$ and one additional spin projection which is conventionally chosen to be along the $z$-direction. $S_z$ has two Eigenvalues $m_\frac{1}{2}=\pm\frac{\hbar}{2}$. The simultaneous Eigenvectors of $S^2$ and $S_z$: $(1,0)^T$ and $(0,1)^T$ form the basis of spin states with well defined quantum numbers $s$ and $m_s$. Eigenvectors of the other spin operators $S_x$ and $S_y$ can be computed and expressed in this basis.
The quoted wikipedia article states that by taking Kronecker products of the spin $\frac{1}{2}$ representation with itself repeatedly, one may construct all higher irreducible representations. What this means in practice is that we can use the Kronecker products to couple spins. The important point here is that when we couple spins the naive basis for states which we get from the direct product of spin states is not an Eigenbasis of the coupled spin operators. Let me illustrate this with an example. We begin with coupling two spin $\frac{1}{2}$: $$ \tilde{S_i}=S_i\otimes \mathrm{Id}_2+\mathrm{Id}_2\otimes S_i $$ which results in $$ \tilde{S}_x=\frac{\hbar}{2} \begin{pmatrix}0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{pmatrix},\quad \tilde{S}_y=\frac{\hbar}{2\mathrm{i}} \begin{pmatrix} 0 & 1 & 1 & 0 \\ -1 & 0 & 0 & 1 \\ -1 & 0 & 0 & 1 \\ 0 & -1 & -1 & 0 \end{pmatrix},\quad \tilde{S}_z=\hbar\begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}. $$ $\tilde{S}_z$ has four eigenvalues: $+1$, $0$, $0$ and $-1$. To fully characterize the coupled states we need to compute the Casimir operator $\tilde{S}^2$: $$ \tilde{S}^2 = \tilde{S}_x^2+\tilde{S}_y^2+\tilde{S}_z^2=\hbar^2 \begin{pmatrix}2 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{pmatrix}. $$ Using the Eigenvectors of $\tilde{S}^2$ we can construct a matrix $U$ which diagonalizes $\tilde{S}^2$: $$ U= \begin{pmatrix} 0 & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 1 & 0 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \Rightarrow U \tilde{S}^2 U^\dagger=\hbar^2 \begin{pmatrix}0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{pmatrix} $$ We can now classify the coupled Eigenstates of $\tilde{S}^2$ by their eigenvalues: we have three "triplet" states with $s=1$ ($s(s+1)=2$) and one "singlet" state with $s=0$. A representation is classified by the eigenvalue of $\tilde{S}^2$. Using $U$ we can split $\tilde{S}_i$ into block diagonal matrices where each block corresponds to one representation with distinct $s$: $$ U\tilde{S}_xU^\dagger=\hbar \left( \begin{array}{c|ccc} 0 & 0 & 0 & 0 \\ \hline 0 & 0 & \frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0 & 0 & \frac{1}{\sqrt{2}} & 0 \\ \end{array} \right),\quad U\tilde{S}_yU^\dagger=\hbar \left( \begin{array}{c|ccc} 0 & 0 & 0 & 0 \\\hline 0 & 0 & -\frac{i}{\sqrt{2}} & 0 \\ 0 & \frac{i}{\sqrt{2}} & 0 & -\frac{i}{\sqrt{2}} \\ 0 & 0 & \frac{i}{\sqrt{2}} & 0 \\ \end{array} \right),\quad U\tilde{S}_zU^\dagger=\hbar\left( \begin{array}{c|ccc} 0 & 0 & 0 & 0 \\\hline 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ \end{array} \right). $$ The $s=1$ blocks of $U\tilde{S}_i U^\dagger$ are the operators of the $s=1$ representation. Due to properties of the Kronecker product and the block diagonal form of $U\tilde{S}_i U^\dagger$ it is clear that the blocks satisfy the defining commutator relation separately thus we succeeded in constructing a $s=1$ representation by coupling two $s=\frac{1}{2}$ spins/representations and computing the Eigenvectors of the Casimir operator of the coupled spins. Higher representations can be computed by coupling more spins/representations: e.g. coupling $s=1$ with $s=\frac{1}{2}$ will yield the $s=\frac{3}{2}$ quartet (and in this case one additional $s=\frac{1}{2}$ doublet).