How to make notation like $Y_{l m_{l}}(\theta, \phi)\chi_{m_s}$ more rigorous as a tensor product?

Question

Sometimes in quantum mechanics we come across notation like $Y_{l m_{l}}(\theta, \phi)\chi_{sm_s}$ where $Y_{lm_l}$ is a spherical harmonic representing the spatial part of some particle wavefunction and $\chi_{sm_s}\in \mathcal{H}_S$ is a spinor or vector representing the spin of a particle. Alternatively, in classicaly electromagnetism we may have $Y_{l, m_{l}}(\theta, \phi)$ represents the spatial distribution of an electric/magnetic field and $\chi_{sm_s}$ represents the local vector part of the field.

There is a sense in which these two "objects" are angular momentum representations with indices $l$ and $s$ and it is known that it is possible to express them in a $j, m_j$ basis using $J= L + S$ and Clebsch-Gordan coefficients.*

I am comfortable with this in the case that we have two vectors like $|l, m_l\rangle \in \mathcal{H}_L$ and $|s, m_s\rangle \in \mathcal{H}_S$ and we are interested in describing bases of the tensor Hilbert space $\mathcal{H}_J = \mathcal{H}_L \otimes \mathcal{H}_S$. In that case I know from the theory of angular momentum addition that there are two related bases for $\mathcal{H}_J$. One is expressed as $|l, m_l\rangle \otimes |s, m_s\rangle$ and one is expressed as $|j, m_j;l, s\rangle$ and the two bases are related by Clebsch Gordan coefficients:

\begin{align} |j, m_j; l, s\rangle =& |l, m_l\rangle \otimes |s, m_s\rangle \langle l, m_l, s, m_s |j, m_j; l, s\rangle\\ =& |l, m_l\rangle \otimes |s, m_s\rangle C_{l, m_l, s, m_s}^{j, m} \end{align}

However, something feels like an abuse of notation (or at least a shortcut) when we say $Y_{lm_l}(\theta, \phi)\chi_{sm_s}$ is a tensor product of this sort. I feel like this is being a bit notationally pedantic, but I guess my issue is that I would grant that $Y_{lm_l}$ (as a function in $L_2(\mathbb{S}^2,\mathbb{C})$) is a vector in a Hilbert space, but it feels like $Y_{lm_l}(\theta, \phi)$ is a scalar because the spherical harmonic has been evaluated. It feels like a shortcut to replace the tensor product by simple scalar-vector multiplication.

What is a more "proper" way to write the tensor product in this case? I could see something like $Y_{lm_l}\otimes \chi_{sm_s}$ making sense but then how do we "evaluate it" at a certain point $(\theta, \phi)$? Would we write something like $(Y_{lm_l}\otimes \chi_{sm_s})(\theta, \phi)$ and understand that the output of this function evaluation is a vector in $\mathcal{H}_S$?

This questions is related to the question/answer/comments at Vector Spherical Harmonics and total angular momentum.

*Sometimes $|l, m_l\rangle \otimes |s, m_s\rangle$ is written in a shorter notation as $|l, m_l, s, m_s\rangle$. Perhaps this is a similar notation abbreviation as the one involving $Y_{lm_l}(\theta, \phi)$ and $\chi_{sm_s}$?

Answer 1

It's been stipulated that $\chi_{s, m_s} = |s, m_s\rangle\in\mathcal{H}_s$ with $\mathcal{H}_s$ a Hilbert space. Let's generalize and just let $|\chi\rangle \in \mathcal{H}_s$.

Now consider the space $L^2(\mathbb{S}^2, \mathbb{C})$ of square integrable complex functions on the 2-sphere. $L^2(\mathbb{S}^2, \mathbb{C})$ is a Hilbert space. Suppose $f\in L^2(\mathbb{S}^2, \mathbb{C})$. In Dirac notation we would write $f = |f\rangle$. The $L^2(\mathbb{S}^2, \mathbb{C})$ Hilbert space has a basis denoted by $|\theta, \phi\rangle$. which decomposes any $|f\rangle$ as

$$ \langle \theta, \phi|f\rangle = f(\theta, \phi) $$

Consider the spherical harmonics $Y_{l, m_l}(\theta, \phi)$ with $l$ fixed and $-l \le m_l \le l$. The set $$ \{Y_{l, m_l} | -l \le m_l \le l\} = \{ | Y_{l, m_l}\rangle | -l \le m_l \le l\} $$ constites a vector subspace of $L^2(\mathbb{S}^2, \mathbb{C})$ which we denote by $\mathcal{H}_l$. Because $|Y_{l, m_l}\rangle \in L^2(\mathbb{S}^2, \mathbb{C})$ we have $$ \langle \theta, \phi | Y_{l, m_l} \rangle = Y_{l, m_l}(\theta, \phi) $$

We can take the tensor product space $\mathcal{H}_j = \mathcal{H}_l \otimes \mathcal{H}_s$. Let $|Y\rangle \in \mathcal{H}_l$. Then an element of $\mathcal{H}_j = \mathcal{H}_l\otimes \mathcal{H}_s$ could be written as

$$ |Y\rangle \otimes |\chi\rangle $$

One natural thing to do would be do calculate the components of this tensor product element in the original basis. We would do this by

$$ \left(\langle Y_{l, m}|\otimes \langle s, m_s|\right)\left(|Y\rangle \otimes |\chi\rangle \right) = \langle Y_{l, m}|Y\rangle \langle s, m_s|\chi\rangle \in \mathbb{C} $$

This tells us how much overlap the vector has with the $|Y_{l, m}\rangle$ spherical harmonic and $|s, m_s\rangle$ spin state. Note that $\langle Y_{l, m}| \in \mathcal{H}_l^*$ and $\langle s, m_s| \in \mathcal{H}_s^*$ and I guess it follows that $\langle Y_{l, m}|\otimes \langle s, m_s| \in \mathcal{H}_j^*$ so this whole "component finding" formalism is pretty natural.

Another way we could have done this is by using $\langle \theta, \phi|$ instead of $\langle Y_{l, m}|$. If we had had $|Y\rangle = |Y_{l, m}\rangle$ then the result would have been

$$ Y_{l, m}(\theta, \phi)\langle s, m_s|\chi\rangle $$

But what I asked for is a little different. I asked for sense-making of the expression $$ Y_{l, m}(\theta, \phi)|\chi\rangle $$

To get this the operator we need to act on $|Y_{l, m}\rangle \otimes |\chi\rangle$ is

$$ \langle \theta, \phi| \otimes \mathbb{I}_s $$

This is a little odd because $\langle \theta, \phi \in \mathcal{H}_l^*$ but $\mathbb{I}_s \in \mathcal{L}(\mathcal{H}_s)$ (the linear operators from $\mathcal{H}_s\rightarrow \mathcal{H}_s$), not $\mathcal{H}_s^*$ (the linear operators from $\mathcal{H}_s \rightarrow \mathbb{C}$). So this is a little bit of an abnormal construction.

Nonetheless, I think it's valid and does give a rigorous definition of what was requested. @JasonFunderberker points out in the comments that these manipulations give rise to an isomorphism between the spaces of the different objecs I have been manipulating here. This isomorphism likely justifies the abuse of notation which identifies $Y_{l, m}(\theta, \phi)$ with $|Y_{l, m}\rangle$.