mathematical physics - Tensor Operators

Motivation.

I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. I then skimmed the Wikipedia page on tensor operators, and felt similarly dissatisfied. Here's why

In these discussions, one essentially defines an indexed set of operators $T_{i_1\cdots i_k}$ to be a "cartesian" tensor operator of rank $k$ provided

$$U(R)\, T_{i_1\cdots i_k}\, U^\dagger(R) = R_{i_1}^{\phantom{i_1}j_1}\cdots R_{i_1}^{\phantom{i_1}j_1}T_{j_1\cdots j_k}$$ for each rotation $R\in\mathrm{SO}(3)$ where $U$ is some unitary representation of $\mathrm{SO}(3)$ acting on a Hilbert space (usually that of some physical system whose behavior under rotations we with to study). Similarly one defines a "spherical" tensor operator of rank $n$ as an indexed set of operators $T^{(n)}_{q}$ with $-n

Based on these standard definitions, I would think that one could define something less "coordinate-dependent" and extended to representations of any group, not just $\mathrm{SO}(3)$, as follows.

Candidate Definition. Let a group $G$ be given. Let $U$ be a unitary representation of $G$ on a Hilbert space $\mathcal H$, and let $\rho$ be a representation of $G$ on a finite-dimensional, real or complex vector space $V$. A $k$-multilinear, linear operator-valued function $T:V^k\to \mathrm{Lin}(\mathcal H)$ is called a tensor operator relative to the pair of representations $U$ and $\rho$ provided

$$\begin{align} U(g) T(v_1, \dots, v_k) U(g)^\dagger = T(\rho(g)v_1, \dots, \rho(g)v_k) \end{align}$$ for all $g\in G$ and for all $v_1, \dots, v_k\in V$.

Notice that if a basis $u_1, \dots, u_N$ for $V$ is given, and if we define the components $T_{i_1,\dots i_k}$ of $T$ in this basis by

$$\begin{align} T_{i_1 \dots i_k} = T(u_{i_1}, \dots, u_{i_k}) \end{align}$$ and if $\rho(g)_i^{\phantom ij}$ denotes the matrix representation of $\rho(g)$ in this basis, then by using multilinearity the defining property of a tensor operator can be written as follows $$\begin{align} U(g) T_{i_1\cdots i_k} U^\dagger(g) = \rho(g)_{i_1}^{\phantom {i_1}j_1}\cdots \rho(g)_{i_k}^{\phantom {i_k}j_k} T_{j_1\cdots j_k} \end{align}$$ So this definition immediately reproduces the cartesian tensor definition above if we take, $V =\mathbb R^3$, $G=\mathrm{SO}(3)$, and $\rho(R) = R$, and similarly for the spherical tensor definition if we take $V=\mathbb C^{2n+1}$, $G=\mathrm{SO}(3)$, $\rho = D^{(n)}$ and $k=1$.

Question.

Is the sort of object I just defined the "proper" formalization/generalization of the notion of tensor operators used in physics; it seems to contain the notion of tensor operator used in the physics literature? Is there any literature on the sort of object I define here? I would think that the answer would be yes since this sort of thing seems to me like a natural generalization a mathematically-minded physicist might like to study.

Answer

OP's candidate definition is a direct transcription of the tensor operator notion used in physics (and e.g. in Sakurai section 3.10) into a manifestly coordinate-independent mathematical construction. Tensor operators are e.g. used in the Wigner-Eckart theorem.

In this answer we suggest the following slight generalization of OP's candidate definition. Let the following five items be given:

1. 
   

   Let $G$ be a group.

   
   


2. 
   

   Let $H$ be a complex Hilbert space.

   
   


3. 
   
   

   Let $\rho: G \to GL(V,\mathbb{F})$ be a group representation.

   
   


4. 
   

   Let $R:G \to B(H)$ be a group representation.

   
   


5. 
   

   Let $T:V\to L(H;H)$ be a linear map.

   
   

Definition. Let us call $T$ for a $G$-equivariant map if

$$\forall g\in G, v\in V :\quad T(\rho(g)v)~=~ {\rm Ad}(R(g))T(v)~:=~R(g)\circ T(v)\circ R(g)^{-1}. \tag{*}$$

OP's candidate definition may be viewed as a special case of definition (*). For instance, if $\rho_0: G \to GL(V_0,\mathbb{F})$ is a group representation, then one may let $\rho: G \to GL(V,\mathbb{F})$ in point 3 be the tensor product representation $\rho=\rho_0^{\otimes m}$ with vector space

$$V~=~V_0^{\otimes m}~=~\underbrace{V_0\otimes \ldots \otimes V_0}_{m \text{ factors}}.$$