I am interested in a characterization of the creation and annihilation operators that is in some sense invariant under $O(n)$ rotations of $\mathbb{R}^n$:

## Background

The Harmonic Oscillator on $\mathbb{R}^n$ is the differential operator

$$ H := \sum_{k=1}^n \left[x_k^2-\frac{\partial^2}{\partial x_k^2}\right] = |x|^2 + \nabla.$$

It is not hard to see that the $L^2(\mathbb{R}^n)$ eigenvalues are exactly $\{n,n+2,n+4,\dots\}$. Furthermore, the annihilation operator is an operator on Schwartz functions on $\mathbb{R}^n$ $$C_k := \frac {1}{\sqrt {2}}\left( x_k + \frac{\partial}{\partial x_k}\right)$$ and the creation operator as its adjoint (with the $L^2$ inner product) $$ C_k^\dagger = \frac {1}{\sqrt {2}}\left( x_k -\frac{\partial}{\partial x_k}\right).$$ If we let $V_{n+2m}$ be the set of eigenfunctions with eigenvalue $n+2m$, we can show that $$C_k^\dagger V_{n+2m} \subset V_{n+2(m+1)}$$ $$C_k V_{n+2m} \subset V_{n+2(m-1)}$$ (where we define $V_r=0$ if $r$ is not an eigenvalue) and $V_n$ is spanned by $e^{-|x|^2/2}$. It turns out that $V_{n+2m}$ is isomorphic to the space of degree $m$ homogeneous polynomials in $n$ variables, which I'll denote $\mathcal{P}^n_m$, by the isomorphism $p \mapsto p(C^\dagger)$ i.e. $x_1x_2 \mapsto C_1^\dagger C_2^\dagger$, etc. All of this and more can be found here starting on page 86 (with some slightly different notation than I've used here).

## Motivation

One of the problems with this whole business is that even though $H$ and thus $V_{n+2m}$ are invariant under rotations of $\mathbb{R}^n$, the $C_k$ and $C_k^\dagger$ are not. We made an arbitrary choice of coordinates when we defined them. This leads to a non-cannonical choice of basis for the eigenspaces, and has been giving me problems in my research. My question is thus:

## Question

Even though there is no cannonical choice of basis for $V_{n+2m}$, is there some characterization of the creation and anihilation operators that is invariant under $O(n)$ rotations of $\mathbb{R}^n$. That is, what is an $O(n)$-invariant characterization of the space of operators $$ \text{span}_\mathbb{R}\{C_1^\dagger,C_2^\dagger,\dots, C^\dagger_n\}$$

If this is not possible, then as an alternative answer, I am interested in insight into how rotations and the operators interact.