fa.functional analysis - Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way

Let me try to answer this question. I apologise if my notation is slightly different, since I will work in some more generality, since the equivariance properties of the creation and annihilation operators are actually more transparent, I believe, relative to the the general linear group instead of the orthogonal group. Also the fact that this is the harmonic oscillator is a red herring. In the case of the harmonic oscillator, we have to introduce more structure, reducing the group of symmetries.
It is also in this case, where the grading to be discussed below coincides (up to a choice of scale) with the energy of the system.

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Let $E$ be an $n$-dimensional real vector space and let $E^*$ denote its dual. Then on $H = E oplus E^* oplus mathbb{R}K$ one defines a Lie algebra by the following relations:
$$ [x,y]= 0 = [alpha,beta] qquad [x,alpha] = alpha(x) K = - [alpha,x] qquad [K,*]=0$$
for all $x,y in E$ and $alpha,beta in E^*$. This is called the Heisenberg Lie algebra of $E$, denoted $mathfrak{h}$.

The automorphism group of $mathfrak{h}$ is the group $operatorname{Sp}(Eoplus E^*)$ of linear transformations of $Eoplus E^*$ which preserve the symplectic inner product defined by the dual pairing:
$$omegaleft( (x,alpha), (y,beta) right) = -alpha(y) + beta(x).$$

Let $mathfrak{a} < mathfrak{h}$ denote the abelian subalgebra with underlying vector space $E oplus mathbb{R}K$. One can induce a $mathfrak{h}$-module from an irreducible (one-dimensional) $mathfrak{a}$-module as follows. Let $W_k$ denote the one-dimensional vector space on which $E$ acts trivially and $K$ acts by multiplication with a constant $k$. Then letting $U$ be the universal enveloping algebra functor, we have that
$$ V_k = Umathfrak{h} otimes_{Umathfrak{a}} W_k$$
is an $mathfrak{h}$-module. The Poincaré-Birkhoff-Witt theorem implies that $V_k$ is isomorphic as a vector space to the symmetric algebra of $E^*$, which we may (as we are over $mathbb{R}$) identify with polynomial functions on $E$.

The subgroup of $operatorname{Sp}(Eoplus E^*)$ which acts on $V_k$ is the general linear group $operatorname{GL}(E)$ and hence $V_k$ becomes a $operatorname{GL}(E)$-module. In fact, $V_k$ is graded (by the grading in the symmetric algebra of $E^*$ or equivalently the degree of the polynomial):
$$V_k = bigoplus_{pgeq 0} V_k^{(p)}$$
and each $V_k^{(p)}$ is a finite-dimensional $operatorname{GL}(E)$-module isomorphic to $operatorname{Sym}^p E^*$.

Every vector $x in E$ defines an annhilation operator: $A(x): V_k^{(p)} to V_k^{(p-1)}$ via the contraction map
$$E otimes operatorname{Sym}^p E^* to operatorname{Sym}^{p-1} E^*$$
whereas every $alpha in E^*$ defines a creation operator: $C(alpha): V_k^{(p)} to V_k^{(p+1)}$ by the natural symmetrization map
$$E^* otimes operatorname{Sym}^p E^* to operatorname{Sym}^{p+1} E^*.$$

Both of these maps are $operatorname{GL}(E)$-equivariant, and this is perhaps the most invariant statement I can think of concerning the creation and annihilation operators.