  We consider operators acting on a UMD Banach lattice $X$ that have the same
algebraic structure as the position and momentum operators associated with the
harmonic oscillator $-\frac12\Delta + \frac12|x|^{2} $ acting on
$L^{2}(\mathbb{R}^{d})$. More precisely, we consider abstract harmonic
oscillators of the form $\frac12 \sum _{j=1} ^{d}(A_{j}^{2}+B_{j}^{2})$ for
tuples of operators $A=(A_{j})_{j=1} ^{d}$ and $B=(B_{k})_{k=1} ^{d}$, where
$iA_j$ and $iB_k$ are assumed to generate $C_{0}$ groups and to satisfy the
canonical commutator relations. We prove functional calculus results for these
abstract harmonic oscillators that match classical H\"ormander spectral
multiplier estimates for the harmonic oscillator $-\frac12\Delta +
\frac12|x|^{2}$ on $L^{p}(\mathbb{R}^{d})$. This covers situations where the
underlying metric measure space is not doubling and the use of function spaces
that are not particularly well suited to extrapolation arguments. For instance,
as an application we treat the harmonic oscillator on mixed norm Bargmann-Fock
spaces. Our approach is based on a transference principle for the Schr\"odinger
representation of the Heisenberg group that allows us to reduce the problem to
the study of the twisted Laplacian on the Bochner spaces
$L^{2}(\mathbb{R}^{2d};X)$. This can be seen as a generalisation of the
Stone-von Neumann theorem to UMD lattices $X$ that are not Hilbert spaces.
