an algebraically rich setting. The approach used by van Neerven and Portal in [@vNPWeylGaussian] is well-adapted to studying the standard Ornstein-Uhlenbeck semigroup, thanks to an exact formula for the semigroup in the $(Q,P)$-calculus.

To generalise the theory developed in [@vNPWeylGaussian], van Neerven and Portal consider in [@vNPWeylPairs] pairs of $d$-tuples of operators on a Banach space, which generate uniformly bounded groups satisfying certain commutation relations, which they refer to as a Weyl pair. This path of generalisation has been fruitful, and proves that the sum of squares of all $2d$ operators comprising a Weyl pair with a bounded Weyl calculus has a bounded Hörmander functional calculus (under some mild conditions).

However, in the Gaussian case, the natural position and momentum operators generate uniformly bounded groups on $L^p$ if and only if $p=2$. In fact, $\exp(i\xi P)$ is bounded on $L^p$ for $p\neq 2$ if and only if $\xi = 0$, in which case $\exp(i 0 P)$ is the identity. Thus the theory developed in [@vNPWeylPairs] cannot possibly be applied in Gaussian situations. In this paper we consider such cases.

We will work on measure spaces of the form $(\mathbb{R}^d, \mathcal{B}, \mu)$, where $\mathcal{B}$ is the Borel $\sigma$-algebra and $d\mu = 