Weyl pair, for any value of $p \in [1, \infty]$. The pair $(X,D)$ will be referred to as the standard pair, and their Weyl calculus is the standard Weyl calculus (see for example [@HormanderWeylCalc]).

The body of the work of \[vN,P\] follows from the next theorem, displaying the semigroup generated by $\frac{1}{2}(Q^2+P^2-d)$ in the joint functional calculus, which turns out to be the correct expression for the Ornstein-Uhlenbeck operator in the case they consider. This formula has been known by physicists for the standard pair $(Q,P)=(X,D)$ (in which case $\frac{1}{2}(X^2+D^2-d)$ is the harmonic oscillator operator minus $d/2$ times the identity), and relies heavily on the simple algebraic nature of $\frac{1}{2}(Q^2+P^2)$ in terms of $(Q,P)$.

\[TheoremSGEasy\] For $t > 0$, let $a_t:\mathbb{R}^{2d} \to \mathbb{C}$ be the function $(x, \xi) \mapsto \left(\frac{2}{1+e^{-t}}\right)^d \exp(- s_t (x^2 + \xi^2))$, where $s_t = \frac{1-e^{-t}}{1+e^{-t}}$. Then $\{a_t(Q,P)\}_{t > 0}$ is the semigroup generated by $\frac{1}{2}(Q^2+P^2 - d)$, on its natural domain inherited from the domains of $Q$ and $P$.

In this paper we wish to develop a different side to the Weyl calculus, in which we do not require the bounded group assumption of Definition \[defnWeylPair\]. The main reason for this is because the key example 