\mathbb{R}^d\}\cup\{ \lambda I; \lambda \in \mathbb{C}, |\lambda|=1\}$ into a (non-commutative) group, in fact a representation of the $(2d+1)$-dimensional Heisenberg group. From here, the Weyl calculus is defined as:

\[defnWeylCalc\] For a Weyl pair $(Q,P)$ on a Banach space, we define the bounded operator $a(Q,P)$ for $a \in \mathcal{S}(\mathbb{R}^{2d})$ via the formula $$a(Q,P) = \frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{2d}} \mathcal{F}a(x, \xi) \exp(i(xQ + \xi P)) dxd\xi,$$ where $\mathcal{F}$ denotes the Fourier transform, normalised such that equality holds above with $Q$ and $P$ replaced by elements of $\mathbb{R}^d$. The map $a \mapsto a(Q,P)$ is called the Weyl pseudodifferential calculus, or the joint functional calculus, for the Weyl pair $(Q,P)$.

That this definition makes sense follows from boundedness of the group $\exp(i(xQ + \xi P))$ and integrability of the Schwartz function $\mathcal{F}a$. As an example of a Weyl pair, consider the position and momentum operators $(X,D)$ on the Euclidean Lebesgue space $L^p(\lambda)$ ($p \in [1, \infty]$) given by $X_if(x) = x_if(x), P_if(x) = -i\frac{\partial}{\partial x_i}f(x)$ equipped with their natural domains. These generate the groups of phase shift and translation respectively, both of which are bounded. That they satisfy the integrated canonical commutation relations is a simple exercise, which once checked shows that $(X,D)$ on $L^p(\lambda)$ are a 