  The concern of this article is a semiclassical Weyl calculus on an infinite
dimensional Hilbert space $H$. If $(i, H, B)$ is a Wiener triplet associated to
$H$, the quantum state space will be the space of $L^2$ functions on $B$ with
respect to a Gaussian measure with $h/2$ variance, where $h$ is the
semiclassical parameter. We prove the boundedness of our pseudodifferential
operators (PDO) in the spirit of Calder\'on-Vaillancourt with an explicit
bound, a Beals type characterization, and metaplectic covariance. An
application to a model of quantum electrodynamics (QED) is added in the last
section, for fixed spin $1/2$ particles interacting with the quantized
electromagnetic field (photons). We prove that some observable time evolutions,
the spin evolutions, the magnetic and electric evolutions when subtracting
their free evolutions, are PDO in our class.
