  We prove new criteria for essential self-adjointness of pseudodifferential
operators which do not involve ellipticity type assumptions. For example, we
show that self-adjointness holds in case that the symbol is $C^{2d+3}$ with
derivatives of order two and higher being uniformly bounded. These results also
apply to hermitian operator-valued symbols on infinite-dimensional Hilbert
spaces which are important to applications in physics. Our method relies on a
phase space differential calculus for quadratic forms on $L^2(\mathbb{R}^d)$,
Calder\'on-Vaillancourt type theorems and a recent self-adjointness result for
Toeplitz operators on the Segal-Bargmann space.
