The symbolic calculus of pseudodifferential operators virtually coincides
 with things physicists do in connection with both quantum mechanics and
gauge field theories.  The traditional pseudodifferential calculus is
 susceptible to two formal improvements -- one to treat Riemannian and
 vector-bundle derivatives in a coordinate-independent way, and one to make
Hermitian and symplectic symmetries manifest.  We review these and report
ongoing progress in doing both at once.