  Path integral representations for generalized Schr\"odinger operators
obtained under a class of Bernstein functions of the Laplacian are established.
The one-to-one correspondence of Bernstein functions with L\'evy subordinators
is used, thereby the role of Brownian motion entering the standard Feynman-Kac
formula is taken here by subordinated Brownian motion. As specific examples,
fractional and relativistic Schr\"odinger operators with magnetic field and
spin are covered. Results on self-adjointness of these operators are obtained
under conditions allowing for singular magnetic fields and singular external
potentials as well as arbitrary integer and half-integer spin values. This
approach also allows to propose a notion of generalized Kato class for which
hypercontractivity of the associated generalized Schr\"odinger semigroup is
shown. As a consequence, diamagnetic and energy comparison inequalities are
also derived.
