  Field theories for the one-body density of an active fluid, such as the
paradigmatic active model B+, are simple yet very powerful tools for describing
phenomena such as motility-induced phase separation. No comparable theory has
been derived yet for the underdamped case. In this work, we introduce active
model I+, an extension of active model B+ to particles with inertia. The
governing equations of active model I+ are systematically derived from the
microscopic Langevin equations. We show that, for underdamped active particles,
thermodynamic and mechanical definitions of the velocity field no longer
coincide and that the density-dependent swimming speed plays the role of an
effective viscosity. Moreover, active model I+ contains the Schr\"odinger
equation in Madelung form as a limiting case, allowing to find analoga of the
quantum-mechanical tunnel effect and of fuzzy dark matter in the active fluid.
We investigate the active tunnel effect analytically and via numerical
continuation.
