  Using an approach based on Doi-Peliti field theory, we study several
different Active Ising Models (AIMs), in each of which collective motion
(flocking) of self-propelled particles arises from the spontaneous breaking of
a discrete symmetry. We test the predictive power of our field theories by
deriving the hydrodynamic equations for the different microscopic choices of
aligning processes that define our various models. At deterministic level, the
resulting equations largely confirm known results, but our approach has the
advantage of allowing systematic generalization to include noise terms. Study
of the resulting hydrodynamics allows us to confirm that the various AIMs share
the same phenomenology of a first order transition from isotropic to flocked
states whenever the self propulsion speed is nonzero, with an important
exception for the case where particles align only pairwise locally. Remarkably,
this variant fails entirely to give flocking -- an outcome that was foreseen in
previous work, but is confirmed here and explained in terms of the scalings of
various terms in the hydrodynamic limit. Finally, we discuss our AIMs in the
limit of zero self-propulsion where the ordering transition is continuous. In
this limit, each model is still out of equilibrium because the dynamical rules
continue to break detailed balance, yet it has been argued that an equilibrium
universality class (Model C) prevails. We study field-theoretically the
connection between our AIMs and Model C, arguing that these particular models
(though not AIMs in general) lie outside the Model C class. We link this to the
fact that in our AIMs without self propulsion, detailed balance is not merely
still broken, but replaced by a different dynamical symmetry in which the
dynamics of the particle density is independent of the spin state.
