  We consider the quantum reaction-diffusion dynamics in $d$ spatial dimensions
of a Fermi gas subject to binary annihilation reactions $A+A \to \emptyset$.
These systems display collective nonequilibrium long-time behavior, which is
signalled by an algebraic decay of the particle density. Building on the
Keldysh formalism, we devise a field theoretical approach for the
reaction-limited regime, where annihilation reactions are scarce. By means of a
perturbative expansion of the dissipative interaction, we derive a description
in terms of a large-scale universal kinetic equation. Our approach shows how
the time-dependent generalized Gibbs ensemble assumption, which is often
employed for treating low-dimensional nonequilibrium systems, emerges from
systematic diagrammatics. It also allows to exactly compute -- for arbitrary
spatial dimension -- the decay exponent of the particle density. The latter is
based on the large-scale description of the quantum dynamics and it differs
from the mean-field prediction even in dimension larger than one. We moreover
consider spatially inhomogeneous setups involving an external potential. In
confined systems the density decay is accelerated towards the mean-field
algebraic behavior, while for deconfined scenarios the power-law decay is
replaced by a slower non-algebraic decay.
