  We develop the formalism for the one-loop no-boundary state in a cosmological
model with fermions. We use it to calculate the reduced density matrix for an
inflaton field by tracing out the fermionic degrees of freedom, yielding both
the fermionic effective action and the standard decoherence factor. We show
that dimensional regularisation of ultraviolet divergences would lead to an
inconsistent density matrix. Suppression of these divergences to zero is
instead performed through a nonlocal Bogoliubov transformation of the fermionic
variables, which leads to a consistent density matrix. The resulting degree of
decoherence is less than in the case of bosonic fields.
