  Whereas boson coherent states with complex parametrization provide an
elegant, and intuitive representation, there is no counterpart for fermions
using complex parametrization. However, a complex parametrization provides a
valuable way to describe amplitude and phase of a coherent beam. Thus we pose
the question of whether a fermionic beam can be described, even approximately,
by a complex-parametrized coherent state and define, in a natural way,
approximate complex-parametrized fermion coherent states. Then we identify four
appealing properties of boson coherent states (eigenstate of annihilation
operator, displaced vacuum state, preservation of product states under linear
coupling, and factorization of correlators) and show that these approximate
complex fermion coherent states fail all four criteria. The inapplicability of
complex parametrization supports the use of Grassman algebras as an appropriate
alternative.
