  We introduce a symmetric Poisson bracket that allows us to describe
anticommuting fields on a classical level in the same way as commuting fields,
without the use of Grassmann variables. By means of a simple example, we show
how the Dirac bracket for the elimination of the second class constraints can
be introduced, how the classical Hamiltonian equations can be derived and how
quantization can be achieved through a direct correspondence principle.
Finally, we show that the semiclassical limit of the corresponding Schroedinger
equation leads back to the Hamilton-Jacobi equation of the classical theory.
Summarizing, it is shown that the relations between classical and quantum
theory are valid for fermionic fields in exactly the same way as in the bosonic
case, and that there is no need to introduce anticommuting variables on a
classical level.
