  We develop the basic ideas and equations for the BRST quantization of
Yang-Mills theories in an explicit Hamiltonian approach, without any reference
to the Lagrangian approach at any stage of the development. We present a new
representation of ghost fields that combines desirable self-adjointness
properties with canonical anticommutation relations for ghost creation and
annihilation operators, thus enabling us to characterize the physical states on
a well-defined Fock space. The Hamiltonian is constructed by piecing together
simple BRST invariant operators to obtain a minimal invariant extension of the
free theory. It is verified that the evolution equations implied by the
resulting minimal Hamiltonian provide a quantum version of the classical
Yang-Mills equations. The modifications and requirements for the inclusion of
matter are discussed in detail.
