We clarify the relationship and difference between theories of the optical-model potential which had previously been developed in the framework of Feshbach's projection operator approach to nuclear reactions and of Green's function theory, respectively. For definiteness, we consider the nucleon nucleus system but all results can readily be adapted to the atomic case. The effects of antisymmetrization are properly taken into account. It is shown that one can develop along closely parallel lines the theories of hole'' and particle'' mean fields. The hole one-body Hamiltonians describe the single-particle properties of the system formed when one nucleon is taken away from the target ground state, for instance in knockout or pickup processes. The particle one-body Hamiltonians are associated with the system formed when one nucleon is elastically scattered from the ground state, or is added to it by means of stripping reactions. An infinite number of particle, as well as of hole, Hamiltonians are constructed which all yield exactly the same single-particle wave functions. Many equivalent'' one-body Hamiltonians can coexist because these operators have a complicated structure: they are nonlocal, complex, and energy-dependent. They do not have the same analytic properties in the complex energy plane. Their real and imaginary parts fulfill dispersion relations which may be different. It is shown that hole and particle Hamiltonians can also be constructed by decomposing any vector of the Hilbert space into two parts which are not orthogonal to one another, in contrast to Feshbach's original theory; one interest of this procedure is that the construction and properties of the corresponding hole Hamiltonian can be justified in a mathematically rigorous way. We exhibit the relationship between the hole and particle Hamiltonians and the mass operator.'' The latter is associated to the timeordered Green's function, rather than to its advanced and retarded parts separately as the hole and particle Hamiltonians. Similarities and differences between the hole and particle Hamiltonians and the mass operator are exhibited by constructing their explicit expressions