quantum field theory - Self-adjoint functionals

Usually, QFT is used in operator representation, that is one can write e.g. $-i\partial_{t}\psi=H\psi$ with H being an operator. And one can ask if H is self-adjoint etc. However, there's also the possibility of the functional representation where H would be a functional. What would be the analog of self-adjointness for functonals?

THX