Non-unitarity of finite dimensional Lorentz group and its implications

In Peskin and Schroeder, section 3.2, it is stated that Lorentz group being non-compact it does not have any finite dimensional, faithful unitary representation. But it has also been said that one shouldn't worry because $\psi$ is not a wavefunction but a field.

Now my question is, (editing as per recommendation by Qmechanic to ask the subquestions in separate posts)

Why do we not have to worry about non-unitarity in the case of a field?