The hermiticity of Hamiltonian comes up as a result of requiring real energy eigenvalues and well-defined inner-product for correlation amplitudes. In the corresponding Lagrangian picture (path-integral), I am not clear about the explicit restriction that the above hermiticity of Hamiltonian impose on the Lagrangian. Its the basic path-integral formulation itself, in addition to the integral depending non-trivially on a special path-integral measure, exponential of Lagrangian and so on that makes it unclear. I don't see why Lagrangian should be hermitian. Any pointer?