Multi-time Green functions for systems evolving in time under unitary transformations. The linear, quadratic and higher order responses

Application of the unitary transformation, containing time dependent “external fields”, to the equation of motion for the density matrix and the Schrödinger equation leads to a self-consistent solution for the transformed density matrix and an expression for the transformed Hamiltonian. Knowing the change of the Hamiltonian and solving an integral equation we can get the equivalent solution for the density matrix. With help of this solution we can express changes of averages of suitable operators in terms of multi-time Green functions. Both equivalent, but of different form, solutions can be presented as infinite powre series in external fields (or a dimensionless parameter). Having two formulae for changes of averages we compare the coefficients of the nth power of the parameter. This gives us relations among multi-time Green functions. For superfluid 3He-B in the frame of linear response it is shown that the rotation of the spin system relative to the orbital one can be presented as rotation in the spin space. For the quadratic response all the basic equations containing the Green functions are written down explicitly.