Following a similar line of reasoning as spin-LSC outlined above, spin-PLDM can be thought of as a natural extension to the fully linearized case of spin-LSC, now incorporating two sets of mapping variables for each electronic state. This extension can be thought of as an extension from a wavefunction representation to a density matrix representation where one set of mapping variables ({Z}) represents the "ket" moving forward in time and the other ({Z }) represents the "bra" moving backward in time. In general, the observable quantities can be constructed as an average of these two non-interacting paths in electronic action, only connected through simultaneous interaction with nuclear motion. The general correlation function can be written as,