Projection Operator Method

Edson Sardella

We consider the dynamics of a system of interacting spins described by the Ginzburg-Landau Hamiltonian. The method used is Zwanzig's version of the projection-operator method, in contrast to previous derivations in which we used Mori's version of this method. It is proved that both methods produce the same answer for the Green's function. We also make contact between the projection-operator method and critical dynamics.

JOSIP SEKE

It is pointed out that the application of the projection-operator method leads to the violation of a consistency condition if the so-called Born approximation is made in the nonMarkovian master equation. Since only the validity of this consistency condition guarantees obtaining non-negative probabilities, its violation makes the applicability of the whole method in this approximation meaningless. Therefore, it is demonstrated that an additional Markov approximation restores the validity of the consistency condition in the master equation, and thus re-establishes the applicability of the projection-operator method. Furthermore, the possibility of obtaining a self-consistent projection-operator method is also examined.

JOSIP SEKE

It is pointed out that the usual projection-operator method yields non-positive-definite reduced density operators for subsystems. In order to obtain non-negative probabilities, a new self-consistent projection-operator method for treating non-Markovian behaviour is developed. Finally, the applicability of the method is shown by an example.