The Green functions are especially useful for summing over the restricted classes of perturbation theory of diagrams, and are very powerful when combined with spectral representations. They play the most important part in the field-theoretical treatment of the many-body problem, and they are flexible enough to describe the effects of retarded interactions and all quantities of physical interest like thermodynamic properties can be derived from them. There are different types of Green functions,or propagators: One particle, two-particle,..., n particle, advanced, retarded, zero-temperature, finite-temperature, real-time, imaginary time, and so on. The Green?s functions enjoy popularity because they yield, in a direct way, the most important physical properties in a systematic way. Among the types, we prefer the retarded double-time Green?s function (GF) for our theoretical work because this implies the principle of causality.