Order Analysis of the Propagator Matrices

In the following, the subscript 1 refers to the hi = a part of the field operator manifold and the subscript 3 to the ha part and so on. Through second order, the inverse of the electron propagator matrix then becomes 

Third-order electron propagator A similar treatment through third order yields G iE) = G-i( ) 

Fourth-order and partial fourth-order electron propagator 

Without including the operator manifold hs, the full fourth-order propagator matrix can be expressed as 

It is generally more important to include the contributions from the hs manifold before increasing the order of the expansion, and one therefore finds it justifiable to study the electron propagator through what has been coined the partial fourth-order , where only the terms formed from the matrices already obtained in third order are retained. 

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