Adiabatic S-matrix formalism

Gell-Mann and Low [23] derived a formula which yields the energy shift due to the interaction (149) in terms of the matrix elements of the operator 5. (0, —00) where S. is the electron operator (108) obtained from I5q(109) with replaced by operator (149). Later Sucher [24] derived a symmetrized version of the energy shift formula, containing the matrix elements of the operator 5y(oo, —00) and which is more suitable for the renormalization procedure. 

For the free atom, in the absence of the external fields, the energy corrections contain only 5-matrix elements of even order. The reason is that the perturbation Hint depends linearly on the operators of the emission or the absorption of photons (see Eq(103)) and these operators should enter peiirwise in the expansion for the energy correction to give the photon vacuum state. Then the equation (151) looks simpler  

It is essential to distinguish the contribution of the irreducible and reducible graphs (5-matrbc elements). In the irreducible graphs the initial or reference state is omitted in the summation over intermediate atomic states. The contribution of the reference state is described by the reducible graphs. The reducible 5- matrix elements can be expressed as a product of the lower-order 5-matrix elements. The calculation of the reducible 5-matrix elements with the use of Eq(151) gives rise to the singular terms I/7, 

1/7 etc. These singularities are cancelled explicitly by the counterterms contained in each set of the square brackets in Eq(151). The remainder after this cancellation is caEed the reference state contribution (RSC). 

For irreducible matrix elements the procedure of the evaluation of the limit 7 -+ 0 can be avoided and the adiabatic formula (151) can be replaced by a simpler one [11]  

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