Renormalization counter terms

Asymptotic Condition.—In Section 11.1, we exhibited the equivalence of the formulation of quantum electrodynamics in the Coulomb and Lorentz gauges in so far as observable quantities were concerned (t.e., scattering amplitudes). We also noted that both of these formulations, when based on a hamiltonian not containing mass renormalization counter terms, suffered from the difficulty that the... 

However, one readily verifies, using invariance techniques similar to those outlined in the last section, that if/( ) is given by Eq. (11-484), it doss not satisfy the criterion that <0 /( ) p, > = 0. It is easy to show that Eq. (11-476) can be satisfied if we add the (mass renormalization) counter term Anufi(x) to the right side of Eq. (11-484). In fact... 

The energy shift of a bound-electron state contracted terms of the second-order S matrix element taking into account the mass-renormalization counter term provided by S l according to... 

The decomposition of the irreducible part of the self-energy wave-function correction term is depicted in Fig. 2. The divergent terms are these with zero and one interaction in the binding potential present, below referred to as zero-potential term and one-potential term , respectively. The charge divergences cancel between both terms. In addition, a mass counter term dm has to be subtracted to obtain proper mass renormalization similar to the case of the free self energy [47] (for our schemes see also [44]). The zero- and one-potential... 

As indicated by subscripts the various terms here denote the reducible loop-after-loop (red), the loop-inside-loop (in), the crossed-loops (cr) contributions and an additional counter term (ac) to the SESE b) graph, respectively. The subscript ren always signifies the renormalized expression, though the expres-... 

The renormalized first-order bound-electron self energy shift for a state a( ) = ( a) can formally be written as the real part of the bound self energy with the mass counter term subtracted... 

In a very general way, divergences in diagrams can be eliminated with the help of counter-terms these counter-terms can be interpreted as resulting from additional interaction terms and the latter are absorbed by renormalization of the partition functions and of the bare interactions. This is exactly what we did in Chapter 10, Section 4.2.6, for the model with purely repulsive two-body interactions, and, in this case, it is easy to see that the process amounts to dimensional regularization. 

When the operator 91 is applied to diagram contributions, the essentially divergent terms, 4( s e S must be interpreted as renormalization terms or counter-terms they are functions of s0, which become infinite when s0 0, and... 

Hence, the counter-terms to be added to the tricritical Lagrangian 90 are monomials Including the renormalized sources art(x) and gft(x) and the renormalized monomials pp, Pp, so that their dimension could not exceed [A] = 3. This implies that the counterterms have the form... 

It involves the trivial renormalization, the subtraction of vacuum expectation values (VEV), and the serious renormalization (see above) which can be handled by the counter-term (CT) technique [4]. 

The necessity for summing over the negative-energy states raises the question of whether we should include QED renormalization effects. The answer is, in general, yes a rigorous treatment would involve virtual pairs and renormalization with counter-terms to remove the divergences in the perturbation sums, which is beyond the scope of relativistic quantum chemistry at present. For electric perturbations, the effect of renormalization is small, because the contribution is already It is therefore... 

Renormalization or introduction of counter terms to avoid the divergences arising from the infinite Dirac sea in the vacuum and from ultraviolet (UV) divergences of QED. 

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