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The Newtonian Mechanics prediction for the Mercury's Precession is actually $532''$ per century. The general result If the central force is attractive, there is a circular orbit of radius $r_0$. This circular orbit is stabble if it correspond to a minimum of the effective potential, i.e. $$U_{ef}''(r_0)>0.$$ Using that $U_{ef}=L^2/2mr^2+U$ and $F(r_0)=-... 1 I would like to stress the difference between 1) Perturbative Renormalization 2) Non-perturbative Renormalization By Perturbative Renormalization I mean removing infinities from the computation of a correlator/amplitude, order by order. This is done by introducing counterterms, i.e. re-writing the bare parameters of the lagrangian as$\lambda_{Bare} = \...

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To understand why renormalization may work you might first consider simpler situations in Classical/Quantum Mechanics. In this case there are explicitly solvable toy models where one can see exactly what happens and why. See my paper "A Toy Model of Renormalization and Reformulation" on arXiv. About how to cope with growing terms, see my short note here.

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Renormalization is always needed when the Hamiltonian is singular. Singular means that the formal expression for the Hamiltonian resulting from the interaction specified is not a self-adjoint operator in a dense domain. Then the dynamics is formally ill-defined and must be renormalized by taking care to represent everything properly as a limit that makes ...

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The trick is in the introduction of a renormalization scale. Once the perturbation theory has been regularized, one obtains a momentum (and cut-off) dependent interaction of the (schematic) form in 4D $$\lambda(p)=\lambda_0+\alpha\lambda_0^2 \ln(\Lambda^2/p^2),$$ where $\lambda_0$ is the bare interaction, and $\alpha$ some numerical factors. What one ...

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Before I try to answer your question, one thing: Does Ryder really calculate the $\mathcal{O}(\lambda^2)$ to the propagator as the first contribution in perturbation theory, because there is actually a $\mathcal{O}(\lambda^1)$ to the propagator and the $\mathcal{O}(\lambda^2)$-loop is as far as I am concerned a two-loop diagram, i.e. having two loop momenta,...

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In general, or as far as I'm aware, the band diagrams and Density of States shown in your question are electronic band structures. You can also see this from the graph's legend which shows that you're dealing with d and p orbitals belonging to W and S. However, looking at the numbers in your question, I can see where the confusion comes from. The direct ...

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First, the idealized clean non-relativistic problem with the $1/r$ Coulomb potential is integrable: the exact wave functions and energy eigenvalues may be precisely written down using elementary functions. It is exactly true that the eigenvalues of the bound states are $-13.6\,{\rm eV}/n^2$ etc. Now, the first simple correction is the proton motion. This is ...

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One always needs to allow the frequency to change, otherwise one gets horrible secular terms. In case of resonances one needs additional tricks. A good mathematical book is ''Perturbation methods in nonlinear systems'' by G.E.O. Giacaglia (Springer 2012). He discusses both the traditional Poincare-Linsted method and more advanced methods based on Lie ...

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