Ultraviolet divergence

All of these approaches, however, have at least this one basic point in common the only reason that a lattice structure is introduced is to formally prevent the appearance of ultraviolet divergences and thus simplify the otherwise difficult calculations that must be made on the continuum. In other words, it is the... 

The only apparent difference of the EDE (1.23) from the regular Dirac equation is connected with the dependence of the interaction kernels on energy. Respectively the perturbation theory series in (1.25) contain, unlike the regular nonrelativistic perturbation series, derivatives of the interaction kernels over energy. The presence of these derivatives is crucial for cancellation of the ultraviolet divergences in the expressions for the energy eigenvalues. 

Leading recoil corrections in Za (of order (Za) (m/M)") still may be taken into account with the help of the effective Dirac equation in the external field since these corrections are induced by the one-photon exchange. This is impossible for the higher order recoil terms which reflect the truly relativistic two-body nature of the bound state problem. Technically, respective contributions are induced by the Bethe-Salpeter kernels with at least two-photon exchanges and the whole machinery of relativistic QFT is necessary for their calculation. Calculation of the recoil corrections is simplified by the absence of ultraviolet divergences, connected with the purely radiative loops. 

Ignoring the momentum dependence of the polarizabilities, one immediately comes to a logarithmically ultraviolet divergent integral [31]... 

The ultraviolet divergence is generated by the diagrams with insertions of two anomalous magnetic moments in the heavy particle line. This should be expected since quantum electrodynamics of elementary particles with nonvanishing anomalous magnetic moments is nonrenormalizable. 

Really the original works [9, 10] contain just the elementary proton ultraviolet divergent result in (11.12) which turns into the ultraviolet finite muonium result in (10.5) if the anomalous magnetic moment k is equal zero. 

The last term in the braces is ultraviolet divergent, but it exactly cancels in the sum with the point proton contribution in (11.12). The sum of contributions in (11.12) and (11.13) is the total proton size correction, including the Zemach correction. According to the numerical calculation in [6] this is equal to AE = —33.50 (55) x lO Ep. As was discussed above, the Zemach correction included in this result strongly depends on the precise value of the proton radius, while numerically the much smaller recoil correction is less sensitive to the small momenta behavior of the proton form factor and has smaller uncertainty. For further numerical estimates we will use the estimate AE = 5.22 (1) X 10 Ep of the recoil correction obtained in [6]. 

In U(l) quantum electrodynamics, the ultraviolet divergence is removed [17] by countering it with a similar term. For the free electron, there is the infinite term... 

The amplitude contribution from the B(3> field occurs in a second-order process using the sum over all possible fluctuations of B(3> in the virtual photon that causes electron-electron interaction. The amplitude due to B(i) has an ultraviolet divergence [17] described by Crowell. This may be removed by regularization techniques. 

Therefore quantum fluctuations in B<3) are accompanied by fluctuations in the transverse electric field. The ultraviolet divergence is probably unimportant [17] because of the co 2 dependence of the fluctuation. The infrared divergence is also damped statistically. The divergences in U(l) electrodynamics [6] can exist as a subset of 0(3) electrodynamics and can be absorbed into integrals that involve photon loop processes associated with quantum fluctuations in B(3 ... 

Here k is the magnitude of the four vector, and k is the magnitude of the spacial part of the 4-vector kk. The integrals in this amplitude suffer from the usual ultraviolet divergence that can be removed through regularization techniques. 

This indicates a number of things. The first is that the quantum fluctuations of the B 3 field are accompanied by fluctuations in the standard electric field. Further, the ultraviolet divergence of the above integral is probably unimportant due to the relationship with the fluctuation. This tends to imply an infrared... 

This calculation demonstrates that the loop fluctuation of a photons, which correlated to a virtual quanta of B field, can be calculated to be finite with out divergence. So the virtual fluctuation of a field does not lead to an ultraviolet divergence, and thus 0(3)h QED is renormalizable by dimensional regularization. 

The implementation of NRQED used here is defined by the Lagrangian and the treatment of infrared and ultraviolet divergences. The Lagrangian has the form [20,21]... 

Ultraviolet divergent integrals were regulated with a momentum cutoff as in the NRQED part of the matching calculation. The total correction to the decay rate is the sum of Eqs. 38 - 42 ... 

Eq. (2) presents the basis for the covariant renormalization approach. The explicit expressions are known for E Ten(E), X u 6 in momentum space. For obtaining these expressions the standard Feynman approach [11,12] or dimensional regularization [13] can be used. They are free from ultraviolet divergencies but acquire infrared divergencies after the renormalization. However, these infrared divergencies, contained in X 1) and cancel due to the Ward identity X -1) = —A1 1 and the use of the Dirac equation for the atomic electron in the reference state a) ... 

A(q) = e q) + q2/(2m ) — q.V being the energy mismatch between the the states in) and q). The second term in the square brackets in Eq. (7b) arises from the coupling-constant renormalization in Eq. (4) and compensates for the ultraviolet divergence of the first term. This compensation is completely analogous to that of the electron mass renormalization in calculations of the radiative shift of an atomic optical transition [Bethe 1947 Cohen-Tannoudji 1992],... 

Equilibrium averages can be calculated from Eq. (2.18) or Eq. (2.20) by integrating over all configurations of the displacement field, provided a short distance cutoff that simulates the effect of the underlying lattice is imposed to eliminate ultraviolet divergences. 

Most of these diagrams contain two intermediate electron propagators and, therefore, double summations over the whole spectrum of the Dirac equation in the external nuclear field. This makes their computation numerically intensive. Both the selfenergy and vacuum-polarization screening corrections are ultraviolet divergent and require renormalization to yield a finite result. 

Vjh is the potential energy Qf a neutral 2-d Coulomb plasma with charge -e for long range it exhibits the 2-d logarithmic Coulomb potential (r- °0) with a soft core cut-off at small distance a to avoid ultraviolet divergencies. The relevant parameters here are identified as... 

In Quantum Electrodynamics, renormalization was developed around 1950, especially by Schwinger, Feynman, and Dyson, in order to eliminate nonphysical ultraviolet divergences. 

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