Mass renormalization

As discussed before, the mass renormalization is a reflection of the fact that the particle traces a distance longer than 2Qq in the total multidimensional coordinate space. 

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... 

Mass renormalization requires [15] that an additional term vj/yvv /5m be added where 5m is the difference between the physical and bare masses [77]. 

Then e is associated with a level shift (or mass renormalization) of the unperturbed level e ° while e" gives us the decay time. 

The evaluation of the scaling functions directly in d = 3 has been advocated in [SD89, SD90], where the method has been shown to work for small momenta. Also the treatment of the additive mass renormalization (corresponding to th introduction of p. J has been carefully considered there. 

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... 

These cases can be contrasted by most uranium intermetallics, which have Fermi surfaces in good agreement with LDA calculations which treat the f electrons as band states (IQ. In the one case where a mixed valent Fermi surface is known (CeSno), it is also in excellent agreement with an LDA f band calculation (T8-19L with a mass renormalization of five due to a self-energy correction resulting from virtual spin fluctuation excitations (2Q). Notice the different dynamic correlations used to explain the mass renormalizations in the f core and f band cases. 

The existence of strong antiferromagnetic correlations in heavy fermion metals has been confirmed by extensive neutron scattering studies ( 5), and these correlations are now thought to be responsible for both the large mass renormalizations and exotic superconductivity... 

If we are interested in d3mamical quantities on the other side (vibrational density of states, infrared or Raman spectra), these quantities will then depend on the fictitious mass /x through the atomic mass renormalization. This dependence of the t3q)ical ionic frequency on fi will be linear at small /x. Thus, for extracting vibrational frequencies from Car-Parrinello MD, the criterium for choosing jj, should certainly be tighter than described in the previous paragraph. 

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],... 

Comparing Eqs (J.liy-ij.19 ) we see that Z t) is essentially the Fourier transform of the spectral density associated with the system-bath interaction. The differences are only semantic, originating from the fact that in Eqs (7.77)-(7.79) we used mass renormalized coordinates while here we have associated a mass nij with each harmonic bath mode j. 

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... 

At least the study of the term Wo will allow us to show the difference of the values of this term between those obtained by the relativistic and nonrelativistic calculations. This difference is weak for the contribution of the discrete spectrum and the low levels of the continuum but becomes considerable (see Note below) for the ones of the continuum of high levels and explains the necessity of the mass renormalization represented by the term Wm-... 

The remaining S(VP)E contribution has been evaluated in the Uehling approximation by Mallampalli and Sapirstein [60] and Persson et al. [7]. This effective self-energy diagram requires charge renormalization as well as mass renormalization. At present only the contribution of the Uehling term of the effective photon... 

Performing the angular integration over directions of the photon momentum, summation over polarizations, and subtracting from Eq (181) the corresponding expression for the free electron (mass renormalization) one obtains ... 

We have shown [46] that if a multipole expansion of the interaction "u is adopted and the mass renormalization procedure carried out at the partial wave level, a convergent representation is obtained for Es which is in precise agreement with the parametric methods which were introduced by Feynman [47] and implemented in the numerical calculations of Mohr [48]. For both the self energy and vacuum polarization, the positive- and negative-energy states are equally important in the calculation of the physical effect, with respect to both the numerical value of the energy shift and the finiteness of the theory. 

Plasma Frequency defined as /2p = Axme jm, where n is the volume density of conduction electrons, e is the charge of an electron, and m is the effective mass renormalized from the free electron mass by lattice and interaction effects. 

Mass renormalization from selfconsistent solution of the self-energy equation... 

After the mass renormalization in a space with d < 4, the short-range interaction potential is assuin< d lo be... 

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