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|Figure 03f is the Feynman diagram in momentum space for the electron self-energy (the insert portrays the unperturbed propagator). Using the Feynman rules, the S matrix can be written down for the transition of an electron from state (p, s) to state (p', s') in the form:|
where the self-energy correction (p) is a 4X4 matrix given by:
-[A + (ip + m)B + ], and m is, at this point, finite but arbitrary.
Figure 03f Electron Self-energy [view large image]
|Counting powers of the off-shell 4-momentum |k| = (k2 - E2)1/2 in the numerator and denominator of the integral suggests that it diverges linearly like d|k|. However, there are various methods to trim the divergence, so that it diverges only logarithmically like d|k|/|k| log(), see "footnote".|
In the S-matrix expansion, each term has a number of the interaction Hamiltonian|
HI -e00 0 = -eRR R eR = (Z2Z31/2)e0 (integration over 3-D space is omitted here).
If the renormalized charge further amalgamates the factors Z1, then the effective charge is :
eeff = eR/Z1 = (Z31/2)e0 ,
since the "Ward's Identities" has shown that Z1 = Z2. The infinity is removed by cutoff in the effective charge. It is interpreted as a small reduction of the charge due to the screening by virtual particles.
Thus in terms of the renormalized quantities, all the divergences disappear from the QED formulation, which yield amazing predication with an accuracy up to one part in trillions (as in "Lamb Shift", and "Gyromagnetic Ratio") even though the methodology had been criticized by many prominent physicists.
which increases as Q increases (or, equivalently, as the probing distance becomes shorter, see Figure 03ha).|
Similar vacuum polarization effects occur in QCD with similar Feynman graph but for gluon exchange between quarks and antiquarks. There is an additional contribution from the gluon self-coupling as shown in Figure 03ha. The effective coupling constant for strong interaction for large Q is now in the form:
where nf = 6 is the effective number of quark flavours for large Q, and -1 is called the
Figure 03ha Effective Coupling Constant [view large image]
|confinement length since s becomes infinite when Q = . The phenomenon of asymptotic freedom in QCD is associated with the fact that s 0 as Q .|
Figure 03hb Lattice Regularization [view large image]