## Lut pup

The 4E2 factor in front is canceled by the (1/2E)2 factor in \ffi\2. The final steps are the same as for the KG case, Sec. 4.7, giving da_ _ /2ZaE\2

.vhere v2 = p2/E2. This famous result is the Mott cross section for the scattering of a spin \ particle. Comparing it with the KG result, Eq. (4.68), we see that it differs by the factor

For large energy, v ~ 1, and the cross section goes to zero in the backward direction. (See Fig. 5.2.) This difference in the backward direction is due to magnetic scattering: the interaction of the magnetic moment of the electron (associated with its spin) with the magnetic field it sees when moving toward the fixed Coulomb field.

### 5.6 NEGATIVE ENERGY STATES

In this section, the role of negative energy states in the Dirac theory is examined. We will treat second order Coulomb scattering as an example, so the discussion will parallel the development given in Sec. 4.7, where the contribution of negative energy states to second order Coulomb scattering of spinless (KG) particles

scattering angle Q

Fig. 5.2 The Coulomb scattering cross section in arbitrary units. The solid line is the cross section for a spin zero particle, and the dashed line is Eq. (5.46) for a spin ^ particle. Note that both cross sections peak strongly in the forward direction but that there is an additional suppression in the backward direction for the spin £ particle.

vas studied. As the results for the Dirac theory are very similar, the discussion here will emphasize the similarities and differences.

### Completeness Relation

Recall that the evaluation of the second order matrix element for the S-matrix required the completeness relation. For the two-component KG theory, the needed relation was given in Eq. (4.71). The corresponding relation for the Dirac equation is

KD(r,r') = {tf&VW&V) + OKr.'V)} = 1 *3(r - r') .

Note that a plus sign stands in front of the negative energy sum; the KG completeness relation, Eq. (4.71), had a minus sign. This difference is due to the different normalization condition satisfied by the positive and negative energy states in the Dirac and KG theories.

Proof: The general proof of (5.48) is identical to the general proof of Eq. (4.71). Even the proof by construction is similar, except that now the matrix is

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