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where p stands for (po,  p). From the orthogonality relations (243), it follows that in .NET framework
where p stands for (po,  p). From the orthogonality relations (243), it follows that PDF417 Decoder In .NET Framework Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Creating PDF 417 In .NET Framework Using Barcode maker for Visual Studio .NET Control to generate, create PDF417 2d barcode image in VS .NET applications. b(p, a) = (4nd 2 )3/4 ep2d2/2 u(a)t(p)w d*(p, a) PDF 417 Reader In VS .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET framework applications. Generating Barcode In .NET Using Barcode printer for VS .NET Control to generate, create barcode image in VS .NET applications. (4nd 2 /4 ep2d2/2v(a)t(p)w
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Ji(t) fI +
d3p m {pi (2n)3 If If ~ aa' [I b(p, a)j2 + 1 a)j2] d(p, [b*(p, a) d*(p, a') e 2iEt ij(a)(p)aiOv(a')(p) (260) It is now time dependent. Besides the group velocity term, there is a real, oscillating term. The frequency of these oscillations is very highlarger than This phenomenon, traditionally called zitterbewegung, is an example of the difficulties due to the negative energy states in the framework of a oneparticle theory. A more striking manifestation is the famous Klein paradox. Let us idealize the localization process by a square potential barrier of height V in the half space Z == x 3 > 0 (Fig. 21). Consider now in the z < 0 half space an incident positive energy plane wave of momentum k > 0 along the z axis: I/Imc{Z) (spinup along the
axis) The reflected wave has the form
I/Iref{z) 1 k ) .kz a e ( ~ m
.kz ( e E: m
1 0 0 (superposition of spinup and spindown positive energy solutions). In the z > 0 half space, i.e., in the presence of the constant potential V, the transmitted wave has a similar form: trans
(z)=ce,qz(E  : + m )+de,qz( V
E V+m
THE DIRAC EQUATION
with an effective momentum q of
Figure 21 Klein's paradox in a square potential.
Writing down the continuity of the solution at z
determines the coefficients a, . .. ,d: b=d=O l+a=c
1 a
(no spinflip) where r ==
E +m k EV+m
As long as IE  Vi < m, q is imaginary and the transmitted wave decays exponentially; beyond a few Compton wavelengths, it is negligible. If we increase V so as to restrict this penetration region, the transmitted wave becomes oscillatory when V 2 E + m. The computation of the transmitted, reflected, and incident currents yields jtrans jme
+ rf
~ref =
line
(1  r)2 = 1_j~rans
1+r line
The conservation of probabilities does indeed look satisfied: Unfortunately, since r < 0, the reflected flux is larger than the incident one! We are again in trouble when we try to localize the particle within a distance of the order of the Compton wavelength. In spite of these difficulties, the Dirac equation and its oneparticle interare very useful and physically sensible as long as we consider external forces which are slowly varying on a scale of a few Compton wavelengths. They provide us with the first relativistic corrections to the Schrodinger picture. This is what we are going to explore at length in the next sections, before returning to a deeper investigation of the meaning of negative energy states. We now realize that the difficulties which led us to disregard the KleinGordon equation have not been really solved. Even though we shall pursue this discussion in the framework of the spin! theory because of its important physical implications, we could as well concern ourselves with the scalar case within the same range of validity. This is another instance where important physical theories were constructed for what seems afterwards to be unconvincing motivations. pr~tation
QUANTUM FIELD THEORY
223 Electromagnetic Coupling
We wish now to study the interactions of a Dirac particle with an external (classical) electromagnetic field characterized by its potential AJl(x). The relevant coupling is obtained from the free Dirac equation through the minimal coupling prescription described in Chap. 1 : (261) The Dirac equation then reads  eJ1 m)ljI(x) = 0 (262) This prescription ensures the invariance of the equation under gauge transformations: ljI(x) + eiea(x) ljI(x) { (263) AJl(x) + Aix)  OJlIX(X) Here e denotes the charge of the particle; it is negative e = lei for the electron. The Lorentz covariance of this equation is clear. If we change our reference frame, the electromagnetic potential transforms as a vector A~(X' = Ax) = (A lVA v(x) and therefore the analysis of Sec. 213 may be extended to the present case. Equation (262) can be rewritten more explicitly as

