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b(p, a) = (4nd 2 )3/4 e-p2d2/2 u(a)t(p)w d*(p, a)
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(4nd 2 /4 e-p2d2/2v(a)t(p)w
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Using the explicit expressions (2-37), we see that the ratio bld* is typically of order Ipl/(m + E) and becomes important when Ipl '" m. If the wave packet is spread out over a distance d 11m, the contribution of momenta Ipi '" m lid is heavily suppressed, and the negative energy components are negligible; the one-particle theory is then consistent. However, if we want to localize the wave packet in a region of space of the same size as the Compton wavelength, that is, d ;:S 11m, negative energy solutions play an appreciable role. This quantitative discussion is in agreement with the heuristic arguments presented at the beginning of this chapter. For a wave packet with negative energy contributions as in (2-57),
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QUANTUM FIELD THEORY
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we compute as above the normalization condition
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p = (po, -
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p), the total current is
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)
(2-60)
It is now time dependent. Besides the group velocity term, there is a real, oscillating term. The frequency of these oscillations is very high-larger than
This phenomenon, traditionally called zitterbewegung, is an example of the difficulties due to the negative energy states in the framework of a one-particle 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. 2-1). 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)
(spin-up 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 spin-up and spin-down 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 2-1 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 spin-flip)
where r ==
E +m k E-V+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 one-particle 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 Klein-Gordon 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
2-2-3 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 : (2-61) The Dirac equation then reads
- eJ1- m)ljI(x) = 0
(2-62)
This prescription ensures the invariance of the equation under gauge transformations:
ljI(x) ---+ eiea(x) ljI(x) {
(2-63)
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. 2-1-3 may be extended to the present case. Equation (2-62) can be rewritten more explicitly as
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