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over all wave numbers k and polarizations A. In a big box of size L, k = (2n/L)n (nx, ny, and nz integers) and
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82 QUANTUM FIELD THEORY
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The last integral is divergent both for small and large frequencies. The ultraviolet divergence when OJ -+ 00 comes from our poor quantum treatment; there is actually a cutoff for distances of order hlmc, that is, frequencies OJ '" mc 2 lh. At the other end, the infrared divergence for small frequencies should be cured by a more accurate treatment of the electromagnetic field in the presence of the charges. The large wavelength modes are sensitive to the low-lying electronic states. This suggests an infrared cutoff of the order OJ '" cia = mc 2 (1/h. Using these crude estimates we expect a value
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and therefore a Lamb shift
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2Pl/2
For the level n = 2 of the hydrogen atom the calculated shift 2S 1/2 L'lE Lamb 660 MHz
in rough agreement with the observed shift of 1057 MHz. The comparison of this term with the Darwin term in Eq. (2-82) shows a reduction factor of order IX In 1/1X. These approximations will be considerably improved in the sequel (see Chap. 7). Nuclear effects The nucleus has a finite size and its charge distribution is not concentrated at a point. For the proton this is represented by a form factor. This affects predominantly the s-wave states, since higher i-wave functions vanish at the origin. An interesting consequence is the isotopic effect, predicted as early as 1932, where different isotopes would have slightly shifted levels.
For light nuclei, the main contribution comes from the mass difference of the various isotopes:
since the reduced mass of the electrons is 11m = lime + 1/mnu cl. For heavy nuclei, however, the finite size of the charge distribution is the leading effect. In a nonrelativistic approximation, we write the correction as
where V(r) is the true potential and -(Zelr) its Coulomb approximation.
THE DIRAC EQUATION
f + 4~: ] ~~it/l(OW fd3xr2~V
~ e it/l(OW
d x [V(r)
where we have inserted ~r2 = 6, integrated by parts, and used the Poisson law: ~ V = - p, P d 3 x = -Ze. According to (2-84), the effect is proportional to Z4. This effect might be used in practice for isotopic separation. A given isotope is excited to some level by a first laser ray and then ionized by a second laser ray. An electrostatic separation may then be achieved. A second consequence is that the critical value Zc ~ 137 beyond which the ground state becomes unstable is pushed to larger values Z ~ 175.
A correct treatment should also include the recoil of the nucleus. This is a difficult problem, which may be tackled by a relativistic two-body equation (Chap. 10). An heuristic reasoning may, however, shed some light on this point. From the predictions of the Schrodinger and Klein-Gordon equations [(2-83) and (2-86)], we see that in both cases they can be interpreted by imposing the following condition on the velocity v:
Two-body relativistic corrections
i!X .. . R = n = posItIve lllteger
Indeed, in the Schrodinger case,
mv 2 m!X 2 E=-=-2
2 2n
whereas for the Klein-Gordon equation v = or
p2/E2,
and the desired result is obtained by replacing n by n - bj as in (2-86). In a two-body problem, let us assume that the relevant velocity is the relative one, namely, the velocity of one of the particles measured in the rest frame of the other. Then
E tot
= (p +
= m + M + (1 _
m~ M.
V 2 )1/2
which is symmetric in the interchange
Using again our empirical rule
we obtain
QUANTUM FIELD THEORY
IX mM 1X4 mM mM = m + M - - - - + - - - [ 3------=2n 2 m + M 8n 4 m + M (m + M)2 1X4 mM 1 6 - n 3 m + M 2j + 1 + O(IX )
This expression reproduces fairly well the recoil effects obtained refined treatment.
a more
2-4 HOLE THEORY AND CHARGE CONJUGATION 2-4-1 Reinterpretation of Negative Energy Solutions
In spite of the successes of the Dirac equation, we must abandon our ostrich policy and face the interpretation of negative energy solutions. As explained previously, their presence is intolerable, since they make all positive energy states unstable in the final analysis. A solution was proposed by Dirac as early as 1930 in terms of a manyparticle theory. Although this shall not be the final standpoint, as it does not apply to scalar particles, for instance, it is instructive to retrace his reasoning. It provides an intuitive physical picture useful in practical instances, and permits fruitful analogies with different situations such as electrons in a metal. Its major assumption is that all the negative energy levels are filled up in the vacuum state. According to the Pauli exclusion principle, this prevents any electron from falling into these negative energy states, and thereby insures the stability of positive energy physical states. In turn, an electron of the negative energy sea may be excited to a positive energy state. It then leaves a hole in the sea. This hole in the negative energy, negatively charged states appears as a positive energy, positively charged particle-the positron. Besides the properties of the positron, its charge Ie I = - e and its rest mass me, this theory also predicts new observable phenomena:
1. The annihilation of an electron-positron pair. A (positive energy) electron falls into a hole in the negative energy sea with the emission of radiation. From energy momentum conservation at least two photons are emitted, unless a nucleus is present to absorb energy and momentum. 2. Conversely, an electron-positron pair may be created from the vacuum by an incident photon beam in the presence of a target to balance energy and momentum. This is the process mentioned above; a hole is created while the excited electron acquires a positive energy.
Thus the theory predicts the existence of positrons which were in fact observed in 1932. Since positrons and electrons may annihilate, we must abandon the interpretation of the Dirac equation as a wave equation. Also, the reasons for
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