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QUANTUM FJELD THEORY
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U(x, x'; r) = C(x, x')r- 2 exp { -
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tr In [(eFr)-l sinh (eFr)]}
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x exp
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[~(X - x')eF coth (eFr)(x -
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+ ~ (J",F"'r + im 2 rJ
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The function C(x, x') is determined by Eqs. (2-140) which lead to
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eF", ] [ iiJ~ - eA"(x) - 2 (x - x')' C(x, x') = 0 [ iO~
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, - eA"(x')
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+ eF", (x 2
x')' ] C(x, x')
The solution has the form C(x, x')
C(x') exp - ie
d~' [A(~) + iF( ~ -
x')]
Since A"W + iF",(~ - x')' has a vanishing curl, the integral is independent of the path of integration. Taking a straight line from x' to x, the second term does not contribute, owing to the antisymmetry of F"" and we may write C(x, x')
C exp -ie
d~' A(~)
The constant C is finally determined by (2-136a) as
i C= - - (4n)2
In summary, the propagator in a constant field reads
SA(X, x') =
[i~x -
ej(x)
+ m] (-i)
dr U(x, x'; r)
(2-141a)
with
U(x, x'; r) =
(4n) r
. 1 ~exp {IX d~" A"W - -tr In [(eFr)-l sinh (eFr)] -ie
+ ~ (x 4
x')eF cotanh (eFr)(x - x')
+ ~ (J",F"'r + i(m 2
ie)r}
(2-146)
Notice the presence of the - ie term [in order to fulfill (2-136b)] and of the phase factor exp - ie Sd~ A. Its role is to make U gauge covariant; when A"(x) -. A"(x) + iJ"A(x),
U(x, x'; r) -. e-ieA(x) U(x, x'; r) eieA(x')
We now turn to the plane wave case. The calculation is quite analogous to the previous one, and we merely sketch the successive steps. We consider a linearly polarized plane wave and use the same notations as in Sec. 2-2-3: A" = e"f(~) with ~ = n' x, n 2 = 0, F", = 4>",!'W where 4>", = n"e,n,e w Notice that iJPF"p = O. Therefore Eqs. (2-143) take the form
(2-147)
THE DIRAC EQUATION
Remarking that (d/dr)(n' n)
0, d~/dr
2n' nand [~, d~/dr]
~(O) -
0, we first solve for ~:
~(r)
= n'x(r) =
</1'"n"
2n'nr
Then we integrate the equation for </1'"n" and obtain
= en'f(~)
where C' is a constant operator which commutes with n' n. Inserting this expression into (2-147) and integrating it, we get
Here D" is a new constant operator commuting with n' n. We then calculate x(r) - x(O) and eliminate
n"(r)
= -"
x (r) - x (0)
+ [~(r) _ ~(OlY
f"(') d~ [2eC"f(~) + e2n"f2(~) +:2 n"</1p,(JP'f'(~) J e
1;(0)
~(r) ~ ~(O) {2eC" fW r)] + e2n"f2[~(r)] + ~ n"</1p,(Jp'f'[~(r)]}
1 en C"=--</1"p[x P (r)-x P (O)]" 2r ~(r) - ~(O)
This allows us to express the constant C" as
f ') d~fm
After computation of the various commutators
Wr), x"(O)]
WO), x"(r)]
2in"r
[x"(r), x"(O)] = -Sir
we may write the hamiltonian H as
(2-14S)
<i5F) denotes the quantity <i5f2)
d~F(~)
~(r) - ~(O)
[fl;(')
d~f(~)
~(r) - ~(O)
f(~)
The evolution operator has the form U(x, x'; r)
C(x x') [(X - x'f -'-2- exp i - - r 4r
" (JP' + r(e 2 <i5F) + m 2) + er _'I"_P'_
f(~')J
where the function C(x, x') is again determined by the relations (2-140), We find C(x, x')
C(x') exp - ie
( Ix
dy" A"(y) -
</1"P(y P~, P n'y -
-X')[fn,Y
duf(u) ---p n' y - s
f(n' y)
where the integral is path independent. For a straight line, the only remaining term in the phase is exp - ie J~, dy" A"(y), and we find, finally, U(x, x'; r) = - (4nfr2 exp i - e
{(X - X')2 -4-r-
+ e 2<i5f2) + m 2 +:2 </1p,uP'
f(~) - f(~') ~ _ ~' - i8 r
(2-149)
dy" A"(Y)}
lO'!JQUANTUM FIELD THEORY
This result recalls the classical result of Sec. 1-1-3. For a periodic function f(~), the term proportional to <pp,a P' is smeared out if we average over a few periods. The net effect is a mass shift
m;ff = m2 + e2<i5f2) = m 2 + e2f2
Such a nonlinear effect is hard to detect; very high intensity beams are required, since for a monochromatic plane wave of frequency wl2n = ciA and of energy density C = E2 = f 2w 2 = phw, we have
~2- =
,1.m 2
--z-z =
e2f2
4naAe w
2 2aAeAp
Here Ae is the Compton wavelength of the electron and p is the number of photons per unit volume in the incident beam. At present the most powerful laser beams do not enable us to reach a sizable value for this ratio.
NOTES
The material of this chapter is very standard and we could hardly escape rephrasing what is found in many excellent textbooks. See, for instance, J. D. Bjorken and S. D. Drell, "Relativistic Quantum Mechanics," McGraw-Hill, New York, 1964, and also A. Messiah, "Quantum Mechanics," vol. 2, North-Holland, Amsterdam, 1962. Hole theory is beautifully described by P. A. M. Dirac in his Solvay report of 1934, reprinted in "Quantum Electrodynamics," edited by 1. Schwinger, Dover, New York, 1958, which contains many of the fundamental papers on relativistic quantum field theory. For the propagator approach see R. P. Feynman, Phys. Rev., vol. 76, pp. 749-769, 1949, and for the proper time method, 1. Schwinger, Phys. Rev., vol. 82, p. 664, 1951. The original solution for a Dirac electron in a plane wave is due to D. M. Volkow, Z. Physik, vol. 94, p. 25, 1935. For a review on the Klein-Gordon equation see H. Feschbach and F. Villars, Rev. Mod. Phys., vol. 30, p. 24, 1958. The nonrelativistic limit of the Dirac equation is studied in L. L. Foldy and S. A. Wouthuysen, Phys. Rev., vol. 78, p. 29, 1950. An overall view on the bound-state spectrum is given in H. A. Bethe and E. E. Salpeter, "Quantum Mechanics of One and Two Electron Atoms," SpringerVerlag, Berlin, 1957. See also M. E. Rose, "Relativistic Electron Theory," John Wiley, New York, 1961. The interesting semiclassical interpretation of the Lamb shift by T. A. Welton is in Phys. Rev., vol. 74, p. 1157, 1948. For the twocomponent neutrino theory see R. P. Feynman and M. Gell-Mann, Phys. Rev., vol., 109, p. 193, 1958. An account of relativistic quantum kinematics is given by P. Moussa and R. Stora in "Analysis of Scattering and Decay," edited by M. Nikolic, Gordon and Breach, New York, 1968. We refer to this work for complementary details on the representations of the Poincare group and their applications.
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