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+ d k kp [dO)(k)e- ik . x + a(O)t(k) e ik . x] 2 2 2(2n)3 + m /1
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[d).)(k)e~).)(k)e-ik.x+a().)t(k)e~).)*(k)eik.x]
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As before e().)(k), for A = 1,2,3 are three orthonormal space-like vectors orthogonal to k(P = /12). We may easily verify that Eqs. (3-143) and (3-140) are satisfied by this expression while the corresponding covariant propagator is 4 2 <01 TA (x)Av(Y) 10) = -i f d k e-ik.(X-y)(gpV - kpkv/~2 + 2 kpkv//1 .) 2 - /1 2 + Ie 2 + Ie p (2n)4 k k - m (3-147)
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m 2 =/1 A
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and the commutator reads
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The reader is invited to study the limits Jl
--->
0 and A. ---> o.
(a) When A. ---> 0, Jl # 0, show that we recover Proca's theory. (b) If A. # 0 and Jl ---> 0, m goes to zero. Show that in this limit the Green functions tend to the values
given in the preceding subsection. In particular, since
(3-149)
We see that the double-pole term tends to 1/(k 2 + is)' as anticipated in Eq. (3-131). We shall return to this zero-mass limit in the following, in particular in Chap. 4. It should also be noticed that in k space the propagators in the Stueckelberg formalism behave as l/k2 for large k, while this is not the case for the massive Proca propagator (3-137). As a supplementary exercise one may check the covariance of the theory and construct the generators p. and M., of the Poincare group. We have not offered here a very thorough description of the kinematical properties of photons. On this point the reader is referred to the works quoted in the notes. Rather, we had in mind to pave the way to recent developments in gauge theories, to be considered in Chap. 12. We close this subsection by describing the principle of a method allowing us to set an upper
QUANTUM FIELD THEORY
limit to the photon mass using terrestrial measurements. This is the so-called constant field method of Schrodinger. Assume as a first approximation that the earth can be simulated by a perfect magnetic point dipole M. The corresponding localized conserved current is such that
j(x)
and can be written j = - ~M x Vi5 3 (x). In the static case, let us derive from Proca's equations the corresponding vector potential such that div A = 0, a condition compatible with these equations. It follows that
a solution of which is
A(x) = - ~ 2
.f 3
d k - - M x k _e,k'x = - M x V( _e_ __ (2n)3 k 2 + Jl2 8nr
The corresponding magnetic induction is
e-p.r e- W B = curl A = - V x (M x V) ~ = [(M' V)V - M~] ~ 8nr 8nr
Let us take the z axis along M with unit vector
z, so that B has the form
The field has been split into two parts. In the first [r(r' z) - jz] has been factored out, corresponding to the angular distribution of the magnetic field in the limit Jl -> O. At constant distance r there appears a uniform added term anti parallel to the earth's dipole. A careful scan of the angular distribution of the magnetic field allows us to set an upper limit to the mass Jl of the order of 4 x 10- 48 g (3 X 10- 15 eV ~ 10- 10 cm -1). More recent measurements using a different method have improved this result by an order of magnitude.
3-2-4 Vacuum Fluctuations
It is instructive to study simple effects arising from the field quantization before embarking on the coupled nonlinear problems. Especially interesting are those which do not seem to rely on the particle interpretation-in the present case, the photon aspect. We give a schematic description of two such situations which have to do with the observability of differences in vacuum fluctuations. We have encountered such phenomena when presenting Welton's interpretation of the Lamb shift in Chap. 2. We can take into account simple macroscopic sources by modifying boundary conditions on the field which was considered up to now in free space. This procedure is to some extent unsatisfactory since it does not describe the microscopic mechanism responsible for these boundary conditions. But it is suitable for elementary calculations. The original observation of Casimir (1948) is that, in the vacuum, the electromagnetic field does not really vanish but rather fluctuates (compare Sec. 3-1-2). If we introduce macroscopic bodies-even uncharged-some work will be necessary to enforce appropriate boundary conditions. Intuition on the sign of this effect is lacking, so that work here is meant in some algebraic sense,
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