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meaning the difference in zero point energies between the two configurations. We had originally disregarded the (infinite) contribution L ihw" to the hamiltonian by arguing that it was unobservable. However, its variation can be measured. Let us illustrate this point for the simple configuration of two large parallel perfectly conducting plates as considered first by Casimir. Of course, we can study different geometries and different materials with similar results (except perhaps for crucial signs). We idealize the plates by two large parallel squares of size L at a distance a (see Fig. 3-1) with a L. Consider the energy per unit surface of the conductor with respect to the vacuum. Its derivative will be a force per unit surface with dimension Me 1 T- 2 (where M is mass, L length, and T time). The only quantities entering the problem are h, c and the separation a (the boundary conditions E perpendicular and B parallel to the plate at the interface do not introduce any dimensional quantity). Of course, the effect is proportional to h, as is the zero point energy. The force per unit surface is therefore proportional to hc/a4 , the only quantity with the required dimension. We shall see that it is attractive. Consider the modes inside the volume L 2 a, where L a and we ignore the edge contributions. As we know, only transverse modes contribute to the energy. If the component k z perpendicular to the plates is different from zero it can only take discrete values kz = nn/a (n = 1,2, ... ) to allow for the nodes on the plates and there are two polarization states. If, however, kz vanishes only one mode survives so that the zero point energy of the configuration is
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Figure 3-1 Casimir effect between two parallel plates.
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As it stands this expression is, of course, meaningless, being infinite. But we must subtract the free value which contributes in this same volume a quantity
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= he fL d k ll
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(2n
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f+oo adk z 2
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he fJ3d k ll =2 (2n
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foo dn 2v/k II + n
2/ 2
Therefore the energy per unit surface is
E - Eo
he 2n
foo k dk (k + L Jk + n n 2
n= 1
/a 2
This quantity is apparently still not defined due to ultraviolet (large k) divergences. However, for wavelengths shorter than the atomic size it is unrealistic to use a perfect conductor approximation. Let us therefore introduce in the above integral a smooth cutoff function f(k) equal to unity for k ;:S km and vanishing for k km where k m is of the order of the inverse atomic size. Set u = a2 k 2 /n; then
= he
:;3 too [ff(~Ju) + ~Ju+nzf(~Ju + n2}
- fo dn
J~ f(~Ju+nz)]
F(l)
g = he
4: [~F(O) +
+ F(2) + ... -
foOO dn F(n)]
Here we have defined
F(n) =
too du Ju+nzf(~Ju+nz)
The interchange of sums and integrals was justified due to the absolute convergence in the presence of the cutoff function. As n -+ 00, F(n) -+ O. We can use the Euler-MacLaurin formula to compute the difference between the sum and integral occurring in the above bracket:
iF(O)
+ F(l) + F(2) + ... -
1 1 dn F(n) = - 2! B 2 F'(O) - 4! B 4 F"'(O)
+ ...
The Bernoulli numbers Bv are defined through the series
Jo Bv V!
and B2 =
-t;, B4 = -fo, .... We have
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We assume that 1(0) = 1, while all its derivatives vanish at the origin, so that F'(O) = 0, F"'(O) = - 4, and higher derivatives of F are equal to zero. All reference to the cutoff has therefore disappeared from the final result
Iff = hen B4 = n he a 3 41 - 720 a3
The force per unit area ff reads
ff = - 240 a4 = - - - yn cm
0.013 d (al'm)4
(3-150)
and its sign corresponds to attraction. This very tiny force has been demonstrated experimentally by Sparnay (1958), who was able to observe both its magnitude and dependence on the interplate distance 1 The above derivation may be criticized on account of the fact that we have seemingly disregarded the effects outside the plates. In the present case they turn out to cancel exactly. The lesson is that vacuum fluctuations manifest themselves under very different circumstances than those encountered in particle creation or absorption. By considering various types of bodies influencing the vacuum configuration we may give an interesting interpretation of the forces acting on them. This is one line of thought that may be kept in mind and in which we may recognize some of the origins of Schwinger's source theory approach to quantum field phenomena.
As another example let us give a brief discussion of Van der Waals forces among neutral atoms or molecules. It can be presented by studying the field fluctuations in the presence of the two systems very much as above. After all, we should be able to give a microscopic description of the Casimir effect among macroscopic bodies using the residual forces among constituents. Rather than actually doing that we shall follow a presentation due to Feinberg and Sucher. A more detailed treatment is given in Chap. 7. The classical energy density of the electromagnetic field is (E 2 + B2)/2, as we recall from Chap. 1. A neutral system may interact with an electromagnetic field. Let aE(aB) be the static electric (magnetic) susceptibility. In a static electric field if p denotes the charge density, Jd 3 x p(x) = 0 because of neutrality. The polarization is defined as P = Jd 3 x p(x)x and the corresponding contribution to the energy due to a change in the electric field bE = - VbV is d 3 x pbV = - d 3 x p(x)x bE = - p. bE if bE is almost constant over the extent of the system. The susceptibility is such that P = aEE for weak fields; hence under a small change bE, the variation of the interaction energy is -aEE' bE = b[ -aE(E 2/2)]. A similar reasoning yields the magnetic contribution, so that for the interaction energy we ha ve
We attempt a relativistic phenomenological quantum generalization of this result. We want to write an interaction part of the lagrangian such that when integrated over three-space it reproduces in a static field and in the nonrelativistic limit, the negative of the above expression. The neutral system may be described by an hermitian scalar field <p, and the only Lorentz scalar quadratic in F. v and in <p having the required property is
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