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The vector potential is then (Power, 1964) a~t and a~ are the creation and destruction operators, respectively, of a photon in the standing-wave mode specified by k and i". There are two values of A. for each value of k, and k and the unit vectors €(l) and €(2) form an orthogonal set. The vectors k and e'A are analogous to the wave vector and polarization vector for a plane-wave mode. With Cartesian subscripts denoting components, and with O>k = kc = Ikl c, the wave vector components kx, ky, and k z are nonnegative integral multiples of ",/L, and = 8112e~ cos(kxX) sin(kyY) sin(kzz) f~y = 8112e; sin(kxx) cos(kyY) sin(kzz) f~ = 8112e~ sin(kxx) sin(kyY) cos(kzz) f~ 44 LARRy SPRUCH Note that The time-dependent vector potential is A(r, t) = L A~(r) COS(ookt +

There is then an interference term in which the particles exchange one photon and one graviton. /(c3 r3 ) This interaction is linear in G and is therefore very much larger than the interaction quadratic in G given just above. 5. Dimensional Analysis The various retardation interaction forms obtained thus far can be obtained without any calculations if we combine certain insights with dimensional analysis. In this approach, however, we give up the possibility of determining the domain of validity of the retardation result obtained and of determining the correct numerical coefficient.

3) where p = 6 for z «Ro and p = 7 for z »Ro. The integration is straightforward, and one obtains E61A ex 1/z2 and E,IA ex 1/z 3, with the coefficients determined, and differentiation gives F IA ex 1/Z3 for z « Ro and FIA ex 1/z 4 for z » R o. For a beautiful treatment which is valid, formally, for any pair of walls, characterized by their real and imaginary frequency-dependent dielectric constants, see Landau and Lifshitz (1960), the original paper by Lifshitz (1955), and Dzyaloshinskii et al.

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