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Ed, 16, 2191 (1978) Kennedy, J P and B Ivan, Designed Polymers by Carbocationic Macromolecular Engineering Theory and Practice, Hanser, Munich, 1992 Kennedy, J P and S Jacob, Acc Chem Res, 31, 835 (1998) Kennedy, J P and J E Johnston, Adv Polym Sci, 19, 57 (1975) Kennedy, J P and E Marechal, Carbocationic Polymerization, Wiley-Interscience, New York, 1982..



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Looking at (5.130), we see that the last two terms in (5.133) should not have been present. The presence of these two terms shows that HA is not gauge invariant. But this is not very surprising, as HA does not include the radiation field. Any system of charged particles subject to acceleration would also involve electromagnetic radiation. Let us then complement HA adding the Hamiltonian of the radiation field, HF, which was derived in 2~ and a,yet u~known c:.ouplin~ ten!l ih. The complete Hamiltonian'is given by H = HA + HI + HF. As H'p = HF, HI must be such that when transformed, it generates terms that cancel the last two terms on the right-hand side of (5.133). In other words, writing HI as a functional of the electromagnetic potentials, HI [A, cPl, it must be such that (5.134)





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To edit a function, click the cell containing the function that you want to edit, click the Formulas tab and click the Insert Function button. Excel displays the function s Function Arguments dialog box, where you can change the cell references or values as needed.

Kennedy, J P and R A Smith, J Polym Sci Polym Chem Ed, 18, 1523, 1539 (1980) Kennedy, J P and R G Squires, Polymer, 6, 579 (1965); J Macromol Sci Chem, A1, 861 (1967) Kennedy, J P, L S Minckler, G Wanless, and R M Thomas, J Polym Sci, A2, 2093 (1964) Kennedy, J P, S Y Huang, and S C Feinberg, J Polym Sci Polym Chem Ed, 15, 2801, 2869 (1977) Kishimoto, Y, S Aoshima, and T Higashimura, Macromolecules, 22, 3877 (1989) Klemarczyk, P, Polymer, 42, 2837 (2001) Knauss, D M, H A Al-Muallem, T Huang, and D T Wu, Macromolecules, 33, 3557 (2000) Kobayashi, M, T Ishizone, and S Nakahama, Macromolecules, 33, 4411 (2000) Koroskenyi, B, L Wang, and R Faust, Macromolecules, 30, 7667 (1997) Kraft, R, A H E Muller, H Hocker, and G.

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As you can easily verify, the minimal-coupling interaction derived in Section 5.2 is the solution of this equation. Without the minimal-coupling interaction, the Hamiltonian is not gauge invariant. Historically, the form of the coupling between matter and radiation was known long before the discovery of the symmetry under gauge transformations of the quantummechanical system of charged particles interacting with the electromagnetic field [315]. This symmetry was discovered by Fock in 1926 [215]. But the idea of gauge invariance as a basic unifying principle predates quantum mechanics and can be traced back [413,414] to Weyl's [636] failed attempt to unify electromagnetism and gravitation in 1919 [315]. Much later, in 1928, Weyl wrote [637]: "Butl now believe that this gauge invariance does not tie together electricity and gravitation, but rather electricity and matter in the manner described above." The quantum nature of the electromagnetic field gives rise to many physical phenomena, some of which we mentioned earlier. In 6 we use the formalism we have presented here to study spontaneous emission. First, we examine spontaneous emission in free space and show how it is triggered by the zero-point fluctuations in the field. Then we describe how a cavity can alter this zero-point field and the consequences it has for spontaneous emission.

V Schulz, Makromol Chem Rapid Commun, 1, 363 (1980) Kraft, R, A H E Muller, V Warzelhan, H Hocker, and G V Schulz, Macromolecules, 11, 1093 (1978) Kresge, E N, R H Schatz, and H-C Wang, Isobutylene Polymers, pp 423 448 in Encyclopedia of Polymer Science and Engineering, 2nd ed, Vol 8, H F Mark, N M Bikales, C G Overberger, and G Menges, eds, Wiley-Interscience, New York, 1987 Kresta, J E, R J Chang, S Kathiriya, and K C Frisch, Makromol Chem, 180, 1081 (1979) Kubisa, P and O Vogl, Polymer, 21, 525 (1980) Kubisa, P, K Neeld, J Starr, and O Vogl, Polymer, 21, 1433 (1980) Kubota, H, V Ya Kabanov, D R Squire, and V T Stannett, J Macromol Sci Chem, A12, 1299 (1978) Kunitake, T and K Takarabe, Macromolecules, 12, 1061, 1067 (1979) Kunkel, D.

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