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calculating how large the cosmological constant must be in order to explain these irregularities) [523]. A more rigorous upper bound than that found in the preceding paragraph was obtained by Anderson et al. [21] using data from the Voyager spacecraft to calculate the fractional change in period of Uranus and Neptune. Their analysis shows that the energy density of the vacuum must be smaller than 10 14 GeV/m 3. To estimate the gravitational effect of the electromagnetic zero-point energy predicted by theory, we can adopt the Planck energy as a cutoff. This is the energy at which the gravitational interaction becomes as strong as the other three fundamental forces of nature (i.e., the scale at which we expect the current theory to break down). This energy is about 10 19 GeV. This yields a zero-point energy density of about 10 121 GeV/m 3 ; that is, the disagreement between this and the experimental upperbound derived in the preceding paragraph is of 107 orders of magnitude: the largest disagreement bet'feen theory and experiment ever seen in the history of physics! In analogy with the ultraviolet catastrophe of blackbody radiation that lead Planck to introduce his constant in physics, this has been called the vacuum catastrophe. In 1967, Zeldovich [663] pointed out that the vacuum zero-point energy could account for the cosmological constant [483,629]. The cosmological constant is a term that Einstein introduced to his field equations in 1917 when he was trying to apply them to the entire universe. At that time it was generally believed that the universe would be static. Without this constant term to counterbalance the gravitational attraction of matter, the universe would collapse. The connection between the vacuum zero-point energy and the cosmological constant is particularly relevant today, because in 1998 new astronomical observations revealed that the expansion of the universe is actually accelerating45 [482, 519, 538]. These observations point to a vacuum zero-point energy of about 70% of the total energy density of the universe. Unfortunately, this is still too low to be accounted for by the current fundamental field theory. Some researchers consider the vacuum catastrophe a sign of the breakdown of our current physical theories, something that could only be resolved within the context of a bigger theory, where gravity and quantum mechanics would be unified [27].

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To review the role of Poisson brackets in classical mechanics, see Goldstein's book [240]. You might also like to read Lanczos's interesting historical account [385] of the origins of the Poisson bracket and its introduction in quantum mechanics, as well as have a look at [62] and [143] . Our introduction to canonical quantization followed closely Dirac's own approach in [159]. For an account of how Dirac hit upon the idea of canonical quantization, there is nothing better than getting it straight "from the horse's mouth" in [158].

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If you are interested in knowing how canonical quantization can be applied to systems with constraints, check out [160].

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For a discussion on the shortcomings of Dirac's formalism in quantum mechanics, see [231]. Even though position was represented by an operator in Section 2.1, time was only a parameter. There have been attempts to promote time to an operator, too [10,85]. But the modem approach in quantum field theory is to make position also a parameter, as in Section 2.4. The lack of a time operator implies that the time-energy uncertainty relation does not have the same status as the positionmomentum uncertainty relation in nonrelativistic quantum mechanics, for it cannot be derived from fundamental commutation relations [649]. We will get back to this issue in 3. Some of the key papers on the old quantum theory are collected in [591]. Key papers on QED can be found in [544]. For the early history of QED and a nice description of Born and Jordan's seminal work [68], see [447]. Key papers from the early days of quantum optics can be found in [359]. The latter reference also has a very pedagogical introduction to field quantization. Two very good introductions to Feynman's version of quantum mechanics are the marvelous book by Feynman and Hibbs [204] and the very interesting paper [430]. For a popular version of Feynman's approach to QED, see [201] (another interesting popular book related to this chapter is Podolny's book about the vacuum [493]). For a more detailed account of Feynman's approach to QED, see [203]. If you would like to know more about Green's functions, reference [642] gives a brief elementary account of them. For a more complete account, see Barton's excellent book [40]. 22 of [2061 shows how cavities can be seen as the high-frequencycounterparts of ordinary electronic lumped-circuit oscillators (e.g., LC oscillators). For a detailed calculation of perfect cavity modes, see Appendix A and also 2 of Haus's book [276]. The formal analogy between Poisson brackets and commutators can also be used to do calculations in classical electrodynamics very similar to the way they are done in quantum electrodynamics (i.e., with mode expansions and using the harmonic oscillator Poisson brackets satisfied by the expansion coefficients). This field oscillator approach is described and applied in [90, 233].

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If you have trouble following the various manipulations of vector expressions involving \l in Section 2.2 and in Appendix A, have a look at Appendix E.

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