The Elegant Universe - Brian Greene [213]
5. For the reader interested in more details of this technical issue we note the following. In note 6 of Chapter 6 we mentioned that the standard model invokes a "mass-giving particle"—the Higgs boson—to endow the particles of Tables 1.1 and 1.2 with their observed masses. For this procedure to work, the Higgs particle itself cannot be too heavy; studies show that its mass should certainly be no greater than about 1,000 times the mass of a proton. But it turns out that quantum fluctuations tend to contribute substantially to the mass of the Higgs particle, potentially driving its mass all the way to the Planck scale. Theorists have found, however, that this outcome, which would uncover a major defect in the standard model, can be avoided if certain parameters in the standard model (most notably, the so-called bare mass of the Higgs particle) are finely tuned to better than 1 part in 1015 to cancel the effects of these quantum fluctuations on the Higgs particle's mass.
6. One subtle point to note about Figure 7.1 is that the strength of the weak force is shown to be between that of the strong and electromagnetic forces, whereas we have previously said that it is weaker than both. The reason for this lies in Table 1.2, in which we see that the messenger particles of the weak force are quite massive, whereas those of the strong and electromagnetic forces are massless. Intrinsically, the strength of the weak force (as measured by its coupling constant—an idea we will come upon in Chapter 12) is as shown in Figure 7.1, but its massive messenger particles are sluggish conveyers of its influence and diminish its effects. In Chapter 14 we will see how the gravitational force fits into Figure 7.1.
7. Edward Witten, lecture at the Heinz Pagels Memorial Lecture Series, Aspen, Colorado, 1997.
8. For an in-depth discussion of these and related ideas, see Steven Weinberg, Dreams of a Final Theory.
Chapter 8
1. This is a simple idea, but since the imprecision of common language can sometimes lead to confusion, two clarifying remarks are in order. First, we are assuming that the ant is constrained to live on the surface of the garden hose. If, on the contrary, the ant could burrow into the interior of the hose—if it could penetrate into the rubber material of the hose—we would need three numbers to specify its position, since we would need to also specify how deeply it had burrowed. But if the ant lives only on the hose's surface, its location can be specified with just two numbers. This leads to our second point. Even with the ant living on the hose's surface, we could, if we so chose, specify its location with three numbers: the ordinary left-right, back-forth, and up-down positions in our familiar three-dimensional space. But once we know that the ant lives on the surface of the hose, the two numbers referred to in the text give the minimal data that uniquely specify the ant's position. This is what we mean by saying that the surface of the hose is two-dimensional.
2. Surprisingly, the physicists Savas Dimopoulos, Nima Arkani-Hamed, and Gia Dvali, building on earlier insights of Ignatios Antoniadis and Joseph Lykken, have pointed out that even if an extra curled-up dimension were as large as a millimeter in size, it is possible that it would not yet have been detected experimentally. The reason is that particle accelerators probe the microworld by utilizing the strong, weak, and electromagnetic forces. The gravitational force, being incredibly feeble at technologically accessible energies, is generally ignored. But Dimopoulos and his collaborators note that if the extra curled-up dimension has an impact predominantly on the gravitational force (something, it turns out, that is quite plausible in string theory), all extant experiments