The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [73]
Figure 5.6 Parts of the extra dimensions in string theory can be wrapped by branes and threaded by fluxes, yielding “dressed-up” Calabi-Yau shapes. (The figure uses a simplified version of a Calabi-Yau shape—a “three-hole doughnut”—and represents wrapped branes and flux lines schematically with glowing bands encircling portions of the space.)
A rough count gives a sense of scale. Focus on fluxes. Just as quantum mechanics establishes that photons and electrons come in discrete units—you can have 3 photons and 7 electrons, but not 1.2 photons or 6.4 electrons—so quantum mechanics shows that flux lines also come in discrete bundles. They can penetrate a surrounding surface once, twice, three times, and so on. But apart from this restriction to whole numbers, there’s in principle no other limit. In practice, when the amount of flux is large, it tends to distort the surrounding Calabi-Yau shape, rendering previously reliable mathematical methods inaccurate. To avoid venturing into these more turbulent mathematical waters, researchers typically consider only flux numbers that are about 10 or less.9
This means that if a given Calabi-Yau shape contains one open region, we can dress it up with flux in ten different ways, yielding ten new forms for the extra dimensions. If a given Calabi-Yau has two such regions, there are 10 × 10 = 100 different flux dressings (10 possible fluxes through the first paired with 10 through the second); with three open regions there are 103 different flux dressings, and so on. How large can the number of these dressings get? Some Calabi-Yau shapes have on the order of five hundred open regions. The same reasoning yields on the order of 10500 different forms for the extra dimensions.
In this way, rather then winnowing the candidates to a few specific shapes for the extra dimensions, the more refined mathematical methods have led to a cornucopia of new possibilities. All of a sudden, Calabi-Yau spaces can clothe themselves with far more outfits than there are particles in the observable universe. For some string theorists, this caused great distress. As emphasized in the previous chapter, without a means of choosing the exact form for the extra dimensions—which we now realize means also selecting the flux outfit that shape wears—the mathematics of string theory loses its predictive power. Much hope had been placed on mathematical methods that could go beyond the limitations of perturbation theory. Yet, when some of those methods materialized, the problem of fixing the form for the extra dimensions only got worse. Some string theorists lost heart.
Others, more sanguine, believe it’s too early to give up hope. One day—perhaps a day that’s just around the corner, perhaps a day that’s far off—we will discover the missing principle that determines what the extra dimensions look like, including the fluxes the shape may be sporting.
Others still have taken a more radical tack. Maybe, they suggest, the decades of fruitless attempts to pin down the form for the extra dimensions are telling us something. Maybe, these radicals brazenly continue, we need to take seriously all of the possible shapes and fluxes emerging from string theory’s mathematics. Maybe, they urge, the reason the mathematics contains all these possibilities is that they’re all real, each shape being the extra-dimensional part of its own separate universe. And maybe, grounding a seemingly wild flight of fancy in observational data, this is just what’s needed to address perhaps the thorniest problem of all: the cosmological constant.
*You can think of this as a grand generalization of the results touched on in Chapter 4, in which different forms for the extra dimensions