The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [52]
This leads to a grand challenge: using pen, paper, possibly a computer, and one’s best understanding of the laws of physics, calculate the particle properties and find results in agreement with the measured values. If we could meet this challenge, we’d take one of the most profound steps ever toward understanding why the universe is as it is.
In quantum field theory, the challenge is insurmountable. Permanently. Quantum field theory requires the measured particle properties as input—these features are part of the theory’s definition—and so can happily accommodate a broad range of values for their masses and charges.12 In an imaginary world where the electron’s mass or charge was larger or smaller than it is in ours, quantum field theory could cope without blinking an eye; it would simply be a matter of adjusting the value of a parameter within the theory’s equations.
Can string theory do better?
One of the most beautiful features of string theory (and the facet that most impressed me when I learned the subject) is that particle properties are determined by the size and shape of the extra dimensions. Because strings are so tiny, they don’t just vibrate within the three big dimensions of common experience; they also vibrate into the tiny, curled-up dimensions. And much as air streams flowing through a wind instrument have vibrational patterns dictated by the instrument’s geometrical form, the strings in string theory have vibrational patterns dictated by the geometrical form of the curled-up dimensions. Recalling that string vibrational patterns determine particle properties such as mass and electrical charge, we see that these properties are determined by the geometry of the extra dimensions.
So, if you knew exactly what the extra dimensions of string theory looked like, you’d be well on your way to predicting the detailed properties of vibrating strings, and hence the detailed properties of the elementary particles the strings vibrate into existence. The hurdle is, and has been for some time, that no one has been able to figure out the exact geometrical form of the extra dimensions. The equations of string theory place mathematical restrictions on the geometry of the extra dimensions, requiring them to belong to a particular class called Calabi-Yau shapes (or, in mathematical jargon, Calabi-Yau manifolds), named after the mathematicians Eugenio Calabi and Shing-Tung Yau, who investigated their properties well before their important role in string theory was discovered (Figure 4.6). The problem is that there’s not a single, unique Calabi-Yau shape. Instead, like musical instruments, the shapes come in a wide variety of sizes and contours. And just as different instruments generate different sounds, extra dimensions that differ in size and shape (as well as with respect to more detailed features we’ll come upon in the next chapter) generate different string vibrational patterns and hence different sets of particle properties. The lack of a unique specification of the extra dimensions is the main stumbling block preventing string theorists from making definitive predictions.
Figure 4.6 A close-up of the spatial fabric in string theory, showing an example of extra dimensions curled up into a Calabi-Yau shape. Like the pile and backing of a carpet, the Calabi-Yau shape would be attached to every point in the familiar three large spatial dimensions (represented by the two-dimensional grid), but for visual clarity the shapes are shown only on grid points.
When I started working on string theory, back in the mid-1980s, there were only a handful of known Calabi-Yau shapes, so one could imagine studying each, looking for a match to known physics. My doctoral dissertation was one of the earliest