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ever-smaller distances (which is why we don’t see these jitters in everyday life). Specifically, the calculations show that it is the wildly energetic jitters over distances shorter than the Planck scale that make the math go haywire (the shorter the distance, the greater the jitters’ energy). Since quantum field theory describes particles as points with no spatial extent, the distances these particles probe can be arbitrarily small, and hence the quantum jitters they feel can be arbitrarily energetic. String theory changes this. Strings are not points—they have spatial extent. This implies that there is a limit to how small a distance can be accessed, even in principle, since a string can’t probe a distance smaller than its own size. In turn, a limit to how small a scale can be probed translates into a limit on how energetic the jitters can become. This limit proves sufficient to tame the unruly mathematics, allowing string theory to merge quantum mechanics and general relativity.

9. If an object were truly one-dimensional, we wouldn’t be able to see it directly since it would offer no surface from which photons could reflect and would have no capacity to produce photons of its own through atomic transitions. So, when I say “see” in the text, that’s a stand-in for any means of observation or experimentation you might use to seek evidence of an object’s spatial extent. The point, then, is that any spatial extent smaller than the resolving power of your experimental procedure will escape your experiment’s notice.

10. “What Einstein Never Knew,” NOVA documentary, 1985.

11. More precisely, the component of the universe most relevant to our existence would be completely different. Since the familiar particles and the objects they compose—stars, planets, people, etc.—amount to less than 5 percent of the mass of the universe, such a disruption would not affect the vast majority of the universe, at least as measured by mass. However, as measured by its effect on life as we know it, the change would be profound.

12. There are some mild restrictions that quantum field theories place on their internal parameters. To avoid certain classes of unacceptable physical behavior (violations of critical conservation laws, violations of certain symmetry transformations, and so on), there can be constraints on the charges (electric and also nuclear) of the theory’s particles. Additionally, to ensure that in all physical processes, probabilities add to 1, there can also be constraints on particle masses. But even with these constraints, there is wide latitude in the allowed values of particle properties.

13. Some researchers will note that even though neither quantum field nor our current understanding of string theory provides an explanation of the particle properties, the issue is more urgent in string theory. The point is a bit involved, but for the technically minded here’s the summary. In quantum field theory, the properties of particles—say their masses, to be definite—are controlled by numbers that are inserted into the theory’s equations. The fact that quantum field theory’s equations allow such numbers to be varied is the mathematical way of saying that quantum field theory does not determine particle masses but instead takes them as input. In string theory, the flexibility in the masses of particles has a similar mathematical origin—the equations allow particular numbers to vary freely—but the manifestation of this flexibility is more significant. The freely varying numbers—numbers, that is, that can be varied with no cost in energy—correspond to the existence of particles with no mass. (Using the language of potential energy curves introduced in Chapter 3, envision a potential energy curve that’s completely flat, a horizontal line. Just as walking on a perfectly flat terrain would have no impact on your potential energy, changing the value of such a field would have no cost in energy. Since a particle’s mass corresponds to the curvature of its quantum field’s potential energy curve around its minimum, the quanta of such fields