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By Root 2006 0

of outstanding mathematical problems. One that I’m particularly fond of has to do with counting the number of spheres that can be packed (in a particular mathematical way) within a given Calabi-Yau shape. Mathematicians had been interested in this question for a long time but found the calculations in all but the simplest cases impenetrable. Take the Calabi-Yau shape of Figure 4.6. When a sphere is packed into this shape, it can wrap around a portion of the Calabi-Yau multiple times, much like a lasso can wrap multiple times around a beer barrel. So, how many ways can you pack a sphere into this shape if it wraps around, say, five times? When asked a question like this, mathematicians had to clear their throats, glance at their shoes, and quickly depart for pressing appointments. String theory flattened the hurdles. By translating such calculations into far easier ones on a paired Calabi-Yau shape, string theorists produced answers that knocked mathematicians back on their heels. The number of five-times-wrapped spheres packed into the Calabi-Yau in Figure 4.6? 229,305,888,887,625. And if the spheres wrap around themselves ten times? . Twenty times? . These numbers proved to be harbingers for a spectrum of results that have opened a whole new chapter in mathematical discovery.17

So, whether or not string theory offers a correct approach to describing the physical universe, it has already established itself as a potent tool for investigating the mathematical one.

The State of String Theory: An Evaluation

Building on the last four sections, Table 4.2 provides a status report for string theory, including some additional observations that I didn’t explicitly call out in the text above. It paints a picture of a theory in progress, one that has produced stunning achievements but has not yet been tested on the most important scale: experimental confirmation. The theory will remain speculative until a convincing link to experiment or observation is forged. Establishing such a link is the great challenge. But it’s not a challenge that’s peculiar to string theory. Any attempt to unite gravity and quantum mechanics enters a domain that’s far beyond the cutting-edge of experimental research. It’s part and parcel of taking on such a supremely ambitious goal. Pushing the fundamental boundaries of knowledge, seeking answers to some of the deepest questions contemplated during the past few thousand years of human thought, is a formidable undertaking, one that won’t likely be completed overnight. Nor in a handful of decades.

In evaluating the state of the art, many string theorists argue that a crucial next step is to articulate the theory’s equations in their most exact, useful, and comprehensive form. Much of the research during the theory’s first couple of decades, through the mid-1990s, was carried out using approximate equations that many were convinced could reveal the theory’s gross features but were too coarse to yield refined predictions. Recent advances, to which we will now turn, have catapulted understanding far beyond what could be achieved by the approximate methods. While definitive predictions have remained elusive, a new perspective has emerged. It’s come from a series of breakthroughs that has opened grand new vistas on the theory’s potential implications, among which are new varieties of parallel worlds.

Table 4.2. A summary status report for string theory.

GOAL: Unite gravity and quantum mechanics

IS GOAL REQUIRED?: Yes.

The primary goal is to meld general relativity and quantum mechanics.

STATUS: Excellent.

A wealth of calculations and insights attest to string theory’s successful merger of general relativity and quantum mechanics.18

GOAL: Unify all forces

IS GOAL REQUIRED?: No.

Unification of gravity and quantum mechanics does not require a further unification with the other forces of nature.

STATUS: Excellent.

While not required, a fully unified theory has long been a goal of physics research. String theory achieves this goal by describing all forces in the same manner—their quanta are strings executing