ATM theorists were striving to generalize this result: to explain the (relatively) smooth ten-dimensional “total space” of the Standard Unified Field Theory—whose properties accounted for all four forces: strong, weak, gravitational, and electromagnetic—as the net result of an infinite number of elaborate geometrical structures.
Nine spatial dimensions (six rolled up tight), and one time, was only what total space appeared to be if it wasn’t examined too closely. Whenever two subatomic particles interacted, there was always a chance that the total space they occupied would behave, instead, like part of a twelve-dimensional hypersphere, or a thirteen-dimensional doughnut, or a fourteen-dimensional figure eight, or just about anything else. In fact—just as a single photon could travel along two different paths at once—any number of these possibilities could take effect simultaneously, and “interfere with each other” to produce the final outcome. Nine space, one time, was nothing but an average.
There were two main questions still in dispute among ATM theorists:
What, exactly, was meant by “all” topologies? Just how bizarre could the possibilities contributing to the average total space become? Did they have to be, merely, those which could be formed with a twisted, knotted sheet of higher-dimensional plastic—or could they include states more like a (possibly infinite) handful of scattered grains of sand— where notions like “number of dimensions” and “space-time curvature” ceased to exist altogether?
And: how, exactly, should the average effect of all these different structures be computed? How should the sum over the infinite number of possibilities be written down and added up when the time came to test the theory: to make a prediction, and calculate some tangible, physical quantity which an experiment could actually measure?
On one level, the obvious response to both questions was: “Use whatever gives the right answers"—but choices which did that were hard to find… and some of them smacked of contrivance. Infinite sums were notorious for being either intractable, or too pliable by far. I jotted down an example—remote from the actual tensor equations of ATMs, but good enough to illustrate the point:
Let S = 1-1+1-1+1-1+1- …
Then S = (1-1) + (1-1) + (1-1) + … = 0 + 0 + 0 … = 0
But S = 1 + (-1+1) + (-1+1) + (-1+1) … = 1 + 0 + 0 + 0 … = 1
It was a mathematically naive “paradox"; the correct answer was, simply, that this particular infinite sequence didn’t add up to any definite sum at all. Mathematicians would always be perfectly happy with such a verdict, and would know all the rules for avoiding the pitfalls—and software could assess even the most difficult cases. When a physicist’s hard-won theory starred generating similarly ambiguous equations, though, and the choice came down to strict mathematical rigor and a theory with no predictive power at all… or, a bit of pragmatic side-stepping of the rules, and a theory which churned out beautiful results in perfect agreement with every experiment… it was no surprise that people were tempted. After all, most of what Newton had done to calculate planetary orbits had left contemporary mathematicians apoplectic with rage.
Violet Mosala’s approach was controversial for a very different reason. She’d been awarded the Nobel prize for rigorously proving a dozen key theorems in general topology—theorems which had rapidly come to comprise a standard mathematical toolbox for ATM physicists, obliterating stumbling blocks and resolving ambiguities. She’d done more than anyone else to provide the field with solid foundations, and the means of making careful, measured progress. Even her fiercest critics agreed that her mathematics was meticulous, beyond reproach.
The trouble was, she told her equations too much about the world.
The ultimate test of a TOE was to answer questions like: “What is the probability of a ten-gigaelectronvolt neutrino fired at a stationary proton scattering off a down quark and emerging at a certain angle?"… or even just: “What is the mass of an electron?” Essentially, Mosala prefixed all such questions with the condition: “Given that we know that space-time is roughly four-dimensional, and total space is roughly ten-dimensional, and the apparatus used to perform the experiment consists, approximately, of the following…”
Her supporters said she was merely setting everything in context. No experiment happened in isolation; quantum mechanics had been hammering that point home for the last hundred and twenty years. Asking a Theory of Everything to predict the chance of observing some microscopic event—without adding the proviso that “there is a universe, and it contains, among other things, equipment for detecting the event in question"—would be as nonsensical as asking: “If you pick a marble out of a bag, what are the odds that it will be green?”
Her critics said she used circular reasoning, assuming from the very beginning all the results she was trying to prove. The details she fed into her computations included so much about the known physics of the experimental apparatus that—indirectly, but inevitably—they gave the whole game away.
I was hardly qualified to come down on either side… but it seemed to me that Mosala’s opponents were being hypocritical, because they were pulling the same trick under a different guise: the alternatives they offered all invoked a cosmological fix. They declared that “before” the Big Bang and the creation of time (or “adjoining” the event, to avoid the oxymoron), there had been nothing but a perfectly symmetrical “pre-space,” in which all topologies carried equal weight… and the “average result” of most familiar physical quantities would have been infinite. Pre-space was sometimes called “infinitely hot"; it could be thought of as the kind of perfectly balanced chaos which space-time would become if so much energy was poured into it that literally everything became equally possible. Everything and its opposite; the net result was that nothing happened at all.
But some local fluctuation had disturbed the balance in such a way as to give rise to the Big Bang. From that tiny accident, our universe had burst into existence. Once that had happened, the original “infinitely hot,” infinitely even-handed mixture of topologies had been forced to become ever more biased, because “temperature” and “energy” now had a meaning—and in an expanding, cooling universe, most of the “hot” old symmetries would have been as unstable as molten metal thrown into a lake. And when they’d cooled, the shapes into which they’d frozen had just happened to favor topologies close to a certain ten-dimensional total space—one which gave rise to particles like quarks and electrons, and forces like gravity and electromagnetism.
By this logic, the only correct way to sum over all the topologies was to incorporate the fact that our universe had—by chance—emerged from pre-space in a certain way. Details of the broken symmetry had to be fed into the equations “by hand"—because there was no reason why they couldn’t have been utterly different. And if the physics resulting from this accident seemed improbably conducive to the formation of stars, planets, and life… then this universe was just one of a vast number which had frozen out of pre-space, each with a different set of particles and forces. If every possible set had been tried, it was hardly surprising that at least one of them had turned out to be favorable to life.