"There is a difference between whether the universe is infinite or just really really really really really really big," Anthony Aguirre said at the recent FQXi conference in Puerto Rico. I'm pretty sure I counted six reallys. With that remark, he encapsulated a major debate going on within physics and cosmology right now. Although the conference theme was officially the physics of information, it could just as well have been the physics of infinity, so often did that little sideways '8' put in an appearance. Is the universe finite or infinite? Is nature capable of a finite or infinite number of possible states? Can spacetime be infinitely subdivided or is it made of finite-size cells? The questions seem undecidable. But maybe the finitude or infinitude makes itself felt every time you do a measurement and every time you stir cream into a coffee cup and can't unstir it out.
The forces of finitude include Max Tegmark, who has been bad-mouthing infinity on Edge.org, in quotes to New Scientist, and in Chapter 11 of his new book. His complaint is what cosmologists call the measure problem: there's no way of unambiguously counting members of an infinite set. If there's no way to count, there's no way to calculate probabilities and therefore no way to relate theory to experiment. The whole empirical framework of science verges on collapse. A finite universe presents no such difficulty. Even Peter Woit, who agrees with Tegmark on little else, finds common ground with him on the measure problem.
The aficionados of infinity include Alan Guth, who argued in Puerto Rico and on Edge.org that a truly infinite universe would neatly explain the arrow of time. When space has no bound, neither does entropy. It keeps on increasing forever, always pointing the way forward for time. The universe need not have begun in a contrived initial state to create the impetus toward increasing disorder.
When two opposing positions can both muster plausible arguments, what you have is less a debate and more a dilemma. If it were up to them, physicists would surely prefer finitude, yet nature seems to have made different plans. The universe is expanding at an accelerating rate and, if it keeps doing so, it is destined to spawn an infinity of baby universes. "It would be cozy if it were finite, but it doesn't seem to be," Aguirre told me. "Eternal inflation gives you an infinite universe, and something like eternal inflation is happening now and probably happened in the past. Nature is rubbing infinity in our face."
Eternal inflation could cease to be eternal if the dark energy that drives it withered away on a timescale of billions of years. But if dark energy were so unstable, Aguirre has argued, we should see signs of its decay somewhere out there. All indications are that time will never end, which means that space probably doesn't, either.
The brain-melting Boltzmann-brain paradox is one reason that infinite space and infinite time go together. If time were infinite yet space finite, the contents of the universe would cycle through their possible configurations over and over and over again. Molecules would occasionally converge to produce a conscious mind that lasted for a split second, but was under the misimpression it was the product of billions of years of cosmic evolution. Indeed, in the vastness of eternity, such flashes of deluded awareness would vastly outnumber brains that had formed the old-fashioned way, and we'd have to conclude that our observations are implanted memories, like fossils that young-Earth creationists think God planted in rock strata to fool us. It's a paradox because an empirical science would lead us to the conclusion that empirical science is a sham.
This sort of argument is what Aguirre had in mind as a genuine distinction between a truly infinite universe and a merely ginormous one. "If it's finite, no matter how big you make it, it still eventually runs into the paradox," he said. Chance fluctuations that are inevitable in finite space are vanishingly unlikely in infinite space. An infinite universe is ever-changing, never doing the same thing twice, as Sean Carroll eloquently described in his prize-winning essay for the first FQXi essay contest.
The arrow-of-time argument that Guth has been developing also has the potential of distinguishing infinite from finite. The basic idea goes back to a provocative paper a decade ago by Carroll and Jennifer Chen (who has since left physics research to work on energy regulation). Whereas their scenario involved an accelerating universe, Guth gave a supersimple example involving a gas in an infinite void. At some moment you can take as t=0, the gas occupies some minimum volume. From then on, the gas will expand without limit. If the void is finite, the gas will eventually cycle back to its starting point. Time has a clear forward progression only if the void is truly infinite.
The overall history of Guth's minimalist universe is fully time-symmetric, as the laws of physics demand. Prior to t=0, the gas was also expanding without limit, albeit backwards in time, and again time has a clear forward progression, the reverse of the arrow on the other side. Only around t=0 does the arrow become ambiguous. If any mortal beings are alive for the crossover, they'll observe curious reversals of fortune such as those that Ken Wharton, who writes science fiction when not doing physics, once imagined in a poignant short story.
Guth's scenario is classical, but similar intimations of infinity arise in quantum physics. Yasunori Nomura has argued that an infinite range of possible states (that is, an infinite Hilbert space) would make the process of quantum decoherence irreversible, explaining the arrow of time in quantum measurement.
In a funny way, then, the arrow of time we observe in daily life may reflect the infinity of space, and human mortality may hinge on the immortality of the universe. But still. Infinity? Can it be a real thing rather than simply our idealization?
Aguirre, for all his advocacy of infinity, is unconvinced by Guth's and Carroll and Chen's arrow-of-time arguments. "They're brushing certain things under the rug," he said. For instance, they take for granted that, if the maximum possible entropy is infinite, it doesn't matter how the universe began. Any possible initial state has finite entropy, so you get the arrow of time for free. But you can't take anything for granted when it comes to infinity. Guth and the others implicitly rule out initial states with infinite entropy. Is that really justified? Such a state is hard to imagine, but that doesn't mean it can't exist, Aguirre said. His musings remind me of one of the strangest concepts in mathematics: the axiom of choice. This is a rule for selecting objects from an infinite collection even when all standard rules fail. The weird thing is that, although mathematicians know that such a rule exists, they don't know what the rule is. Worse, they know they'll never know what it is. A state of infinite entropy may likewise exist even if it is impossible to specify.
Other speakers in Puerto Rico proposed ways to evade the paradoxes that imply infinity. Carroll himself argued that Boltzmann brains go away when you take care to distinguish quantum from thermal fluctuations. Andy Albrecht contended that the measure problem evaporates when you think of all probabilities as inherently quantum. Even a tossed coin, he said, ultimately lands on one side or the other because of quantum indeterminism. If so, probabilities aren't defined in terms of repeated trials, and the inability to count elements of an infinite set is a red herring.
At this rate, physicists may not have to wonder about infinity. Their discussions may go on long enough to prove the point one way or the other.