have been thinking for years that our universe might be just one bubble
amid countless bubbles floating in a formless void. And when they say
“countless,” they really mean it. Those universes are damned hard to
count. Angels on a pin are nothing to this. There’s no unambiguous way
to count items in an infinite set, and that’s no good, because if you
can’t count, you can’t calculate probabilities, and if you can’t
calculate probabilities, you can’t make empirical predictions, and if
you can’t make empirical predictions, you can’t look anyone in the eye
at scientist wine-and-cheese parties. In a Sci Am article
last year, cosmologist Paul Steinhardt argued that this counting
crisis, or “measure problem,” is reason to doubt the theory that
predicts bubble universes.
Other cosmologists think they just need to learn how to count better. In April I went to a talk by Leonard Susskind
(silhouetted in the photo above), who has been arguing for a decade
that you don’t need to count all the parallel universes, just those that
are capable of affecting you. Forget the causally disconnected ones and
you might have a shot at recovering your empiricist credentials.
“Causal structure is, I think, all important,” Susskind said. He
presented a study he did last year with three other Stanford physicists, Daniel Harlow, Steve Shenker, and Douglas Stanford. I didn’t follow everything he said, but I was enamored of a piece of mathematics he invoked, known as p-adic
numbers. As I began to root around, I discovered that these numbers
have inspired an entire subfield within fundamental physics, involving
not just parallel universes but also the arrow of time, dark matter, and
the possible atomic nature of space and time.
Lest you think that the whole notion of parallel universes was
ill-starred to begin with, cosmologists have good cause to think our
universe is just one member of a big dysfunctional family.
The universe we see is smooth and uniform on its largest scales, yet it
hasn’t been around long enough for any ordinary process to have
homogenized it. It must have inherited its smoothness and uniformity
from an even larger, older system, a system permeated with dark energy
that drives space to expand rapidly and evens it out—the process known
as cosmic inflation. Dark energy also destabilizes the system and causes
universes to nucleate out like raindrops in a cloud. Voilà, our
bubbles are nucleating all the time. Each gains its own endowment of
dark energy and can give rise to new bubbles—bubbles within bubbles
within bubbles, an endless cosmic effervescence. Even our universe has a
dab of dark energy and can birth new bubbles. The space between the
baby bubbles expands, keeping them isolated from one another. A bubble
has contact only with its parent.
The process produces a family tree of universes. The tree is a
fractal: no matter how closely you zoom in, it looks the same. In fact,
the tree is a dead ringer for one of the most famous fractals of all,
the Cantor set. In a simplified case, if you start with a single universe, by the Nth generation, you have 2N
of them. You label each universe by a binary number giving its position
in the structure. After the first bubble nucleation, you have two
universes, the inside and outside of the bubble: 0 and 1. In the first
generation, universe 0 spawns 00 and 10, and universe 1 spawns 01 and
11. Then, universe 00 gives birth to 000 and 100, and so it goes.
The process goes on forever, approaching a continuum of universes
(the red line at the top of the diagram) indexed by numbers with an
infinity of bits. The fun thing is that these numbers are not
standard-issue infinite-digit numbers like 1.414… (√2) or 3.1415… (π),
which mathematicians call “real” numbers—the ones you find on a
grade-school number line. Instead they are so-called 2-adic numbers with
very different mathematical properties. In a more general setup, each
universe could fork into p universes rather than just two, hence the general term p-adic.
Mathematicians came up with p-adic numbers in the late 19th
century as an alternative way, besides real numbers, to fill in the
spaces between integers and integer fractions to make an uninterrupted
block of numbers. In fact, Russian mathematician Alexander Ostrowski
showed that p-adics are the only alternative to the reals.
Unfortunately, mathematicians have done a good job of smothering the beauty beneath formal definitions,
theorems, lemmas, and corollaries that dot every ‘i’ but never tell you
what they’re spelling out. (My mathematician friends, too, complain
that math texts are as compelling to read as software license
agreements.) It wasn’t until I heard Susskind’s description in terms of
counting parallel universes that I had a clue what p-adics were or appreciated their sheer awesomeness.
What differentiates p-adics from reals is how distance is defined. For them, distance is the degree of consanguinity: two p-adics
are close by virtue of having a recent common ancestor in their family
tree. Numerically, if two points have a common ancestor in the Nth generation, those points are separated by a distance of 1/2N. For instance, to find a common ancestor of the numbers 000 and 111, you have to go all the way back to the root of the tree (N=0).
Thus these numbers are separated by a distance of 1—the full width of
the multiverse. For the numbers 000 and 110, the most recent common
ancestor is the first generation (N=1), so the distance is 1/2. For 000 and 100, the distance is 1/4.
To put it another way, if someone gives you two p-adic
numbers, you determine the distance between them using the following
procedure. Line them up, one on top of the other. Compare the rightmost
bits. If they’re different, stop! You’re done. The distance is 1. If
they’re the same, shift to the left and compare the next bits over. If
they’re different, stop! The distance is 1/2. Keep going until you find
the first bit that is different. This bit—and none other—determines the
distance rule messes with your mind. Two parallel universes that look
nearby can be far apart because they lie on different branches of the
tree. Likewise, two points that look far apart might be nearby. In the
figure at left, universe ‘B’ is closer to universe ‘C’ than to ‘A’. What
is more, the number 100 is smaller than the number 10, since it is
closer to the far left side of the multiverse. With p-adics, you gain precision by adding digits to the left side of the number rather than to the right. Accordingly, mathematician Andrew Rich and undergraduate Matthew Bauman have dubbed them “leftist numbers.” p-adics can be added, subtracted, multiplied, and divided like any other self-respecting number, but their leftist proclivities change the rules and make arithmetic unexpectedly easier. To add two p-adics,
you start with the most significant digit (on the right) and add them
one by one toward the least significant digits (on the left). With
reals, on the other hand, you start with the least significant digit,
and you’re out of luck if you have a number such as π with an infinite
number of digits.
The weirdness doesn’t stop there. Consider three p-adic
numbers. You can think of them as the three corners of a triangle.
Oddly, at least two sides of the triangle must have the same length; p-adics,
unlike reals, don’t give you the liberty to make the sides all
different. The reason is evident from the tree diagram: there is only
one path from one number to the other two numbers, hence at most two
common ancestors, hence at most two different lengths. In the jargon, p-adics are “ultrametric.” On top of that, distance is always finite. There are no p-adic infinitesimals, or infinitely small distances, such as the dx and dy you see in high-school calculus. In the argot, p-adics are “non-Archimedean.” Mathematicians had to cook up a whole new type of calculus for them.
Prior to the multiverse study, non-Archimedeanness was the main
reason physicists had taken the trouble to decipher those mathematics
textbooks. Theorists think that the natural world, too, has no infinitely small distances; there is some minimal possible distance,
the Planck scale, below which gravity is so intense that it renders the
entire notion of space meaningless. Grappling with this granularity has
always vexed theorists. Real numbers can be subdivided all the way down
to geometric points of zero size, so they are ill-suited to describing a
granular space; attempting to use them for this purpose tends to spoil
the symmetries on which modern physics is based.
By rewriting their equations using p-adics instead, theorists think they can capture the granularity in a consistent way, as Igor Volovich of the Steklov Mathematical Institute in Moscow argued in 1987. The resulting dynamics might even explain dark matter and the mechanics of cosmic inflation. Naturally,
having found a new toy to play with, physicists immediately wonder how
to break it. Susskind and his colleagues took the tree of parallel
universes, lopped off some of its branches, and figured out how it would
deform the p-adics. Those pruned branches represented
infertile baby universes: those born with zero dark energy or a negative
density of the stuff. Just as pruning a real tree might seem
destructive but actually helps it to grow, pruning the tree of universes
mucks up its symmetry but does so in a good cause: it explains, the
team argued, why time is unidirectional—why the past is different from
the future. p-adics are a case study of how a concept mathematicians invented for its own beauty might turn out to have something to do with the real world. What a bonus that they may be more real than the reals. Photograph courtesy of Gary Smaby. Bubble figure courtesy of George Musser. Tree figures courtesy of Daniel Harlow, Stanford University.
About the Author: George Musser is a contributing editor at Scientific American.
He focuses on space science and fundamental physics, ranging from
particles to planets to parallel universes. He is the author of The Complete Idiot's Guide to String Theory.
Musser has won numerous awards in his career, including the 2011
American Institute of Physics's Science Writing Award. Follow on Twitter