Into the Eleventh Dimension
The quest for a theory linking all matter and all forces led physicists
deep into hyperspace, where they got horribly lost.
But suddenly the way ahead has become clear,
says superstring theorist.
Is there a Final Theory in physics? Will we one day have a
complete theory that will explain everything from subatomic particles, atoms
and supernovae to the big bang?
Einstein spent the last 30 years of his life
in a fruitless quest for the fabled unified field theory. His approach has
since been written off as futile.
In the 1980s, attention switched to
superstring theory as the leading
candidate for a final theory. This revolution began when physicists realised
that the subatomic particles found in nature, such as electrons and quarks,
may not be particles at all, but tiny vibrating strings.
theory was a stunning breakthrough. It became one of the fastest growing
and most exciting areas of theoretical physics, generating a feverish outpouring
of thousands of papers. Then, in the early 1990s, progress seemed to grind
to a halt. People became discouraged when they failed to find the answers
to two key questions: where do strings come from, and is our Universe among
the many solutions of superstring theory? But now the Internet is buzzing
again as papers pour in to the bulletin board at Los Alamos National laboratory
in New Mexico, the official clearing house for superstring papers.
The trigger for this excitement was the discovery of "M-theory",
which may answer those two vital questions about superstrings. "I may be
biased on this one, but I think it is perhaps the most important development
not only in string theory, but also in theoretical physics at least in the
past two decades," says Harvard physicist Cumrun Vafa. M-theory led John
Schwarz of Caltech, one of the founders of superstring theory, to proclaim
a "second superstring revolution". And it inspired a spellbinding three-hour
lecture by another leading exponent,Edward Witten of the Institute for Advanced
Study at Princeton, New Jersey. The aftershocks of the breakthrough have
spread to other disciplines, too. "The excitement I sense in the people in
the field and the spin-offs into my own field of mathematics...have really
been quite extraordinary," says Phillip Griffiths, director of the Institute
for Advanced Study." I feel I've been very privileged to witness this first
In one dazzling stroke, M-theory has come close to solving
superstring theory's two long-standing questions , leaving many theoretical
physicists (myself included) gasping at its power. M-theory, moreover, may
even force string theory to change its name because, although many features
of M-theory are still unknown , it does not seem to be a theory purely of
strings. Other strange beasts seem to emerge, including various types of
membranes. Michael Duff of Texas A&M University is already giving talks
with the title " The theory formerly known as strings".
M-theory does not prove the final correctness of superstring
theory. Not by any means. Proving or disproving its validity may take years
more. But it still marks an astonishing breakthrough. Remember that some
of the finest minds of this century have been stumped by the problem of creating
a "Theory of
Everything". Einstein summed up the problem when he said: Nature shows
us only the tail of the lion. But I do not doubt that the lion belongs to
it even though he cannot at once reveal himself because of his enormous size."
The tail" is what we see in nature , which can be described by the four
fundamental forces -gravity, electromagnetism and the strong and weak nuclear
forces. The lion is the ultimate theory that will unify them in one short
Today, physicists believe that the first force, gravity, can
be described by Einstein's general relativity, based on the smooth warping
of the fabric of space- time. This is an elegant theory that describes the
macroscopic world of black holes,
quasars and the big bang. But gravity has stubbornly refused to unite with
the other three forces , which are described by quantum
theory. Here, instead of the smooth fabric of space-time, we have the
discrete world of packets of energy, or quanta.
The form of quantum theory that goes furthest in describing
matter and its interactions is the Standard Model, which is based on a bizarre
bestiary of particles such as quarks , leptons and bosons (see Diagram).
The Standard Model may be one of the most successful theories in science,
but it is also one of the ugliest. Its inadequacy is betrayed by some 19
arbitrary constants not derived by any kind of theory that have to be put
in "by hand" to make the equations work.
Capturing the "lion", which unites these two great theories,
would be a crowning achievement for physics. But while Einstein was first
to set off on this noble hunt, tracking the footprints left by the lion,
he ultimately lost the trail and wandered off into the wilderness.
Today, however, physicists are following a different trail-the one leading to superstring theory. Unlike previous proposals, it has survived every blistering mathematical challenge ever hurled at it. Not surprisingly, the theory is a radical-some might say crazy-departure from the past, being based on tiny strings vibrating in 10-dimensional space-time.
To understand how going to higher dimensions can help to unify
lower dimensions, think back to how the Romans used to fight wars. Without
radio communications and spy planes, battles were horribly confused, raging
on many fronts at the same time. That's why the Romans always leapt into
"hyperspace"- the third dimension-by seizing a hill- top. From this vantage
point, they were able to survey the two-dimensional battlefield as a single,
Missing: the Standard Model works well, but still has big gaps. Could superstrings complete the picture?
Leaping to higher dimensions can also simplify the laws of
nature. In 1915, Einstein changed completely our notion of gravity by leaping
to the extra dimension of time. In 1919, the German mathematician Theodor
Kaluza added a fifth dimension and in so doing unified space-time with Maxwell's
equations for electromagnetism. This triumph was largely forgotten amid the
frenzy of interest generated by quantum mechanics. Only in the 1980s did
physicists return to this idea to create superstring theory.
In superstring theory, the subatomic particles we see in nature
are nothing more than different resonances of the vibrating superstrings,
in the same way that different musical notes emanate from the different modes
of vibration of a violin string. (These strings are very small-of the order
of 1035 metres.)
Likewise, the laws of physics -the forces between charged particles, for example-are the harmonies of the strings; the Universe is a symphony of vibrating strings. And when strings move in 10-dimensional space-time, they warp the space-time surrounding them in precisely the way predicted by general relativity. So strings simply and elegantly unify the quantum theory of particles and general relativity. Better still, gravity is not an inconvenient add-on. "Unlike conventional quantum field theory, string theory requires gravity," Witten has said. "I regard this fact as one of the greatest insights in science ever made."
But, of course, all this takes place in 10 dimensions. Physicists
retrieve our more familiar 4-dimensional Universe by assuming that, during
the big bang, 6 of the 10 dimensions curled up (or "compactified") into a
tiny ball, while the remaining four expanded explosively, giving us the Universe
we see. What has consumed physicists for the past ten years is the task of
cataloguing the different ways in which these six dimensions can compactify.
Their task has been especially difficult because mathematicians have not
worked out the topology and properties of these higher-dimensional universes.
The physicists have had to blaze the trail and invent entirely new areas
of mathematics. These efforts have revealed millions of compactifications,
each of which yields a different pattern of quarks, electrons and so on.
As we have seen, the first frustrating problem with superstring
theory is that physicists do not understand where strings come from. To make
matters worse, there are five string theories that unify quantum theory with
relativity. This is an embarrassment of riches. Each competing theory looks
quite different from the others. One, called Type 1 string theory, is based
on two types of strings : "open strings", like short strands with two ends,
and "closed strings", in which the ends meet to form a ring. The other four
have only closed strings. Some, such as Type 2b, generate only left- handed
particles, which spin in only one direction [Ref
I.Asimov "Left Hand
of the Electron"]. Others, such as Type 2a, have left and right-handed
Today's excitement has grown from the finding that if we postulate
the existence of a mysterious M-theory in 11 dimensions we can show that
the five competing string theories are actually different versions of the
same thing. Like a Roman general surveying the battlefield from the third
dimension, physicists today stand on the hilltop of the 11th dimension and
see the five superstring theories below, unified into a simple, coherent
picture, representing different aspects of the same thing.
The first step towards this advance came two years ago when Witten and Paul Townsend of the University of Cambridge showed that Type 2a string theory in 10 dimensions was equivalent to M-theory in 11 dimensions with one dimension curled up. Since then, all five theories have been shown to be equivalent. So at last physicists know where superstrings come from : they originate in the 11th dimension from M-theory.
M-theory also predicts that strings coexist with membranes
of various dimensions. For example , a particle
can be defined as a zero-brane (zero-dimensional object). A string is a
one-brane, an ordinary membrane like a soap bubble
is a two-brane, and so on. (Using p to represent the dimension of the
object, one wag dubbed this motley collection
) When these p-branes vibrate or pulsate , they create new resonances , or
particles, which were missed in earlier formulations of superstrings. The
name "M-theory" was coined by Witten: M perhaps stands for "membrane" or
the "mother of all
strings", or possibly "mystery" Take your pick.
To see how this all fits together, imagine three blind men
hot on the trail of Einstein's lion. Hearing it race by, they give chase
and desperately grab at it. Hanging onto the tail for dear life, one feels
its one-dimensional form and loudly proclaims, "It's a string. The lion is
a string." The second man grabs the lion's ear. Feeling a two-dimensional
surface , he calls out "No, no, the lion is really a two-brane." The third
blind man, hanging on to the lion's leg, senses a three-dimensional solid,
and shouts , "You're both wrong. The lion is a three- brane !" They are all
right. Just as the tail, ear and leg are different parts of the same lion
, the string and various p-branes appear to be different limits of M-theory.
Townsend calls it "p-brane democracy".
The acid test for any theory is that it must fit the data.
No matter how original and elegant superstring theory is , it will stand
or fall on whether it describes the physical Universe. Either it is a Theory
of Everything, as its advocates hope , or it is a theory of nothing. There
is no in-between. So theoretical physicists must answer the second question
: is our Universe, with its strange collection of quarks and subatomic particles,
among the solutions of superstring theory? This is where it runs into an
embarrassing problem, which is that physicists have been unable to find all
its four-dimensional solutions. The mathematics have been fiendishly
difficult-too hard for anyone to solve completely.
In general, there are two types of solutions. So far, only
the first class, called "perturbative" solutions have been found. Across
all branches of physics, theorists faced by an equation they cannot solve
reach for well-established ways to find approximate solutions. In superstring
theory, millions of these perturbative solutions have been catalogued. Each
one corresponds to a different way in which to curl up 6 of the 10 dimensions.
However, none of them precisely reproduces the pattern of quarks , leptons
and bosons in the Standard Model, although some come close.
So, many believe that the Standard Model may be found among
the second class of solutions, the "non-perturbative" solutions. But
non-perturbative solutions are generally among the most difficult of all
solutions in physics. Some physicists despaired of ever finding non- perturbative
solutions of superstring theory. After all, even the non-perturbative solutions
of simple four-dimensional theories are completely unknown , let alone those
of a complicated 10-dimensional theory.
How does M-theory help to solve this intractable problem?
The answer lies in a startling tool called "duality". Simply put, in M-theory
there is a duality, or simple mathematical relationship , between the
perturbative and non-perturbative regions. This allows us at last to take
a peek at this "forbidden zone".
To see how duality works, consider Maxwell's theory of electricity and magnetism , for example. Physicists have known for decades that if they interchange the electric field E and magnetic field B in Maxwell's equations , and also swap the electric charge e and magnetic charge g, then the equations stay the same. That is, nothing happens to Maxwell's theory if we make the dual transformation: E«B and e«g.
In fact, in Maxwell's theory, the product e times g is a constant: so small e corresponds to large g. This is the key. Suppose an equation includes a mathematical function that depends on g2 and which cannot be solved exactly. The standard mathematical trick is to approximate a solution with a perturbation expansion: g2+ g4 + g6... and so on. So long as g is less than 1, each successive term in the series is smaller than the last, and the overall value converges on a single figure.
But if g is greater than 1 then the total gets larger
and larger, and the approximation fails. This is where duality comes in.
If g is large, then e is less than 1. Using perturbation ,
we get the series e2+ e4 +
e6 which gives a sensible value. Ultimately, this means
that using perturbation on e can solve problems in the non-perturbative
region of g.
Duality in Maxwell's theory is rather trivial. But in M-theory,
we find another duality:
g«1/g. This relationship,
though simpler, turns out to be incredibly powerful. When I first saw it,
I could hardly believe my eyes. It meant that a string theory defined for
large g, which is usually impossible to describe using present-day
mathematics, can be shown to be equivalent to another type of string theory
for small g, which is easily described using perturbation theory.
Thus, two different string theories can be dual to each other.
In the non- perturbative region of string theory was another string theory.
This is how, in fact, we prove the equivalence of all five string theories.
Altogether, three different types of duality called S, T and U have been
discovered , which yield an intricate web of dualities linking string theories
of various dimensions and types. At an incredible pace , physicists have
now mapped almost all the solutions and dualities that exist in 10, 8 and
Before M-theory, finding the non-perturbative solutions in
these dimensions would have been considered impossible. Now the problem is
trivial. For example , let us say that two theories A and B are dual to each
other in 10 dimensions. If we compactify both theories in the same way, then
we obtain theories A' and B'. But now we know something new: that A' is also
dual to B'. Thus, the non-perturbative behaviour of A' is given by B'. By
elaborating this process, we get an almost complete understanding of the
different possible universes down to 6 dimensions. Thus, M-theory solves
entire classes of problems that were previously thought to be unsolvable.
It even gives us valuable new details about quantum effects in black holes.
But there are many loose ends. For example, what precisely
is M-theory? So far, we only know fragments of the theory (the low-energy
part). We are still waiting for someone to come up with a full description
of M-theory Last year, Vafa shocked physicists by announcing that there may
be a 12-dimensional theory lurking out there, which he called "F-theory"
(F for father).
More important, we are still far from mapping all the dualities
of four dimensions. If everything works out as hoped , we should find that
one of these four- dimensional universes contains the Standard Model and
thus describes the known Universe. But there are millions of these solutions,
so sifting through them to find the one we are after will take many years.
So will the final theory be in 10, 11 or 12 dimensions? According
to Schwarz, the answer may be none of these. He feels that the true theory
may not have a fixed dimensionality, and that 11 dimensions only emerge once
we try to solve it. Townsend takes a similar view, saying, " The
whole notion of dimensionality is an approximate
one that only emerges in some semiclassical context."
So does this means that the end is in sight-that some day soon
we will be able to work out the Standard Model from first principles? When
I put this question to some leading physicists in this field they were still
cautious. Townsend likened our present state of knowledge to the old quantum
era of the Bohr atom, just before the full elucidation of quantum mechanics.
"We have some fruitful pictures and some rules," he says. "But it's also
clear that we don't have a complete theory."
Witten, too, believes we are on the right track. But he says
we will need a few more "revolutions" like the present one to finally solve
the theory. "I think there are still a couple more superstring revolutions
in our future, at least," says Witten. "If we can manage one more superstring
revolution a decade , I think that we will do all right." From Harvard, Vafa
adds: "I hope this is the light at the end of the tunnel'. But who knows
how long the tunnel is?"
Personally,I am optimistic. For the first time, we can see the outline of the lion , and it is magnificent. One day, we will hear it roar.
Kaku is professor of theoretical physics at the City University of New
York and author of Hyperspace: A Scientific Odyssey through the 10th Dimension,
Oxford University Press.