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BIG BANG
Can
gravity
take a quantum leap?
Feature, 10 September 1994, page one
Physicists have always believed it
would be possible to create a grand
theory of everything, except that
gravity got in the way. Robert
Matthews investigates two
approaches to solving the problem
What keeps us stuck to the Earth? A simple
enough question, you might think, but one that
is still waiting to be answered as physicists
struggle to understand the nature of gravity.
The problem, they say, is that gravity stands
apart from the other fundamental forces of
nature and reconciling them has proved
fiendishly difficult. After decades of work,
however, scientists believe the pieces of the
jigsaw are coming together. Their new theories
of gravity could explain why the very notions of
time and space exist and why particles cannot
be point-like objects.
The origins of these theories lie in the events of
1915, when Albert Einstein announced his
theory of gravity, known as the general theory
of relativity. Gravity, said Einstein, was the
result of matter curving the fabric of
space-time. The idea revolutionised the
Newtonian view of gravity as some form of
ill-defined 'influence'.
Einstein showed that this concept of gravity
could encompass all the successes of Newton's
theory, and lead to new predictions besides. To
date, all the predictions - from the bending of
light by the Sun's gravitational field to the
orbital behaviour of incredibly dense stars -
have proved to be right.
Yet, for all its successes, general relativity
stands apart from that other central pillar of
modern physics, quantum theory. Itself
astonishingly successful, quantum theory has
brought the three other fundamental forces of
the cosmos - the subatomic 'weak' and 'strong'
forces, and the electromagnetic force - under
its unifying banner. It has revealed the unity of
the particles of matter in the Universe, and
unveiled unexpected connections between
matter and all the forces that govern it. All,
that is, except one - gravity. Earlier this year,
some of the world's leading theoretical
physicists gathered over a six-month period at
the Isaac Newton Institute at the University of
Cambridge to attempt to solve some of these
problems.
The lack of unity between gravity and the
quantum world stems from the fundamentally
different approaches to physics of general
relativity and quantum theory. Put simply,
general relativity has well-defined, smooth,
rolling plains of space-time, while quantum
theory has steps, jumps and fuzziness.
Nonetheless, physicists remain convinced that
one day the most successful theory of gravity
can be combined with the most successful
theory of everything else. In short, they believe
it is possible to create a quantum theory of
gravity.
Few Clues
The question is how. For many years, there
were few clues and even less progress in the
search for a route through to this unifying
theory. Now real progress is being made, on two
fronts. Both originate from the brave but
ill-fated first attempts to build a quantum
theory of gravity in the 1950s and 1960s. The
idea was simple: take the best classical theory of
gravity we have, namely general relativity, and
quantise it using the same methods that have
worked for the theory of everything else, from
hydrogen atoms to subatomic particles.
Unfortunately, gravity was not to yield so easily.
The results produced by these standard
quantisation methods were ludicrous, shot
through with infinities. Such mathematical
horrors had been met before in quantum
theory, and techniques had been found to deal
with them. This time, the remedies failed.
Feature, 10 September 1994, page two
This discouraging start forced theorists to
think hard about how they might reach their
goal. Essentially, two strategies emerged, of
which by far the best known is the 'field theory'
approach. This draws its inspiration from the
past successes quantum theorists have had in
viewing the fundamental forces as the result of
the exchange of 'carrier' particles. In quantum
field theory, the electromagnetic force, for
example, is viewed as the result of the exchange
of packets of electromagnetic energy known as
photons.
It turns out that, on this view, the long-range
and attractive force of gravity is due to the
exchange of massless particles known as
gravitons. A quantum theory of gravity then
becomes a matter of working out what
gravitons do as they flit about in space-time,
interacting with photons, electrons and
everything else. In short, field theorists believe
that, ultimately, gravity can be treated as just
another fundamental force. But those who
have adopted the other route to quantum
gravity, the 'general relativists', believe that if
Einstein taught us anything it is that gravity is
not just another force.
According to Einstein, gravity is the result of
the curvature of space and time, that is,
curvature of the very arena in which physics
takes place. Unlike the other forces, therefore,
gravity cannot be viewed as acting somehow
'within' a space-time arena: gravity is itself
part of that arena. So it is the height of naivety,
argue the general relativists, to presume that a
quantum theory of gravity is just about
gravitons acting in space-time. What, for
starters, is that space-time?
In a quantum theory of gravity, then, it is
dangerous to use ideas like space and time
casually. Of course, this is far easier said than
done. For years, general relativists have been
trying to get to grips with these profound
problems, and in particular with their
manifestation in the notorious Wheeler-de
Witt equation. Conceived in the late 1960s, this
is the quantum gravity analogue of the
well-known Schrodinger equation, the
fundamental equation of wave mechanics.
Schrodinger's equation describes the behaviour
of particles in terms of so-called wave
functions, mathematical entities which can be
used to calculate the probability of an electron
being at a certain place at a certain time.
Exactly what the Wheeler-de Witt equation
describes is less clear - in fact this is one of the
major problems facing general relativists. One
way of thinking about it is as an equation
governing how the wave function of the
geometry of space itself changes with time. But
this description instantly raises the awkward
idea of time somehow standing outside
everything else - a trap theorists are constantly
trying to avoid in quantum gravity.
Equation from hell
Not surprisingly, actually solving the
Wheeler-de Witt equation to get a prediction
about, say, the evolution of the Universe is even
more daunting. Technically a partial
differential equation in infinite dimensions, it is
truly the Equation from Hell.
Not until the 1980s did the general relativists,
struggling to extract something from the
Wheeler-de Witt equation, or the field
theorists, struggling with their infinities, have
much to show for their endeavours. In the
event, the first to break through their impasse
were the field theorists. Just 10 years ago
Michael Green, then at Queen Mary College,
London, and John Schwarz of the California
Institute of Technology showed that a new way
of viewing subatomic particles could lead to a
sensible quantum theory of gravity.
Elementary particles such as the electron are
usually thought of as point-like objects. This is
clearly an idealisation, however. Since the early
1970s, a number of physicists had found that
some very interesting things emerge if particles
are viewed instead as manifestations of tiny but
nonetheless finite-sized objects called strings.
These strings are certainly unusual creatures.
Typically they measure just 10-35 metres
across. Even if you could shrink to this size, you
probably wouldn't know what you were looking
at as strings are 10-dimensional objects.
What's more, they are under a colossal tension
which keeps them tightly wrapped up. With
these bizarre properties, however, strings can
do something miraculous. Quantum field
theory based on strings does not just allow
gravity to coexist with the other forces, it
virtually demands its presence. Gravitons, the
carrier particles of gravity, just fall out of string
equations.
Feature, 10 September 1994, page three
Green and Schwarz discovered something
even more startling when they combined
strings with a concept known as
supersymmetry. Supersymmetry
mathematically unifies the particles of matter,
such as electrons, with the particles that carry
forces, such as gravitons. In 1984, Green and
Schwarz showed that supersymmetric strings,
or superstrings, lead to a theory which demands
the existence of gravity while avoiding the
dreadful mathematical problems that had
confounded all previous theories of quantum
gravity.
Following this discovery, hundreds of theorists
jumped aboard the superstring bandwagon.
Excitement heightened when, within a year or
so, a team from Princeton University
discovered that superstring theory could unite
gravity with the other three fundamental
forces. By the late 1980s, even newspapers were
running articles about superstrings being the
key to a 'theory of everything', a single theory
that could account for all the forces and
particles in the Universe.
Despite such heady progress after so many years
in the doldrums, some theorists remained
sceptical. Just because one theory had been
found to be free of some mathematical
problems, that hardly proved it was the theory.
It was also far from clear how one could ever
test a theory based on such incredibly small
objects as superstrings.
The general relativists were also quick to
deflate any hubris. Superstring theory is based
on a specific choice of space-time arena in
which the action takes place. But can this
choice be justified? More importantly, why does
superstring theory work at all - where does it
come from? Unlike other quantum field
theories, it is not based on some profound
physical understanding. Rather, it has been
'plucked out of the air'. Its successes seem to be
mere approximations to some deeper, more
fundamental theory. But what is that theory?
In recent years superstring theory has dropped
out of the headlines. The reason is that
theorists have been working hard to answer the
criticisms levelled at the theory. Happily, their
tenacity is being rewarded. Perhaps the most
surprising development has been the tentative
backing from research being done by particle
physicists. The unimaginably small scale of
superstrings suggests that a similarly
unimaginably large amount of energy would be
needed to probe their properties. However, if
they really do play a role in unifying all the
forces of nature that we see at work today, then
they might be expected to produce effects at
relatively everyday energies too.
Theorists believe that at a certain very large
energy (or, equivalently, temperature), all the
forces combine into a single 'superforce'.
Superstring theory gives an estimate for that
temperature: it is about 1030 kelvin. Only in
the very early Universe have such temperatures
been reached, and there is no prospect of
reaching them on Earth today.
However, in 1990, researchers at CERN, the
European centre for high-energy physics near
Geneva, found a way around this. Using the
Large Electron Positron Collider (LEP), they
measured how today's fundamental forces
change their strengths as the energy was
stepped up. LEP could not possibly go right up
to the unification energy, but plotting its results
on a graph showed that the forces do indeed
appear to be heading for unification at about
1030 K - just as superstring theory predicts.
God-given constant
Furthermore, that prediction provides a link
between superstring theory and a well-known
feature of conventional low-energy quantum
theory, known as the fine structure constant.
So-called because of its use in unravelling the
fine details of atomic spectra, this number is
used to characterise the electromagnetic force,
and is usually thought of as 'God-given'.
According to Graham Ross at the University of
Oxford, there are now signs that superstring
theory will allow the value of this constant to
be worked out from first principles. This
suggests that superstrings really can provide an
insight into why our Universe is the way it is.
Superstring theory also seems to have other
implications for more 'accessible' scales of
physics. In 1990, mathematical analysis of its
structure revealed a property known as
'modular symmetry'. Roughly speaking, this
means that one can swap any term involving
distance with another term involving the
reciprocal of distance. This in turn implies that
superstring theory contains a minimum
distance below which physics looks the same as
it did at greater distances. There is, then, a kind
of mathematical 'bounce' in superstring theory,
and it takes place at about 10-35 metres.
One consequence of this is an explanation of
why particles are not mere points: one simply
cannot reach even tinier distances without
suffering a 'bounce'. But the existence of a
minimum distance also gives an explanation of
a long-standing problem. Quantum field
theorists are all too familiar with infinities
dogging their calculations in other,
nongravitational problems. Superstring theory
shows that, in essence, these infinities are
caused by the mistaken assumption that there is
no minimum distance in physics
Feature, 10 September 1994, page four
Over the past year or so, string theorists have
been claiming that their theory can be used to
tackle tough problems in areas apparently far
removed from gravity. For example, even
relatively straightforward questions in quantum
chromodynamics (QCD), the theory that
explains the action of the strong nuclear force,
are notoriously hard to answer. Green, now at
the University of Cambridge, is among those
using the ideas involved in string theory to
simplify calculations in QCD.
Superstring theorists are currently very excited
by the emergence of an answer to a
long-standing criticism of their approach to
gravity. If the theory is right, there can be only
one superstring theory. Frustratingly, however,
an infinite number of possibilities seem to be
allowed. Defenders of the theory have always
argued that these infinite possibilities are really
just different solutions to a single theory. In the
past few months, Nathan Berkovitz of King's
College London and others have shown that
this hunch seems to be correct. All the different
varieties do indeed seem to be manifestations of
one single superstring theory.
Among the field theorists who have kept faith
with superstrings, then, morale remains high.
Their past successes, such as proof that
superstring theory does not suffer from
mathematical horrors, have held up, and the
new results have been encouraging. But their
colleagues in the general relativity camp, who
formed the bulk of the theorists attending the
workshop at the Isaac Newton Institute, also
have breakthroughs to celebrate. Many of those
who dropped in at the institute have played a
role in these developments, which stem from
progress in dealing with the dreaded
Wheeler-de Witt equation.
Celebrations in the other camp
For many years, the complexity of the
Wheeler-de Witt equation forced theorists to
use a 'stripped down' version, in which all but a
handful of the infinite dimensions of the
original were thrown away. This approach has
been used to get a handle on questions such as:
how was the Universe we see chosen from the
presumably infinite number of candidates?
Only the most sanguine of theorists, however,
could have total confidence in conclusions
drawn from such a heavy-handed approach.
In 1986, Abhay Ashtekar at Pennsylvania
State University devised a set of variables that
allowed physicists to extract exact solutions
from the full Wheeler-de Witt equation.
Known as Ashtekar variables, they are a little
like the change of variables trick used by
students to solve far simpler differential
equations. In 1988, theorists used the variables
to extract the first exact solution from the
equation. Unfortunately, nobody quite knows
what it means. Now, however, attempts are
being made to see if the full equation can give
answers to definite mysteries, such as how the
Universe emerged from the quantum chaos of
its birth to undergo the period of rapid
'inflation' that is seen as the key to
understanding today's Universe. Ashtekar's
breakthrough has also helped general relativists
to apply the full equation to other real-world
problems. Last year, Lee Smolin at
Pennsylvania State University and Carlo
Rovelli at the University of Pittsburgh
succeeded in finding solutions in cases where
gravity interacts with particles such as
electrons. These interactions also cast light on
the vexed question of how to deal with 'time' in
quantum gravity. In certain circumstances,
they allow a time-like concept to emerge
automatically from the Wheeler-de Witt
equation, rather than it having to be inserted
by hand.
Feature, 10 September 1994, page five
These successes by the general relativists may
seem slight compared to the achievements of
the field theorists. General relativists
emphasise, however, that their progress is much
more solidly based. Solutions to the full
Wheeler-de Witt equation are not mere
approximations. Rather, they are full-blown
results from a real theory of quantum gravity -
though admittedly not necessarily the correct
one. As such, they can be used to study events in
conditions as varied as those in the very early
Universe to those in laboratories now.
Green and his fellow field theorists accept that
their superstring approach is ultimately just an
approximation to some grander theory, and
that they would dearly like to know what that
theory is. Until then, they will be on much
shakier ground when applying their results to
such extreme situations as the centres of black
holes or the birth of the Universe. However,
field theorists like to stress that by treating
gravity as just another force, they are much
more likely to find connections with the other
fundamental forces. But after battling with
gravity for so long, both camps have at least
one feature in common: a reverence for the
subtleties of nature in general, and gravity in
particular. Field theorists such as Ross say that
despite their recent progress, the goal of a
theory of everything still seems a long way off.
The general relativists tend to be even more
reserved. Ashtekar admits that there is an
undercurrent of pessimism among many
researchers in the face of so many technical
difficulties, although he himself remains bullish
about the chances of ultimate success.
Ultimately, the biggest barrier to the
construction of a theory of quantum gravity
may not be the mathematics, but the
interpretation of the mathematics. For
example, general relativists are still struggling
to understand the meaning of the Wheeler-de
Witt equation and its solutions, while field
theorists have yet to understand just what it is
that their superstrings wriggle about in.
Veteran gravity theorist Chris Isham of
Imperial College, London, sees even deeper
issues starting to loom - issues long thought to
be the preserve of philosophy. Perhaps most
central of all is the question of whether space
and time are merely constructs of our personal
experience, as Immanuel Kant argued some
200 years ago.
The mere suggestion that such fundamental
concepts cannot be relied on in the
construction of a theory would fill most
physicists with horror. Yet those who dare to
tackle the mystery of gravity are learning to
live with such possibilities. As Isham puts it:
'The shadow of Kant is hanging over all of us.