¿ìè¶ÌÊÓƵ

Stop a particle physicist in the street and you will soon find yourself
hearing how simple, symmetrical and altogether elegant is this thing we
call the Universe. All the diverse materials around us reduce to the permutations
of a handful of microscopic building blocks obeying elementary rules governing
who can do what to whom and with whom.

Yet when we get back to everyday reality, we are only too aware that
the world is surely nothing of the sort. Our daily lives, the workings of
our businesses, national economies, local ecologies or weather systems are
anything but simple. Rather, they resemble a higgledy-piggledy mess of complexity
governed by a concatenation of interlinked processes that possess neither
symmetry nor elegance. As complexity becomes more organised, so, like crime,
the range of phenomena that issues from it blossoms and grows with unpredictable
subtlety. And indeed, if it were a biologist whom you had stopped in the
street, you would have been told nothing about simplicity and symmetry.
The biologist would have waxed eloquent about the interlinked complexity
of the outcomes of natural selection that we see around us. As these are
neither planned nor guided, there is no reason for them to be simple; there
are so many more ways to be complicated. Their primary characteristic is
persistence, or stability, rather than simplicity. So what are we to make
of this apparent dichotomy: is the world simple or is it complicated?

Ever since the early Greeks began the serious contemplation of natural
things, there have existed two different emphases in thinking about what
is important for our understanding of the Universe. The ‘Platonic’ tradition
emphasises the timeless and unchanging aspects of the world as being the
most fundamental. For Plato himself, these were the ‘forms’-the invariant
blueprints-of which all observed things were merely shadowy examples. The
observed happenings were, therefore, less fundamental than the unchanging
blueprints that governed them. For scientists of the post-Newtonian era,
these unchanging aspects were the famous conserved quantities of physics-energy,
linear momentum, angular momentum and electric charge. By this approach,
the Platonic emphasis seeks to expunge the process of change and the notion
of time from the description of things.

This desire can be realised to a surprising extent. The traditional
laws of Nature dictating change can always be recast as equivalent statements
that some ‘conserved’ quantities such as total energy, momentum or electric
charge remain unchanged in all physical processes, just as the ‘rule’ for
weaving a periodic pattern on a carpet could be replaced by the requirement
that a particular aspect of the carpet’s appearance is the same from different
points on its surface. The laws of Nature weave a network of physical happenings
into a pattern in space and time whose recurrences and unchanging aspects
are direct reflections of the symmetries in the laws themselves.

This Platonic approach has reached its zenith during the past 15 years
in the study of particle physics. For while you can replace laws of Nature
governing changes in space and time by statements that certain quantities
remain unchanging, these statements can, in turn, be replaced by the dicta
that certain patterns or ‘symmetries’ are preserved in Nature.

Particle physicists have developed this connection between laws and
invariance still further in the creation of ‘gauge theories’. In these theories,
the symmetry preserved is of a more abstract geometrical character so that
laws of Nature remain the same when particles change position in space (See
Box). For example, the invariance with respect to arbitrary accelerations
requires the force of gravity to exist. The requirement that such powerful
invariances are preserved demands the existence of the forces of Nature
and dictates the way in which particles interact with each other. They allow
us to offer answers to questions such as: ‘Why do certain forces of Nature
exist?’ rather than merely provide descriptions of how they act in the Universe.

All the known forces of Nature-gravity, the strong nuclear force, the
weak force and electromagnetism-are described by varieties of gauge theory
founded upon the immutability of some pattern when any change occurs. They
are founded, therefore, upon the Platonic assumption that symmetry is fundamental.
The ultimate expression of that faith is the search for a Theory of Everything
within which all these separate theories of the four different forces of
Nature can be subsumed and unified into a single description of the ultimate
symmetry, or law of Nature, from which all else follows.

Whether that ultimate theory is a conventional gauge theory or a superstring
theory, in which the most fundamental entities are lines or loops of energy
rather than points, makes no difference to the appeal to symmetry. The success
achieved beating a path towards one all-embracing symmetry-the Theory of
Everything-through the complex jungle of experience lies behind claims of
physicists that the Universe is simple and deeply symmetrical.

But there is a second tradition in the study of Nature that, until recently,
has been less popular than the Platonic search for the invariants of Nature.
The Aristotelian perspective laid emphasis upon the observable happenings
in the world rather than the unobservable invariants behind it. As a result,
the process of temporal change was regarded as fundamental. It is no accident
that the original advocates of such an emphasis drew their intuition more
from the study of living things than the purposeless pendulums of the physicist.
For the advocates of this approach, the world looks complicated and messy.
They do not expect to explain away all aspects of that complexity by appeal
to simple ‘laws’ acting behind the scenes.

To understand the real difference between the simple Platonic view of
things and the complicated Aristotelian perspective, we need to appreciate
one important fact about the world: symmetrical laws of Nature need not
have outcomes with the same symmetries as the laws themselves. If we place
a pencil on its point and allow it to fall then it will fall in some direction.
The laws governing the fall of the pencil do not have any special preference
for one direction over any other. They are symmetrical with respect to directions
in space. But the pencil must fall in some particular direction and, in
so doing, the underlying symmetry of the governing law of Nature is broken
in the observed outcome of the law. Were this not so, then every outcome
of the law of Nature would have to carry the full invariance of the laws.
Such a world would be in a straitjacket. We could not be sitting in the
spot that we happen to occupy at the moment unless the laws of Nature had
a special favouritism for that spot.

Thus we see that outcomes are much more complicated things than laws
of Nature. Moreover, we do not observe the laws of Nature: we observe only
the outcomes of those laws, and, from the heap of broken symmetries before
us, we must work backwards to reconstruct the pristine laws behind the appearances.
Sometimes, this is very easy to do but often it is impractical because the
direction of the symmetry breaking is sensitive to the whims of the environment.
But we have learnt one important lesson. This process of symmetry breaking
explains how we can reconcile the existence of the observed complexity of
Nature with underlying laws that are simple.

Particle physicists earn their living by studying the laws of Nature,
and their claims for the simplicity and symmetry of the world point to the
economical forms that can be found for the laws of Nature. The life scientist,
or the economist, by contrast, hardly worries about any ‘laws’ of Nature.
The focus, there, is entirely upon the complicated outcomes of the underlying
laws. This state of affairs is perhaps responsible for the surprising lack
of success that accomplished mathematical physicists so often have when
they turn their attention to the problems of the life sciences. Accustomed
to pristine symmetry and mathematical beauty, they discover that the higgledy-piggledy
results of natural selection possess neither of those desirable features.
Instead, they are faced with understanding outcomes that are separated from
the underlying ‘simple’ laws of physics by a long sequence of hidden symmetry
breakings.

We have seen, therefore, that the world can be both simple and complicated
in important ways and the aspect that impresses most will depend upon whether
you are most concerned with the laws of Nature or with their outcomes. An
interesting historical example where this division was clear but unrecognised
can be found in the 18th and 19th centuries. There, one finds examples of
‘design arguments’ for the existence of God from those examples of order
and apparent contrivance in Nature from which humanity seems to benefit.
There were always two varieties of such arguments. The oldest pointed to
the existence of specific situations in the natural world (the design of
the eye, or the way in which animal habitats appeared tailor-made for their
inhabitants, for example) where the outcomes were advantageous, as evidence
of Divine providence. The other style of design argument, popularised first
by Newton and his followers, pointed to the invariant laws of Nature as
the primary evidence for a Deity. These two examples reveal design arguments
based, in the first case, upon the outcomes of the laws of Nature, stressing
the harmonious irrelations between particular examples of complex symmetry
breakings; and in the second, upon the simplicity, invariance and symmetry
of the underlying laws.

During the 20th century, the Platonic approach has dominated fundamental
physics. Since the mid-1970s, when the concept of gauge invariance and symmetry
was found to be a master key that could unlock the secrets of the world
of elementary particles, the laws of Nature have been regarded as more interesting
than their outcomes. This is not altogether surprising; laws are simpler
to study and you might imagine that once in possession of laws you could
understand and predict their outcomes. But in the past few years, physicists,
mathematicians and computer scientists have realigned their focus of attention
upon the outcomes, having come to appreciate that sequences of events exist
which cannot be replaced by timeless invariants in the Platonic manner.

The Aristotelian perspective has re-emerged in the study of complexity
in the abstract-that is, as a general phenomenon not necessarily tied to
a particular complicated physical situation. Suppose we consider some sequence
of outcomes (numbers issuing from a computer, for example): what do we mean
by saying that the sequence is ordered in some way? It means that our brains
have picked upon some pattern that enables us to abbreviate the sequence
in our minds. For example, if we were shown a list of the first 2000 even
numbers, we could store that information by an abbreviated formula; we would
not need to carry around the entire list. If it is possible to store the
information in a sequence in an abbreviated form shorter than the sequence
itself, then the sequence is regarded as non-random and we call it ‘compressible’.
If no such abbreviation exists, then the sequence admits no representation
other than the complete explicit print-out of itself. In this case, we call
it ‘incompressible’. When dealing with incompressible sequences, the Platonic
approach is of little use. The lack of an abbreviated representation means
that there exists no symmetry or invariance whose simple preservation is
equivalent to the data content of the sequence-for that would be a compression
of the sequence. The outcomes contain a level of complexity that requires
nothing less than their explicit listing to capture their full information
content.

Besides elevating the study of outcomes to something that is not necessarily
included within the study of natural laws, this notion of compressibility
gives simple ways of characterising many of our intellectual activities.
We recognise a possible new definition of ‘science’ as being simply the
search for compressions: the laws of Nature are the compressions of our
sense data. The discovery of a Theory of Everything would be the ultimate
compression. Moreover, the apparent success of this process hinges upon
two superficial features of things: the physical world that we observe seems
to be surprisingly amenable to compression, and the brain is remarkably
good at effecting compressions when presented with events.

In a predominantly incompressible world, we would not have scientists
but archivists who simply recorded every observed event. The compressibility
of many aspects of the world saves us from this ‘Bureaucracy of Everything’.
We can use a simple law of motion to describe the motion of heavenly bodies
instead of having to keep a record of their positions and velocities at
all times. Clearly, this compressibility and the brain’s remarkable ability
to make sense of complicated things is an important necessary condition
for our own existence. We could not survive as ‘intelligent’ observers or
readers of ¿ìè¶ÌÊÓƵ in a world where no compressions were possible,
or with brains that produced imaginary or erroneous compressions. A certain
level of predictability and innate predictive power is required for the
successful evolution and survival of living things.

It has clearly proved advantageous to over-develop our capability to
recognise patterns (on the basis, for example, that if you see tigers in
the trees when there are none, your friends will merely call you paranoiac,
whereas if you fail to see tigers in the trees when there are, then your
continued survival must be rather doubtful). But we must beware of the fact
that our brains are altogether too good at finding patterns where none exists.
As a result, we tend to see canals on Mars and all manner of exotic things
lurking in inkblots.

Yet, the brain cannot gather all the information potentially on offer
to it. That would be as impractical as gathering none-would we really want
to receive information about every last electron orbital when we looked
at a painting? The brain overcomes this problem by storing only a part of
all the information available to the senses. Our physiological makeup helps
to effect this truncation by placing limits on the intensities of light
and sound that we can respond to.

However, this does also warn us that the brain would effect a compression
of the observed information even if one did not truly exist. Furthermore,
many aspects of the scientific enterprise set out to truncate the information
available so as to produce a compression, for example, by random sampling
to obtain a representative opinion poll. This teaches something more about
different sciences. In the so-called ‘hard’ sciences, the most important
characteristic is that their subject matter searches for simple idealisations
of complicated situations which can underpin very accurate approximations
to the true state of affairs. If we wish to develop a detailed mathematical
description of a star such as the Sun, then it serves as a very good approximation
to treat the Sun as being spherical with the same temperature all over its
surface. Of course, no real star possesses these properties precisely. But
many stars are such that some collection of idealisations like this can
be made and a very accurate description still results. Subsequently, the
idealisations can be relaxed slightly and we can proceed step by step towards
a more realistic description that allows for the presence of small asphericities,
then to further realism, and so forth.

By contrast, many of the ‘soft’ sciences which seek to apply mathematics
to such things as social behaviour, prison riots, or psychological responses,
fail to produce a significant body of sure knowledge because their subject
matters are far less compressible and do not readily provide obvious and
useful idealisations from which we can proceed towards better and better
approximations to reality.

Our world is both simple and complicated in well defined ways. Until
very recently, the study of the complexity possessed by sequences of events
was ignored in the rush to classify the behaviour of the simplest natural
phenomena in terms of universal laws. In the future, the balance may well
be redressed. In redressing it, we will be faced with a fascinating question:
whether the observed laws of Nature are merely inevitable consequences of
deeper rules (or compressions) governing the unfolding of information and
abstract complexity, or whether the abstract symmetries followed so faithfully
by the particle physicist will turn out to dictate the rules which govern
the generation of complexity and information. Is the Universe ultimately
a computer or a kaleidoscope? Or neither?

John Barrow is professor of astronomy at the University of Sussex. His
latest book, Theories of Everything: the Quest for Ultimate Explanation,
is to be published by Oxford University Press this month.

* * *

This is the age of the gauge symmetry

Today’s best working descriptions of all the known forces of nature-gravity,
electromagnetism, the weak and the strong nuclear forces-are all ‘gauge
theories’. Such theories are predicated entirely upon symmetries. They permit
anything to happen so long as some abstract pattern is preserved in the
process. The requirement that a particular ‘gauge symmetry’ is maintained
by a force of Nature places very powerful mathematical restrictions upon
its possible properties.

Suppose we take some object-your hand will do-and alter its position
in some way so that every point of it is transformed in an identical fashion.
The result is simple: the hand just moves as a whole to a new location in
space: but it looks the same in every respect. A change like this where
every point is changed in the same way is called a ‘global symmetry’.

But physicists wish the laws of Nature to remain unchanged under far
more general circumstances. They believe it is unnatural to expect invariance
of things under changes that are required to be the same everywhere. If
a change happens on the other side of the Universe, how can events here
and now know instantaneously how they must behave in order to keep in step
and maintain universal symmetry? The finite speed of light seems to forbid
it.

To avoid this difficulty we require an invariance of the laws of Nature
under changes that could be different at different places and times. This
more powerful requirement is called a ‘local gauge symmetry’. At first sight,
it appears an impossibility. If every bit of my hand is allowed to move
in a different and arbitrary manner then the structure of my hand cannot
be preserved. It will be dispersed into bits and pieces going their separate
ways. There is only one way in which the form of the hand can remain intact
and unaltered in the presence of such unbridled changes. Forces must necessarily
exist with a particular form that constrains how different parts of the
hand can move. In our illustration, we might imagine elastic bands placed
around our fingers to prevent their dispersal. This analogy indicates that
the imposition of such local gauge symmetry actually dictates what forces
must exist between the particles involved.

The reason why particle physicists have been so enamoured of gauge theories
can be traced to these remarkable properties. Given a particular symmetry
pattern, the requirement that it is preserved can be used to generate a
complete theory of the forces of Nature. The invariance requires the force
to exist. Moreover, it generally dictates rigid constraints upon the types
of particle that can exist and the particular forces that can act between
them. The gauge age has reduced laws of the microworld to symmetries in
a systematic and powerful fashion. Yet gauge theories will not be the last
word. While they have many appealing features and make numerous successful
experimental predictions, they possess unsavoury mathematical defects. Theorists
have discovered that this can be miraculously cured by replacing the basic
point-like entities, which gauge symmetry controls, by line-like entities
called strings. The interactions between strings remain controlled by laws
of Nature respecting special symmetries in face of all possible changes.

Whereas the laws of classical physics were recognised only in retrospect
as being associated with the maintenance of particular symmetries, those
of quantum physics can be derived ab initio from the belief in the universality
of symmetry.