From GREG PENDLEBOOTH
During the past two decades, physicists have used quantum mechanics
to develop a ‘Standard Model’ of how the Universe behaves at the subatomic
level. They would like to be able to use this model to calculate physically
observable values in order to compare them with the results from experiments
such as the ones being carried out by the Large Electron Positron (LEP)
particle accelerator at CERN in Geneva. Unfortunately, many of the detailed
calculations of the Standard Model are so difficult that physicists cannot
do them directly. Instead, they are turning to simulations on supercomputers
based on a technique called lattice gauge theory in order to help them to
refine and understand their beliefs .
According to the Standard Model, the Universe contains many different
types of particles which make up matter and the fundamental forces. Each
particle is distinguished by a unique set of properties defined by quantum
numbers in the way that houses are distinguished by their street addresses.
The most fundamental distinction between particles is based on a quantum
number called spin. Particles whose quantum spin can take only values that
are whole numbers, such as 1, 0 or 2, are called bosons, whereas particles
with half-integer spins – such as 1/2 – are called fermions. Bosons and
fermions behave quite differently.
The most familiar boson is the photon, which is the particle responsible
for electromagnetic phenomena such as light and radio waves. An important
property of bosons is the laws of the Universe allow many identical bosons
to exist with the same energy. This is the reason lasers work – by pulsing
a crystal or a gas, a laser can pile up many photons on top of one another
in the same state of energy to create a concentrated burst of light.
On the other hand, the Universe does not allow two fermions to exist
in the same state. Any two fermions of the same type must differ either
in position or in some other quantum number. This fundamental property of
fermions is called the Pauli exclusion principle, and is one ofthe things
that makes calculations so difficult in the Standard Model.
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All known fundamental fermions are either leptons, such as the electron
and its antiparticle, the positron, or quarks, such as the up and charm
quarks (see Figure 1a). Fermions interact by exchanging bosons (see Figure
1b), each of which is responsible for a different fundamental force: photons
mediate the electromagnetic force, which affects all electrically charged
particles; gluons mediate the strong nuclear force, associated with quarks,
and so on.
Just as an electromagnetic field is associated with photons, a field
called a gauge field is associated with each other type of boson. This is
an example of the wave/particle duality which is one of the basic ideas
in quantum mechanics. To describe the way the Universe behaves on very small
scales, physicists have to use the mathematics that treats particles as
a handful of pebbles thrown in the air, together with mathematics and describes
the waves those particles create when they land in a pool of water. Electrons,
and all the other constituents of matter, are waves as well as particles,
just as the photons responsible for the electromagnetic field are particles
as well as waves.
The most important characteristic of quantum mechanics is its mixture
of determinism and uncertainty. Real fermions do not exchange bosons at
fixed times, or in a fixed order. Instead, there is a probability associated
with each possible type of event, and particles interact in accordance with
those probabilities. Exactly when a particular event happens is uncertain,
but taken together the statistical properties of events that occur must
obey certain laws. Most importantly, different types of events can interfere
with one another. If a certain event could happen in one of two ways, the
total probability of the event happening is not simply the sum of the probabilities
of each alternative. The two probabilities can add and subtract from each
other in the same way that water waves reinforce or cancel each other on
the surface of a lake.
Whenever physicists want to do calculations in quantum mechanics, they
must take into account all possible alternative ways for events to take
place – for example, all possible paths a photon could take between a lamp
and piece of writing paper. This approach of summing over all possibilities
was developed by the American physicist Richard Feynman in the 1950s. Because
these alternatives interfere, however, physicists cannot simply calculate
a probability for each alternative independently and then add the results
together. This makes it very difficult, if not impossible, to perform exact
calculations for anything other than trivial problems.
The traditional (and up to a point highly successful) way to get around
these difficulties is to classify the possible ways an interaction might
take place according to the number of times the particles involved might
interact with each other, and then ignore those possibilities that are very
improbable. This is similar to calculating the probability of a traffic
accident on the motorway by assuming that once two cars have been involved
in a collision, they are unlikely to collide again during the same trip.
A more accurate model would take into account double collisions, but ignore
triple collisions, and so on. Each time we take into account the possibility
of a larger number of repeated collisions, we are perturbing our previous
answer by some small amount. This sort of analysis is called perturbative
theory.
For example, when two electrons interact, they do so by exchanging one,
two or more photons. These possibilities can be shown as Feynman diagrams
(see Figures 1b and 2a). The measurable strength of the interaction between
electroncs and photons is called electromagnetic coupling constant, and
has a value of approximately 1/137. The probability of a particular Feynman
diagram occurring is inversely proportional to the number of events that
take place; in other words, the number of vertices in the diagram. The chances
of a two-photon interaction happening is 1/137 times the chances of a one-photon
interaction, while a triple interaction is 137 times less likely still.
Because 1/137 x 1/137 is very small, and (1/137)3 is smaller still, physicists
can obtain very accurate results using perturbative analysis.
In order to make their calculations work, physicists have to introduce
a value called the cutoff. This is the maximum energy any electron is allowed
to have. Although the use of a cutoff is an approximation, electrons with
very large energies are so rare that cutting them off does not affect the
results; if such a cutoff is not used, on the other hand, intermediate calculations
spuriously produce infinitely large results.
The electromagnetic coupling constant’s value depends on a quantity
that you cannot measure in an experiment, a parameter called the care coupling
constant. Because you cannot observe it, a physicist can choose a value
for it depending on the complexity of the diagram being used so as to get
back the right value for the observable coupling constant. If this is done
correctly, the results of the calculation do not depend either on the value
of the bare coupling constant chosen or on the cutoff. This process of making
results independent of any arbitrarily chosen parameters is called renormalisation.
As long as physicsts restrict their attention to quantum electrodynamics,
or QED, and consider only electrons and photons, the Standard Model works
well enough to predict the values of certain physical constants to 10 decimal
places. But when physicists turn their attention to nuclear particles, the
model becomes much more difficult to work with.
A nuclear particle such as a proton is made up of three quarks held
together by gluons, the bosons responsible for the strong nuclear force.
Just as electrons have an electric charge, quarks have a colour charge that
governs the way in which they interact. (The introduction of the colour
charge gives this field its name, quantum chromodynamics or QCD.) Unfortunately,
the gluons that bind quarks together also have colour charges, which means
that gluons interact with gluons as well as with quarks. This makes QCD
much more complicated than QED, where photons interact only with electrons,
and not with other photons.
In addition, the coupling constant describing the strength of gluon-quark
interactions is much larger than the coupling constant for photon-electron
interactions. This means that in QCD, unllike QED, very complex Feynman
diagrams are just as important to the way quarks and gluons behave as simple
diagrams. Faced with this, physicists resort to a different type of calculation
and enlist the help of large supercomputers.
The four-dimensional ‘space-time’ of the Universe in which particles
interact is first broken up into discrete grid, or lattice. A program can
then keep track of the fermions at each point in the lattice. In this approach,
the gauge fields holding the fermions together are treated as values on
the links connecting the lattice points. Instead of summing over possible
Feynman diagrams, the program sums over possible configurations of the lattice,
weighted according to their probability.
One way to think about the difference between the traditional and lattice
approaches represented by QED and QCD respectively is shown in Figure 2.
In QED, and approximation is to ignore all those diagrams that are more
complicated than some threshold, and add together only the simpler diagrams.
In QCD, on the other hand, the approximation made is to add together a sample
of the different possible configurations of the system. This is why calculations
in QCD are much more work than those in QED – the number of configurations
that can safely be ignored is much smaller.
The sampling process works like this. A simulation program generates
configurations at random, with the correct probabilities, and sums over
those. This is called a Monte Carlo simulation, because of the ‘dice-rolling’
involved in generating lattice configurations. Because each different configuration
of the lattice is taken as a representative of a large number of similar
actual configurations, the weighting associated with the lattice configuration
must somehow reflect how many actual configurations it is representing,
and how probable they are.
Simplifying quantum chromodynamics
Fermions cause problems in Monte Carlo QCD calculations because of the
exclusion principle. When each lattice configuration is generated, the program
must make sure that no two fermions are identical. The easiest way to ensure
this is to introduce a mathematical representation called a matrix which
encapsulates the interaction between the fields at each lattice point. Each
time we change the matrix, we calculate a second matrix, each of whose values
depends on every value in the first matrix. Doing this gives each fermion
a chance to interact with every other fermion. This calculation is very
expensive, and one of the things that makes QCD so thirsty for computer
time – a lattice with eight points on each side contains 84
= 4096 points in total (because space-time is four-dimensional), so the
matrix contains 4096 x 4096, or approximately 16 million values.
The alternative to doing these matrix calculations is just to ignore
fermions when generating configurations and use only a gauge field, in which
gluons interact only with other gluons. Such a simulation is called ‘quenched’.
A quenched approximation to QED, for example, would not allow a photon to
split into an electron and positron (which is theoretically possible) when
travelling between two points. This would mean that photons would not be
able to interact, which is a good approximation in many circumstances.
In QCD, the quenched approximation can be used to try to calculate the
mass of a particle such as the proton, in order to compare the predictions
of the theory against very accurate experimental results. To do this, a
lattice full of interacting gluons is generated, and the three quarks making
up the proton are introduced at a single point. The simulation then measures
how the three quarks move around the lattice subject to the forces exerted
by the gluons, without allowing the quarks to change the gluons or the forces
they exert.
A good way to think about the difference between quenched and unquenched
calculations is to consider the pinball toy shown in Figure 3a. Marbles
that are released one by one at the top of the toy bounce randomly off the
pins and collect in the bins at the bottom. If enough marbles are used,
the bins fill up in a good approximation to a normal curve (see Figure 3b).
If we think of marbles as quarks, and the field of pins as the lattice full
of gluons, this corresponds to the quenched approximation.
The unquenched model would then be where marbles were allowed to bend
pins when they collided with them, so that the next marble to come along
would find a slightly different environment. This marble would therefore
tend to bounce in a slightly different way, and would bend the pin some
more itself, so that the final distribution of marbles could be very different
(see Figure 3c).
One thing makes the quenched approximation interesting is that physicists
can actually do useful simulations with it. A quenched simulation typically
runs more than 100 times faster than an unquenched simulation, and in some
cases will give almost the same answer. In particular, one area of simulation
that is attracting a lot of interest is that of phase transitions in QCD.
Many familiar materials undergo transitions from one phase to another –
water can change to ice, petrol to vapour, and so on. The corresponding
transitions in QCD are of interest because they probably happened immediately
after the big bang, when the Universe was born.
At temperatures close to absolute zero, or in the absence of quarks,
gluons tend to bunch together to form objects called glueballs. As the system
is heated up, glueballs eventually ‘melt’ to form a sea of glue. Knowing
exactly how this happens could tell us a great deal more about the early
moments of the Universe, and by implication about why the Universe looks
as it does today. There is also the possibility of checking phase transition
calculations against experiment. If two very heavy nuclei, such as those
of two uranium atoms, are collided at high speeds, they might reach high
enough temperatures and densities to recreate those early moments. By looking
at the particles produced from such collisions, physicists can hope to characterise
the properties of the phase transition that occurs as that high temperature
drops.
The powerful QCD engines
Obviously, the larger the lattice used in a calculation, the more realistic
the results of the calculation will be. Similarly, the smaller the spacing
between lattice points, the more accurate the calculations on a lattice
of a particular size grows faster than the size of the lattice; doubling
the lattice size increases the amount of calculation by a factor of 16 or
more. As a result, since the early 1980s, physicists interest in QCD calculations
have been designing and building supercomputers in order to do their sort
of physics.
One thing that these ‘QCD engines’ have in common is their use of parallelism
to achieve high speed at a (relatively) low cost. Conventional computers,
from PCs to mainframes contain a single processor and a store of memory.
Computers of this type are often called von Neumann computers, after the
pioneering computer scientist and mathematician John von Neumann. Today,
however, many researchers and manufacturers are building parallel computers,
in which many processors work together, or in parallel, on a single problem.
Parallel computers can be much more cost-effective than conventional
von Neumann designs. To understand why, consider trying to move an ever-increasing
number of people around Britain by train. The wrong way to do this would
be to put all the engineering effort into designing and building a single
very fast train. Eventually, some physical limit on the speed of the train
would be reached, and even if it was not, the time taken to get people on
and off the train would eventually be greater than the time taken to make
the trip.
A better approach is to run many trains at the same time on different
tracks. Although these trains would be slower, they would also be cheaper,
and in most cases the total time for an individual’s journey would be reduced.
This immediately leads to some problems which the single train approach
would not have (avoiding collisions, timetabling, and so on), but can carry
a much higher volume of traffic.
Several British firms have been in the forefront of development of parallel
computing. Such companies as AMT (a spin-off from ICL), and Meiko of Bristol
have been building parallel computers as powerful and flexible as those
available anywhere. While some groups, notably at the University of Edinburgh,
have had successes with such general-purpose machines, much ‘hard-core’
QCD research is now being done on purpose-built computers.
The most powerful such machine in the world today is a 64-processor
computer built by Norman Christ’s group at Columbia University in New York.
This machine is a descendent of the 16-processor model they completed in
April 1985, which set the standard for QCD computing. The current Columbia
machine can do almost 6.5 billion calculations per second, or 6.5 gigaflops,
on lattices with 24 elements on each side. Elsewhere in New York, the GF11
supercomputer at IBM’s John Watson Research Center has 576 processors, and,
when completed, should be able to reach speeds of more than 10 gigaflops.
Outside the US, two of the most significant groups are at the University
of Tsukuba in Japan, where a 7-gigaflop machine called QCDPAX is currently
being doubled in size, and in Italy, where the APE (bee) project has brought
together researchers from several universities to build a 1-gigaflop machine.
The APE collaboration is working on a new machine which should have a theoretical
peak speed of 100 gigaflops in a year’s time.
Here in Britain the Science and Engineering Research Council has recently
provided 600,000 Pounds to add to the facilities in the Edinburgh’s Parallel
Computing Centre (EPCC) at the University of Edinburgh by installing equipment
potentially capable of delivering 2 gigaflops sustained performance. Researchers
at Edinburgh have been doing QED and QCD calculations on one of Meiko’s
large parallel computers, called a Computing Surface, since 1987. The SERC
money has been used to buy a new machine containing 32 of Meiko’s new high-performance
boards. Each of these boards carries two Intel i860 processors, which together
deliver a peak performance of more than 70 megaflops. This new computing
power is being used by particle physicists from Cambridge, Glasgow, Liverpool,
Oxford, and Southampton Universities as well as Edinburgh to investigate
the fundamental nature of matter.
QCD researchers have been at the forefront of developing a new type
of science, in which simulation replaces explicit calculation. The Standard
Model is a good physicists theory, but it cannot produce numerical results
to test against experiment using pen and paper alone. The only way to check
it is to do long, complicated, and expensive calculations of the type described
above. This has turned many physicists into programmers, just as Newton’s
invention of calculus turned natural philosophers into mathematicians 300
years ago.
* * *
Simulating problems on a computer
There are two fundamentally different ways to use a computer to do a
scientific calculation. The first is to write down a set of equations that
completely describes the phenomenon you are interested in, and then use
the computer to solve these equations. The second, which is becoming more
important as scientists try to investigate larger and more complicated problems,
is to write a program which ‘pretends’ to be the phenomenon of interest.
For example, suppose you wish to investigate the way in which the numbers
of foxes and rabbits in a field depend on one another. The first approach
would be to use a program to solve the equations that describe the relationships
between the number of foxes and rabbits, how good foxes are at catching
rabbits, and the rate at which foxes and rabbits breed.
If these equations are very complex, however, it may not be possible
to solve them exactly. In this case, the scientist can turn to simulation,
and write a program that keeps track of the movement and actions of individual
foxes and rabbits. The advantage of this approach is that the scientist
now needs to write down only the rules governing the actions of individual
rabbits. The advantage of this approach is that the scientist now needs
to write down only the rules governing the actions of individual animals,
which are often much simpler than the rules governing the interactions of
large numbers of animals. The disadvantage is that the simulation program
may be much larger than the equation-solving program, and need much more
computer time to produce results.
Greg Pendlebooth is a composite of several researchers in the department
of physics and the Edinburgh Parallel Computing Centre at the University
of Edinburgh.
