By Lars Brink*

One of the basic features in physics is the occurrence of forces that keep matter together. There are for example, the forces that keep the cells together to build up the human body, and there is the gravitational force that keeps us on the ground and the moon in orbit around the earth. We can ourselves exert forces when we push something and, by engineering, get some of the energy content in oil to produce a force on the wheels of a car to move it. From the macroscopic point of view we can imagine many different kinds of forces, forces that act at impact but also forces that act over a distance such as the gravitational one. In physics, though, we try to systematise and to find as many general concepts as possible. One such systematisation is to find out the ultimate constituents of matter. Another is to find out the forces that act between them. In the first case, we have been able to divide up matter into atoms and the atoms into nuclei and electrons, and then the nuclei into protons and neutrons. By colliding protons with protons or protons with electrons, particle physicists have uncovered that all matter can be built from a number of quarks (a concept introduced by Murray Gell-Mann in the 60's) and leptons (electrons and neutrinos and their heavier cousins). In the same process physicists have uncovered four basic forces that act between these matter particles - gravitation, electromagnetism, the strong and the weak nuclear force. Only the first two can be directly seen in the macroscopic world so let us first describe them.

The first quantitative theory of
gravitation based on observations was formulated by Isaac Newton
in 1687 in his *Principia.* He wrote that the gravity force
that acts on the sun and the planets depends on the quantity of
matter that they contain. It propagates to large distances and
diminishes always as the inverse of the square of the distance.
The formula for the force *F* between two objects with
masses *m*_{1} and *m*_{2} a distance
*r*away is thus

*F=Gm _{1}m_{2}/r^{2},*

where *G* is a constant of
proportionality, the gravitational constant. Newton was not fully
happy with his theory since it assumed an interaction over a
distance. This difficulty was removed when the concept of the
gravity field was introduced, a field that permeates space.
Newton's theory was very successfully applied to celestial
mechanics during the 18th and the beginning of the 19th century.
For example J.C. Adams and U.J.J. Leverrier were able to
conjecture a planet outside of Uranus from irregularities in its
orbit and subsequently, Neptune was found. One problem remained
though. Leverrier had in 1845 calculated that Mercury's
orbit precesses 35'' per century in contrast to the
Newtonian value that is zero. Later measurements gave a more
precise value of 43''. (The observed precession is
really 5270''/century, but a painstaking calculation
to subtract the disturbances from all the other planets gives the
value of 43''.) It was not until 1915 that Albert Einstein could
explain this discrepancy.

Galilei was the first to observe that
objects seemingly fall at the same speed regardless of their
masses. In Newton's equations the concept of mass occurs in
two different equations. The second law says that a force
*F* on a body with mass *m* gives an acceleration
*a* according to the equation *F=ma.* In the law of
gravity, the force of gravity *F* satisfies *F=mg,*
where *g* depends on the other bodies exerting a force on
the body (the earth usually, when we talk of the gravity force).
In both equations *m* is a proportionality factor (the
inertial mass and the gravitational mass) and there is no obvious
reason that they should be the same for two different objects.
However, all experiments indicate that they are. Einstein took
this fact as the starting point for his theory of gravitation. If
you cannot distinguish the inertial mass from the gravitational
one you cannot distinguish gravitation from an acceleration. An
experiment performed in a gravity field could instead be
performed in an accelerating elevator with no gravity field. When
an astronaut in a rocket accelerates to get away from earth he
feels a gravity force that is several times that on earth. Most
of it comes from the acceleration. If one cannot distinguish
gravity from acceleration one can always substitute the gravity
force by being in an accelerating frame. A frame in which the
acceleration cancels the gravity force is called an inertial
frame. Hence the moon orbiting the earth can instead be regarded
to be in an accelerating frame. However this frame will be
different from point to point since the gravity field changes.
(In the example with the moon the gravity field changes direction
from one point to another.) The principle that one can always
find an inertial frame at every point of space and time in which
physics follows the laws in the absence of gravitation is called
the *Equivalence Principle.*

The fact that the gravitational force can be thought of as
coordinate systems that differ from point to point means that
gravity is a geometric theory. The true coordinate system that
covers the whole of space and time is hence a more complex one
than the ordinary flat ones we are used to from ordinary
geometry. This type of geometry is called *Non Euclidean
Geometry.* The force as we see it comes from properties of
space and time. We say that space-time is curved. Consider a ball
lying on a flat surface. It will not move, or if there is no
friction, it could be in a uniform movement when no force is
acting on it. If the surface is curved, the ball will accelerate
and move down to the lowest point choosing the shortest path.
Similarly, Einstein taught us that the four-dimensional space and
time is curved and a body moving in this curved space moves along
a *geodesics* which is the shortest path. Einstein showed
that the gravity field is the geometric quantity that defines the
so-called proper time, which is a concept that takes the same
value in all coordinate systems similar to distance in ordinary
space. He also managed to construct equations for the gravity
field, the celebrated *Einstein's equations,* and with
these equations he could compute the correct value for the
precession for the orbit of Mercury. The equations also give the
measured value of the deflection of light rays that pass the sun
and there is no doubt that the equations give the correct results
for macroscopic gravitation. Einstein's theory of
gravitation, or *General Relativity,* as he called it
himself is one of the greatest triumphs of modern science.

It was James Clark Maxwell who, in 1865,
finally unified the concepts of electricity and magnetism into
one theory of electromagnetism. The force is mediated by the
electromagnetic field. The various derivatives of this field lead
to the electric and the magnetic fields, respectively. The theory
is not totally symmetric in the electric and the magnetic fields
though, since it only introduces direct sources to the electric
field, the electric charges. A fully symmetric theory would also
introduce magnetic charges, (predicted to exist by modern quantum
theory but with such huge magnitudes that free magnetic charges
must be extremely rare in our universe). For two static bodies
with charges *e*_{1} and *e*_{2} the
theory leads to *Coulomb's Law* giving the force

where again *k* is a proportionality
constant. Note the resemblance with Newton's law for
gravity. There is one difference though. While the gravitational
force always is attractive, the electromagnetic one can also be
repulsive. The charges can either have negative signs such as for
the electron or be positive as for the proton. This leads to the
fact that positive and negative charges tend to bind together
such as in the atoms and hence, screen each other and reduce the
electromagnetic field. Most of the particles in the earth screen
each other in this way and the total electromagnetic field is
very much reduced. Even so we know of the magnetic field of the
earth. Also in our bodies most charges are screened so there is a
very minute electromagnetic force between a human being and the
earth. The situation is very different for the gravity field.
Since it is always attractive, every particle in the earth
interacts with every particle in a human body, setting up a force
with is just our weight. However, if we compare the
electromagnetic and the gravitational forces between two
electrons we will find that the electromagnetic one is bigger by
a factor which is roughly 10^{40}. This is an
unbelievably large number! It shows that when we come to
microcosm and study the physics of elementary particles we do not
need to consider gravity when we study quantum electrodynamics,
at least not at ordinary energies.

When examining Maxwell's equations one finds that the
electromagnetic field travels with a finite velocity. This means
that *Coulomb's Law* is only true once the
electromagnetic field has had time to travel between the two
charges. It is a static law. One also finds that the
electromagnetic field travels as a wave just in the same way as
light does. It was Rømer who discovered that the velocity of
light is finite and Newton and Huygens who discovered that light
travels as waves in the late 17th century, and by the end of the
19th century the velocity of light was well established and seen
to agree with the velocity of the electromagnetic field. Hence it
was established that light is nothing but electromagnetic
radiation. In 1900 Max Planck proposed that
light is quantised in order to explain the black body radiation.
However, it was Albert Einstein who was the first to really
understand the revolutionary consequences of this idea when he
formulated the *photoelectric* effect. The electromagnetic
field can be understood as a stream of corpuscular bodies to be
called *photons* that make up the electromagnetic field. The
revolutionary aspect of this idea was that a stream of particles
also could behave as a wave and there was much opposition to the
idea from many established scientists of the day. It was not
until 1923 when Arthur Compton
experimentally showed that a light quanta could deflect an
electron just like a corpuscular body would do it, that this
debate was over.

If we think about the electric force between two charges as the electromagnetic field mediating it over a distance, we can now get a more fundamental picture as a stream of photons sent out from one particle to hit the other. This is a more intuitive picture than a force acting over a distance. Our macroscopic picture of a force is that something hits a body that then feels a force. In the microscopic world this is then again a way to understand a force. However, it is more complex. Suppose there are two charged particles that interact. Which particle is sending out a photon and which is receiving the photon if the two particles are identical as quantum mechanics tells us about fundamental particles? The answer must be that the picture should include both possibilities. The discovery that the electromagnetic field is quantised started the development of quantum mechanics and led us to a microcosm that is just built up by point-like objects and where forces occur when two particles hit each other.

Quantum mechanics as such led to many new
revolutionary concepts. One of the most important ones is
*Heisenberg's Uncertainty Relation* formulated by
Werner
Heisenberg in 1927, which states that one cannot measure
position and momentum or energy and time exactly simultaneously.
For a nucleus, one can either determine the position of an
electron and know nothing of its momentum or know its momentum
and nothing about its position. In the picture showing the force
field between two charges, we should think of it as photons
travelling from one charge to another. Hence the energy cannot be
determined better than what the uncertainty relation tells us
because of the uncertainty in the determination of the time.
Hence the special relativity relation for light that the photon
is massless which translates into the relation that the
energy^{2}=momentum^{2}c^{2} need not be
satisfied. If we put the energy and the tree-dimensional momentum
together into the four-momentum we see that it is not constrained
by the masslessness condition, we say that the photon is virtual
and consequently has a (virtual) mass. We can thus interpret the
process above as either a certain photon going from particle 1 to
particle 2 with a certain four-momentum or as one from particle 2
to particle 1 with the opposite four-momentum. When two charges
are far away the uncertainty relation gives little freedom and
the photon is closer to masslessness, We know that
*Coulomb's law* seems to be valid at the longest
distances so it must be set up by the photons close to
masslessness. If two charges are close there should be more terms
to the force. Incidentally in order to measure the velocity of
light the photons must interact. Hence there is a slight
uncertainty in its mass and a slight uncertainty in its velocity.
However, we measure always the same velocity for light which
means that at the macroscopic distances that we measure, the
virtuality and hence the mass of the photon is essentially zero
to a very good accuracy. It is then consistent to say that the
velocity of light is constant.

The full description of the electromagnetic
force between elementary particles was formulated by Sin-Itiro Tomonaga,
Richard Feynman
and Julian
Schwinger in independent works in the 1940's. They
formulated *Quantum ElectroDynamics (QED).* This is a theory
that takes full account of quantum physics and special relativity
(which is the underlying symmetry of *Maxwell's
Equations*). It is very elegantly formulated by so-called
*Feynman diagrams,* where the elementary particles exchange
photons as was described above and where each diagram constitutes
a certain mathematical expression that can be obtained from some
basic rules for the propagation of virtual particles and from the
interaction vertices. The simplest diagram for the interaction
between two electrons is

This diagram in fact leads to
*Coulomb's law.* Feynman now instructs us that we can
combine any line for a propagating electron (or when it travels
backwards, the positron) and any line for a propagating photon
tied together with the vertex where an electron line emits a
photon to make up new diagrams. Every other diagram differing
from the one above constitutes quantum corrections to the basic
force. It was through the work of the three scientists above that
it was shown that every such diagram can be made to make sense to
give finite answers. It is said that *QED* is
*renormalisable.* The strength of the force as in
*Coulomb's law* is governed by the magnitude of the
vertex which is the electric charge *e* in QED and for the
diagram above it is proportional to the square of *e* and is
the *Fine Structure Constant* = 1/137. Since this is a small number
it makes sense to write the amplitude in a series of terms with
higher and higher powers of since that factor will be smaller and
smaller for ever increasing complexity of the diagram. The higher
order terms are higher quantum corrections and the
*perturbation expansion* that we have defined will have
smaller and smaller terms as we go to higher quantum
corrections.

Since there were only two basic forces
known in the beginning of the 20th century, gravitation and
electromagnetism, and it was seen that electromagnetism is
responsible for the forces in the atom, it was natural to believe
that it was also responsible for the forces keeping the nucleus
together. In the 1920's it was known that the nuclei
contain protons, in fact the hydrogen nucleus is just a proton,
and somehow it was believed that electrons could be involved in
keeping the protons together. However, an idea like this has
immediate problems. What is the difference between the electrons
in the nucleus and the ones in orbit around the nucleus? What is
the consequence of Heisenberg's uncertainty relation if
electrons are squeezed into the small nucleus? The only support
for the idea, apart from there being no other known elementary
particles, was that in certain radioactive decays electrons were
seen to come from the nucleus. However, in 1932 James Chadwick
discovered a new type of radiation that could emanate from the
nuclei, a neutral one and his experiment showed that there are
indeed electrically neutral particles inside the nuclei, which
came to be called neutrons. Soon after Eugene Wigner explained
the nuclei as a consequence of two different nuclear forces. The
*Strong Nuclear Force* is an attractive force between
protons and neutrons that keep the nucleus together and the
*Weak Nuclear Force* is responsible for the radioactive
decay of certain nuclei. It was realized that the strength of the
two forces differed a lot. The typical ratio is of the order of
10^{14} at ordinary energies.

A natural idea now was to search for a mechanism like the one in electromagnetism to mediate the strong force. Already in 1935 Hideki Yukawa proposed a field theory for the strong interaction where the mediating field particle was to be called a meson.

However, there is a significant difference between the strong
force and the electromagnetic one in that the strong force has a
very short range (typically the nuclear radius). This is the
reason why it has no classical counterpart and hence had not been
discovered in classical physics. Yukawa solved this problem by
letting the meson have a mass. Such a particle was also
subsequently seemingly found from cosmic rays by Carl Anderson. The
discovery of nuclear fission in the late 1930's led to an
enormous interest in nuclear physics and in the war years most
physicists worked on problems with fission so it was not until
after the war that Yukawa's ideas were taken up again. It
was then realized that the particle found by Anderson could not
be the meson of strong interactions, since it interacted far too
little with matter, and it was then shown that this particle, now
called the muon, is a heavy cousin of the electron. However, the
meson, now called pion, was finally discovered in cosmic rays by
Cecil Powell in
1947 and its properties were measured. A new dilemma now
appeared. When the big accelerators started to operate in the
1950's, the pions were produced vindicating Yukawa's
theory, but when his field theory was scrutinised according to
the rules set up by Feynman, it was shown that indeed the theory
is renormalisable but the coupling constant is huge, larger than
one. This means that a diagram with several interactions will
give a larger contribution than the naive one with the exchange
of only one pion, which is the one though that does gives a rough
picture of the scattering of two protons. The perturbation
expansion does not make sense. Also the scattering of protons
produced new strongly interacting particles beside the pion,
which were named hadrons. Indeed a huge menagerie of elementary
particles were discovered, some of them with a life time of some
10^{-8} to 10^{-10} s and some with a lifetime of
10^{-23} s. This problem was solved by Murray Gell-Mann
when he proposed that all the strongly interacting particles are
indeed bound states of even more fundamental states, the
*quarks.* This idea was eventually experimentally verified
in the Stanford experiments in the years around 1970 led by
Jerome Friedman,
Henry Kendall
and Richard
Taylor. To understand the forces inside the nucleus one
really had to understand the field theory for quarks. Before
describing the forces between quarks we have to discuss the other
nuclear force, the weak one.

In 1896 Henri Becquerel
discovered that uranium salts emit a radiation; they are
*radioactive.* His work was followed up by Marie and Pierre
Curie who discovered that several atoms disintegrated by sending
out radioactivity. With the discovery of the neutron it was
realized that this phenomenon is another aspect of a force at
work. It was found that the neutron decays into a proton and an
electron and a then hypothetical particle proposed by Wolfgang
Pauli, which came to be called the neutrino (really the
antineutrino). Since in the nucleus the mass of the nucleons are
virtual the process can also go the other way in which a proton
decays into a neutron, a positron and a neutrino. The first to
set up a model for this interaction was Enrico Fermi in which it
was supposed that the interaction was instantaneous among the
matter particles. In the late 1950s Fermi's theory
was modified to account for parity violation by Marshak and
Sudarshan and by Feynman and Gell-Mann. Parity violation of the
weak interactions had been postulated by Tsung-Dao Lee and
Chen Ning Yang
in 1956 and experimentally verified by Wu and collaborators the
year after. (The weak interactions can distinguish between left
and right.)

However, the model introduced had severe problems. It is not renormalisable so it cannot really make sense as a general theory. On the other hand the model worked extremely well for many processes. How could one reconcile these two facts? During the 1960's new field theoretic descriptions were proposed and to reconcile the facts above one introduced mediating particles that were extremely heavy. For low energy processes such a particle can only propagate a very short distance and in practice it will look as if the interaction takes place in one point giving the model above for the energies that at the time could be probed. The scheme used, the so-called ‘Non-Abelian Gauge Theories' were used by Sheldon Glashow, Steven Weinberg and Abdus Salam in independent works to suggest a model that would generalise the model above. Such a field theory is a generalisation of QED in which there are several mediating particles which also can have self interactions. In the beginning of the 1970's this scheme of models were proven to be renormalisable and hence good quantum theories by Gerhard ‘tHooft and Tini Veltman. Overwhelming experimental evidence for the model was gathered in the 1970's and finally in 1983 the mediating particles were discovered at CERN in an experiment led by Carlo Rubbia and Simon van der Meer. Indeed the mediating particles are very heavy, almost 100 times the mass of the proton.

A remarkable feature of the SLAC experiments that verified the existence of quarks was 'scaling'. The cross sections for the deep inelastic scattering of electrons on protons depended on fewer kinematical variables for higher energies. The cross sections scaled. This phenomenon was theoretically suggested by James Bjorken and the data showed it clearly. Richard Feynman explained it by assuming that the protons consisted of point-like constituents. To explain scaling these constituents must have a coupling strength that decreases with energy, opposite to the case of QED. This was called 'asymptotic freedom'. It was quite difficult to believe that a quantum field theory could be asymptotically free since the energy dependence of the coupling constant is due to the screening from pairs of virtual particles. Relativistic quantum mechanics allow for such pairs if they do not live too long. This is due to Heisenberg's uncertainty principle and the fact that energy is the same as mass according to Einstein's famous formula.

Asymptotic freedom must mean that the quark charges are antiscreened, which as said was hard to believe to exist in a quantum field theory. However, in 1973, David Gross, David Politzer and Frank Wilczek simultaneously found that for a non-abelian gauge field theory the requirement of asymptotic freedom is satisfied if there are not too many quarks. The key to the solution was that the vector particles mediating the force, the gluons, do indeed antiscreen. This can be understood since the charges of the quarks and the gluons, the "colour charges" satisfy more complicated relations than the simpler electric charges. There are three different colours and their anticolours. While the quarks have a colour charge, the gluons have a colour and an anticolour charge. Hence virtual gluons can line up with charges screening each other while the strength of the field increases.

The discovery of asymptotic freedom opened up for a non-abelian gauge field theory for the interactions among quarks and it was called QuantumChromodynamics, QCD. Over the years this theory has been very successfully tested at the large accelerators and it is now solidly established as the theory of the strong interactions.

The success of non-abelian gauge theories showed that
all the interactions could be unified in a common framework. This led to
the so-called Standard Model in which all the matter particles are treated
together, i.e. the electron and its heavier partners the muon and the tau-particle
and the corresponding neutrinos, which all have only weak interactions, together
with the quarks which can have both strong and weak interactions. The force
particles, i.e. the mediators, are then the photon for electromagnetism,
the *W* and *Z *particles for the weak force and the gluons for the strong force.
Even though the Standard Model unifies the interactions there are differences
in the details. The photon and the gluons are massless particles while the
*W* and *Z* particles have a mass. The *photon* leads to Coulomb's law for
large distances while the *gluons* lead to a confining force between the quarks.
This is in fact due to the asymptotic freedom, which can also be interpreted
to say that the coupling strength increases with lower energy, which quantum
mechanically also means that it increases with distance. In fact this
increase is like the one for a spring, such that the quarks are permanently
bound in the hadrons. Even so the properties of the gluons have been firmly
established by experimenters.

In the standard model above there is no
mentioning of the gravitational force. It has been said that it
is so tremendously weak that we do not need to take it into
account at particle experiments. However, on general grounds
there must be a quantum version of the gravity force that acts at
small enough distances. If we try to just copy the quantisation
of the electromagnetic field in terms of photons we should
quantize the gravity field into so-called *gravitons.*
However, the procedure of Feynman, Tomonaga and Schwinger does
not work here. Einstein's gravity is non-renormalisable.
Where is the problem? Is it Einstein's theory or quantum
mechanics that is not complete? The two great conceptual
milestones of the 20th century, Quantum Mechanics and
Einstein's General Relativity are simply not consistent
with each other. Einstein thought for his whole life that quantum
mechanics is indeed incomplete, but so many tests of it have by
now been made that physicists are instead trying to generalise
Einstein's theory. The remarkable success with the Standard
Model has also shown that the idea of unification of the forces
is a valid one. Why are there four different forces or are they
really different? They do indeed, show up as different forces in
the experiments we do, but the Standard Model shows that the
electromagnetic and the weak forces are unified for energies
above 100 GeV. Similarly the model shows that also the strong
force seemingly so different unifies with the other one at
energies above 10^{15}GeV. Can the gravitational one be
fit into this scheme?

It can be shown that at energies of the
order of 10^{19} GeV the gravity force will be as strong
as the other ones, so there should be a unification of all the
forces at least at that energy, which is an energy so
unbelievably high that it has only occurred in our universe at a
time 10^{-42} s after the Big Bang. However, physics
should also be able to describe phenomena that occurred then, so
there should be a unified picture which also includes gravity.
Such a scheme has now been proposed, *The Superstring Model*
in which particles are described by one-dimensional objects,
strings. This model indeed gives Einstein's theory for low
energies and can be made compatible with the Standard Model at
the energies where it has been probed. It is also a finite
quantum theory so a perturbation theory for gravity based on the
Superstring Model is indeed consistent. It is still too early to
say if this is the final 'theory of everything', but
there is no paradox or inconsistency in the model as far as has
been understood. Finally the model makes one more unification,
namely of the matter particles and the force particles, having
just one sort of particles. This is also the ultimate goal of
physicists, to have one unified force and one unified kind of
particles.

*** **Lars Brink was born in 1943. He has
been a professor of theoretical elementary particle physics since
1986 at Chalmers University of Technology in Göteborg,
Sweden. He was a fellow in the Theory Group at CERN, 1971-73 and
was an scientific associate at Caltech, 1976-77. He has been a
visiting scientist for longer and shorter periods at CERN,
Caltech, ITP in Santa Barbara as well as many other institutions
around the world at numerous occasions. He was vice dean of
physics at Chalmers, 1987-93 and was the chairman of the board of
Nordita in Copenhagen, 1990-93 and a member of the board in
1993-97. He is the chairman of the board of International Center
for Fundamental Physics in Moscow since 1993. He was elected as
member of the Swedish Royal Academy of Sciences in 1997 and as
member of its Nobel Committee for Physics in 2001. He is also the
coordinator of the EU Network "Superstring Theory" since the year
2000. His scientific work has been mainly in elementary particle
theory especially in the attempts to unify all fundamental
interactions. He is one of the pioneers of the Superstring Theory
and has also been much involved in supersymmetric quantum field
theories. One highlight here is the construction and proof of the
first finite quantum field theory in four spacetime
dimensions.

First published 9 August 2001

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