**Quantum Gravity**

Quantum Gravity is reputed to be one of the most difficult puzzles of science. In practical terms it is probably of no direct relevance and may even be impossible to verify by experiment. But for physicists it is the holy grail which may enable them to complete the unification of all fundamental laws of physics.

The problem is to put together general relativity and quantum mechanics into one self consistent theory. The difficulty is that the two parts seem to be incompatible, both in concept and in practice. Conceptually, it is the nature of space and time which present fundamental differences. A direct approach, attempting to combine general relativity and quantum mechanics, while ignoring conceptual differences, leads to a meaningless quantum field theory with unmanageable divergences.

There have, in fact, been many attempts to create a theory of quantum gravity. In this article I will first outline the nature of general relativity and quantum mechanics with emphasis on their similarities and differences. Then I will briefly review some of the main stream approaches to quantum gravity. Finally I will talk about some ways in which these ideas now seem to be converging.

**General Relativity**

General relativity is Einstein’s monumental theory of gravity. It is based on two fundamental principles:

The principle of relativity which states that all basic laws of physics should take a form which is independent of any reference frame, and

The principle of equivalence which states that it is impossible to distinguish the effects of gravity from the effects of being in an accelerated frame of reference.

Einstein struggled with the consequences of these principles for several years, constructing many thought experiments to try to understand what they meant. Finally he learnt about Riemann’s mathematics of curved geometry and realised that a new theory could be constructed in which the force of gravity was a consequence of the curvature of space-time.

In constructing that theory, Einstein was not significantly influenced by any experimental result which was at odds with the Newtonian theory of gravity. He knew, however, that Newtonian gravity was inconsistent with his theory of special relativity and he knew there must be a more complete self consistent theory. A similar inconsistency now exists between quantum mechanics and general relativity and, even though no experimental result is known to violate either theory, physicists now seek a more complete theory.

In the decades that have followed Einstein’s discovery, a number of experimental confirmations of general relativity have been found but there still remains a possibility that it may not be accurate on very large scales, or under very strong gravitational forces. In any case, it is sure to break down under the conditions which are believed to have existed at the big bang where quantum gravity effects were important.

One of the most spectacular predictions of general relativity is that a dying star of sufficient mass will collapse under its gravitational weight into an object so compressed that not even light can escape its pull. These objects are known as black holes. Astronomers now have a growing list of celestial objects which they believe are black holes because of their apparent high density. The accuracy of Einstein’s theory may be stringently tested in the near future when gravitational wave observatories such as LIGO come on-line to observe such catastrophic events as the collisions between black holes.

**Quantum Mechanics**

The Quantum theory was founded before Einstein began his theory of relativity and took much longer to be completed and understood. It was Planck’s observations of quanta in the spectrum of black body radiation which first produced signs that the classical theories of mechanics were due for major revisions.

Unlike general relativity which was essentially the work of one man, the quantum theory required major contributions from Bohr, Einstein, Heisenberg, Schroedinger, Dirac and many others, before a complete theory of quantum electrodynamics was formulated. In practical terms, the consequences of the theory are more far reaching than those of general relativity. Applications such as transistors and lasers are now an integral part of our lives and, in addition, the quantum theory allowed us to understand chemical reactions and many other phenomena.

In the 1960’s and 70’s, further discoveries in quantum field theory have led to successful theories of the nuclear reactions and, in consequence, almost all ordinary physical phenomena can now be attributed to quantum interactions, even if the exact mechanisms are not always fully understood. The electromagnetic and weak nuclear interactions are unified into one force while the strong nuclear interaction is a force of a similar nature known as a gauge theory. Together these forces and all observed particles are combined into one self consistent theory known as the standard model of particle physics.

Despite such spectacular success, confirmed in ever more detail in high energy accelerator experiments, the quantum theory is still criticised by some physicists who feel that its indeterministic nature and its dependency on the role of observer suggest an incompleteness.

Unification

Since Newton set the foundations of physics, progress has come mostly in the form of unification. Maxwell unified electricity, magnetism and light into one theory of electromagnetism. Einstein unified space, time and gravity into one theory of general relativity. More recently, the nuclear forces have been (partially) unified with the electromagnetic force by Weinberg and others.

According to conventional wisdom among physicists, the process of unification will continue until all physics is unified into one neat and tidy theory. There is no a priori reason to be so sure of this. It is quite possible that physicists will always be discovering new forces, or finding new layers of structure in particles, without ever arriving at a final theory. It is quite simply the nature of the laws of physics as we currently know them that inspires the belief that we are getting closer to the end.

After physicists discovered the atom, they went on to discover that it was composed of electrons and a nucleus, then that the nucleus was composed of protons and neutrons, then that the protons and neutrons were composed of quarks. Should we expect to discover that quarks and electrons are made of smaller particles? This is possible but there are a couple of reasons to suppose not. Firstly there are far fewer particles at this level than there ever were at higher levels. Secondly, their interactions are described by a clean set of gauge bosons through renormalisable field theories. Composite interactions, such as pion exchange, do not take such a tidy form. These reasons in themselves are not quite enough to rule out the possibility that quarks, electrons and gauge bosons are composite but they reduce the number of ways such a theory could be constructed. In fact all viable theories of this type which have been proposed are now all but ruled out by experiment. There may be a further layer of structure but it is likely to be different. It is more common now for theorists to look for ways that different elementary particles can be seen as different states of the same type of object. The most popular candidate for the ultimate theory of this type is superstring theory, in which all particles are just different vibration modes of very small loops of string.

Note added: Just a few weeks after writing this, experimenters at Fermilab announced the discovery of evidence for structure within quarks!

Physicists construct particle accelerators which are sort of like giant microscopes. The higher the energy they can produce, the smaller the wavelength of the colliding particles and the smaller the distance scale they probe. In this way physicists can see the quarks inside protons, not through direct pictures but through scattering data. Other things that happen as the energy increases is that new heavy particles are formed and forces become unified. It is impossible to be sure about what will happen the next time a new, more powerful accelerator is built, but physicists can make theories about it.

In the next decade new accelerator experiments at CERN will probe beyond the electro-weak scale. There is some optimism that new physics will be found but nothing is certain.

**Planck Scale**

At first sight it might seem ridiculous to suppose that we can invent valid theories about physics at high energies before doing experiments. However, theorists have already demonstrated a remarkable facility for doing just that. The standard model of particle physics was devised in the 1960’s and experimentalists have spent the last three decades verifying it. The reason for this success is that physicists recognised the importance of certain types of symmetry and self-consistency conditions in quantum field theory which led to an almost unique model for physics up to the electro-weak unification energy scale, with only a few parameters such as particle masses to be determined.

The situation now is a little different. Experimentalists are about to enter a new scale of energies and theorists do not have a single unique theory about what can be expected. They do have some ideas, in particular it is hoped that supersymmetry may be observed, but we will have to wait and see.

Despite these unknowns there are other more general arguments which tell us things about what to expect at higher energies. When Planck initiated the quantum theory he recognised the significance of fundamental constants in physics, especially the speed of light (known as c) and his newly discovered Planck constant (known as h). Scientists and engineers have invented a number of systems of units for measuring lengths, masses and time, but they are entirely arbitrary and must be agreed by international convention. Planck realised that there should be a natural set of units in which the laws of physics take a simpler form. The most fundamental constants, such as c and h would simply be one unit in that system.

If one other suitable fundamental constant could be selected, then the units for measuring mass, length and time would be determined. Planck decided that Newton’s gravitational constant (known as G) would be a good choice. Actually there were not many other constants, such as particle masses known at that time, otherwise his choice might have been more difficult. By combining c, h and G Planck defined a system of units now known as the Planck scale. He calculated that the Planck unit of length is very small, about 10-35 (ten to the power of minus 35) metres. To build an accelerator which could see down to such lengths would require energies about 1015 times larger than those currently available. Note that units of speed and energy can be built from the three basic Planck units but to measure temperature and charge as well we have to also set Boltzmann’s constant and the charge on the electron to one unit. In this way we can devise a fundamental system of measurement for all physical quantities.

Physicists have since sought to understand what the Planck scale of units signifies. Those who work with particles believe that at the Planck scale all the four forces of nature, including gravity, are unified. Physicists who specialise in general relativity have a different idea. In 1955 John Wheeler argued that when you combine general relativity and quantum mechanics you will have a theory in which the geometry of space-time is subject to quantum fluctuations, He computed that these fluctuations would become significant if you could look at space-time on length scales as small as the Planck length. Sometimes physicists talk about a space-time foam at this scale but we don’t yet know what it really means. For that we will need the theory of quantum gravity.

Without really knowing too much for certain physicists guess that at the Planck scale all forces of nature are unified and quantum gravity is significant. It is at the Planck scale that they expect to find the final and completely unified theory of the fundamental laws of physics.

**The Small Scale Structure of Space-Time**

It seems clear that to understand quantum gravity we must understand the structure of space-time at the Planck length scale. In the theory of general relativity space-time is described as a smooth continuous manifold but we cannot be sure that this is correct for very small lengths and times. We could compare general relativity with the equations of fluid dynamics for water. They describe a continuous fluid with smooth flows in a way which agrees very well with experiment. Yet we know that at atomic scales water is something very different and must be understood in terms of forces between molecules whose nature is completely hidden in the ordinary world. If space-time also has a complicated structure at the tiny Planck length, way beyond the reach of any conceivable accelerator, can we possibly hope to discover what it is?

If you asked a bunch of mathematicians to look for theories which could explain the fluid dynamics of water, without them knowing anything about other physics and chemistry, then they would probably succeed in devising a whole host of mathematical models which work. All those models would probably be very different, limited only by the imagination of the mathematicians. None of them would correspond to the correct description of water molecules and their interactions. The same might be true of quantum gravity. Nevertheless, the task of putting together general relativity and quantum mechanics together into one self consistent theory has not produced a whole host of different and incompatible theories. The clever ideas which have been developed have enigmatic things in common. It is quite possible that all the ideas are partially correct and are aspects of one underlying theory which is within our grasp. It is time now to look at some of those ideas.

**Attempts to do Quantum Gravity**

The most direct way to try to quantise quantum gravity is to use perturbative quantum field theory. This is a procedure which has been applied with great success to electrodynamics. To do the same thing for gravity it is necessary to first construct a system of non-interacting gravitons which represent a zero order approximation to quantised gravitational waves in flat space-time. These hypothetical gravitons must be spin two massless particles because of the form of the metric field in general relativity.

The next step is to describe the interactions of these gravitons using the perturbation theory of quantum mechanics, which are defined by a set of Feynman diagrams derived from Einstein’s gravitational field equations. For electrodynamics this can be made to work, but only after conveniently cancelling divergent anomalies which appear in the calculations. For gravity this simply cannot be done. The resulting quantum field theory is said to be unrenormalisable and is incapable of giving any useful result.

Because quantum gravity is an attempt to combine two different fields of physics, there are two distinct groups of physicists involved. These two groups form a different interpretation of the failure of the direct attack. The relativists say that it is because gravity cannot be treated perturbatively. To try to do so destroys the basic principles on which relativity was founded. It is, for them, no surprise that this should not work. Particle physicists say that if a field theory is non-renormalisable then it is because it is incomplete. The theory must be modified and new fields must be added to cancel divergences.

**Supersymmetry**

The first significant progress in the problem of quantum gravity was made by particle physicists. They discovered that a new kind of symmetry called supersymmetry was very important. particles can be classed into two types; fermions such as quarks and electrons, and bosons such as photons and Higgs particles. Supersymmetry allows the two types to intermix. With supersymmetry we have some hope to unify the matter fields with radiation fields.

Particle physicists discovered that if the symmetry of space-time is extended to include supersymmetry, then it is necessary to supplement the metric field of gravity with other matter fields. Miraculously these fields led to cancellations of many of the divergences in perturbative quantum gravity. This has to be more than coincidence. At first it was thought that such theories of Supergravity might be completely renormalisable. After many long calculations this hope faded.

A funny thing about supergravity was that it works best in ten dimensional space-time. This inspired the revival of an old theory called Kaluza-Klein theory, which suggests that space-time has more dimensions than the four obvious ones. The extra dimensions are not apparent because they are curled up into a small sphere with a circumference as small as the Planck length. This theory provides a means to unify the gauge symmetry of general relativity with the internal gauge symmetries of particle physics.

The next big step taken by particle physicists came along shortly after. Green and Schwarz realised that a theory which had originally been studied as a theory of the strong nuclear force was actually more interesting as a theory of gravity. This was the beginning of string theory. Combining string theory and supergravity to form superstring theory quickly led to some remarkable discoveries. A small set of string theories in ten dimensions were perfectly renormalisable. This was exactly what they were looking for.

It seemed once again that the solution was near at hand, but nature does not give up its secrets so easily. The problem now was that there is a huge number of ways to apply Kaluza-Klein theory to the superstring theories. Hence there seem to be a huge number of possible unified theories of physics. The perturbative formulation of string theory makes it impossible to determine the correct way.

Recently there has been renewed hope for string theory from the discovery that different string theories are connected. They may all be parts of one unique theory after all.

**Canonical Quantum Gravity**

While particle physicists were making a lot of noise about superstring theory, relativists have been quietly trying to do things differently. Many of them take the view that to do quantum gravity properly you must respect its diffeomorphism symmetry. The Wheeler-DeWitt equation together with a Hamiltonian constraint equation, describe the way in which the quantum state vector should evolve according to this canonical approach.

For a long time there seemed little hope of finding any solutions to the Wheeler-DeWitt equation. Then in 1986 Ashtekar found a way to reformulate Einstein’s equations of gravity in terms of new variables. Soon afterwards a way was discovered to find solutions to the equations. This is now known as the loop representation of quantum gravity. Mathematicians were surprised to learn that knot theory was an important part of the concept.

The results from the canonical approach seem very different from those of string theory. There is no need for higher dimensions or extra fields to cancel divergences. Relativists point to the fact that a number of field theories which appear to be unrenormalisable have now been quantised exactly. There is no need to insist on a renormalisable theory of quantum gravity. On the other hand, the canonical approach still has some technical problems to resolve. It could yet turn out that the theory can only be made fully consistent by including supersymmetry.

As well as their differences, the two approaches have some striking similarities. In both cases they are trying to be understood in terms of symmetries based on loop like structures. It seems quite plausible that they are both aspects of one underlying theory. Other mathematical fields are common features of both, such as knot theory and topology. Indeed there is now a successful formulation of quantum gravity in three dimensional space-time which can be regarded as either a loop representation or a string theory. A number of physicists such as Lee Smolin are looking for a more general common theory uniting the two approaches.

**Black Hole Thermodynamics**

Although there is no direct empirical input into quantum gravity, physicists hope to accomplish unification by working on the requirement that there must exist a mathematically self consistent theory which accounts for both general relativity and quantum mechanics as they are separately confirmed experimentally. It is important to stress the point that no complete theory satisfying this requirement has yet been found. If just one theory could be constructed then it would have a good chance of being correct.

Because of the stringent constraints that self consistency enforces, it is possible to construct thought experiments which provide strong hints about the properties a theory of quantum gravity has to have. There are two physical regimes in which quantum gravity is likely to have significant effects. In the conditions which existed during the first Planck unit of time in our universe, matter was so dense and hot that unification of gravity and other forces would have been realised. Likewise, a small black hole who’s mass corresponds to the Planck unit of mass provides a thought laboratory for quantum gravity.

Black holes have the property that the surface area of their event horizons must always increase. This is suggestively similar to the law that entropy must increase, and it led Bekenstein to conjecture that the area of the event horizon of a black hole is in fact proportional to its entropy. If this is the case then a black hole would have to have a temperature and obey the laws of thermodynamics. In the 1970’s Stephen Hawking investigated the effects of quantum mechanics near a black hole using semi-classical approximations to quantum gravity. He discovered the unexpected result that black holes do emit thermal radiation in a way consistent with the entropy law of Bekenstein.

This forces us to conclude that black holes can emit particles and eventually evaporate. For astronomical sized black holes the temperature of the radiation is minuscule and certainly beyond detection, but for small black holes the temperature increases until they explode in one final blast. Hawking realised that this creates a difficult paradox which would surely tell us a great deal about the nature of quantum gravity if we could understand it.

The entropy of a system can be related to the amount of information required to describe it. When objects are thrown into a black hole the information they contain is hidden from outside view because no message can return from inside. Now if the black hole evaporates, this information will be lost in contradiction to the laws of thermodynamics. This is known as the black hole information loss paradox.

A number of ways on which this paradox could be resolved have been proposed. The main ones are,

The lost information escapes to another universe

The final stage of black hole evaporation halts leaving a remnant particle which holds the information.

There are strict limits on the amount of information held within any region of space to ensure that the information which enters a black hole cannot exceed the amount represented by its entropy.

Something else happens which is so strange we can’t bring ourselves to think of it.

The first solution would imply a breakdown of quantum coherence. We would have to completely change the laws of quantum mechanics to cope with this situation. The second case is not quite so bad but it does seem to imply that small black holes must have an infinite number of quantum numbers which would mean their rate of production during the big bang would have been divergent. It might be possible to find a way round this but anyway, it is an ugly solution!

Assuming that I have not missed something out, which is a big assumption, we must conclude that the amount of entropy which can be held within a region of space is limited by the area of a surface surrounding it. This is certainly counterintuitive because you would imagine that you could write information on bits of paper and the amount you could cram in would be limited by the volume only. This is false because any attempt to do that would eventually cause a black hole to form. Note that this rule does not force us to conclude that the universe must be finite because there is a hidden assumption that the region of space is static which I did not mention.

If the amount of information is limited then the number of physical degrees of freedom in a field theory of quantum gravity must also be limited. Inspired by this observation, Gerard ‘t Hooft, Leonard Susskind and others have proposed that the laws of physics should be described in terms of a discrete field theory defined on a space-time surface rather than throughout space-time. They liken the way this might work to that of a hologram which holds a three dimensional image within its two dimensional surface.

Rather than being rejected as a crazy idea, this theory has been recognised by many other physicists as being consistent with other ideas in quantum gravity.

**Quantised Space-time**

Although there has been considerable progress on the problem of quantising gravity, it seems likely that it will not be possible to complete the solution without some fundamental change in the way we think about space-time. All the approaches I have described suggest that the Planck units of length and time define a minimum scale of measurement. Indeed the same conclusion can be reached using fairly general arguments based on the Heisenberg uncertainty principle applied to the metric field of gravity.

One possibility would be that space-time is some kind of lattice structure at small scales. A regular cubic lattice structure is generally regarded as an unacceptable alternative because it destroys space-time symmetry. A random lattice is more plausible. Numerical studies of statistical randomly triangulated surfaces are quite encouraging. The Regge calculus describes such a discretisation of gravity and is akin to topological lattice quantum field theories as models of quantum gravity in three dimensions.

As far back as 1947, Synder attempted to quantise space-time by treating space-time co-ordinates as non-commutating operators. The original formulation was unsuccessful but recent work on quantum groups have initiated a revival of this approach. This approach also leads to a discrete interpretation of space-time. Another related topic is non-commutative geometry in which space-time itself is regarded as secondary to the algebra of fields which can be generalised to have non-commuting products.

Still this seems to be not quite radical enough to account for quantum gravity. Some physicists believe that we must modify our views

sufficiently to allow for dynamical changes in the number of space-time dimensions.

To face the quantum gravity challenge we need new insights and new principles like those which guided Einstein to the correct theory of gravity.

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