(x,t). The equation is as follows:| W |
It has been apparent since early times that there is something different about the mathematical properties of the real numbers and the quantities of measurement in physics at small scales. Riemann himself remarked on this disparity even as he constructed the formalism which would be used to describe the space-time continuum for the next century of physics in 1876.
In mathematics numbers have unphysical properties like being an exact ratio of two integers. When you measure a distance or time interval you cannot declare the result to be a rational or irrational number no matter how accurate you manage to be. Furthermore it appears that there is a limit to the amount of detail contained in a volume of space. If we look under a powerful microscope at a grain of dust we do not expect to see minuscule universes supporting the complexity of life seen at larger scales. Structure becomes simpler at smaller distances. Surely there must be some minimum length at which the simplest elements of natural structure are found and surely this must mean space-time is discrete rather than continuous?
This style of argument tends to be persuasive only to those who already believe the hypothesis. It will not make many conversions. After all, the modern formalism of axiomatic mathematics leaves no room for Zeno's paradox. In the fifth century BC the philosopher Parmedies and his disciple Zeno of Elea tried to discredit the senses by posing paradoxes about the divisibility of space-time. In a race between the Archiles and the tortoise, the tortoise was given a head start. To catch him up Archiles must first half the distance between them, then half the remaining distance again. No matter how many times he halves the distance he will not have caught the tortoise. If space and time are infinitely divisible Archiles cannot pass the tortoise according to Zeno. Such thoughts influenced the atomists of ancient Greece, and a more complete philosophy of atomic space and time was developed by the Kalam of Bahgdad from the 9th century.
But axiomatic mathematics has dispelled Zeno's paradox. It is possible to talk about limits and infinity without reaching any mathematical contradiction and it can be proven that the sum of an infinite number of halving intervals is finite. Although some philosophers such as Bertrand Russell persisted with such arguments and developed a detailed and general philosophy of atomism, there are few physicists who would agree that logic and philosophy alone can tell us whether or not space and time are discrete.
However, experimental facts are a different matter and the discovery of quantum theory with its discrete energy levels and the Heisenberg uncertainty principle led physicists to speculate that space-time itself may be discrete as early as the 1930s. In 1936 Einstein expressed the general feeling that the success of the quantum theory points to a purely algebraic method of description of nature and the elimination of continuous function and space-time continuum from physics.
Heisenberg himself noted that the laws of physics must have a fundamental length in addition to Planck's constant and the speed of light, to set the scale of particle masses. At the time it was thought that this length scale would be around 10-15m corresponding to the masses of the heaviest elementary particles known at the time. Searches for non-local effects in high energy particle collisions have now given negative results for scales down to about 10-19m and today the consensus is that it must correspond to the much smaller Planck length at 10-35m.
The belief in some new space-time structure at small length scales was reinforced with the discovery of ultraviolet divergences in Quantum Field Theory. From 1929 it was found that infinite answers appear when you sum up contributions to a physical quantity from waves of ever smaller wavelength. In 1930 Viktor Ambarzunmian and Mitrij Dmitrevich Iwanenko were the first of many physicists to propose that space should be treated as discrete to resolve the problems. Even after it was found possible to perform accurate calculations by a process of renormalisation in 1948 many physicists felt that the method was incomplete and would break down at smaller length scales unless a natural cut-off was introduced.
Another aspect of the quantum theory which caused disquiet was its inherent indeterminacy and the essential role of the observer in measurements. The Copenhagen interpretation seemed inadequate and alternative hidden variable theories were sought. It was felt that quantum mechanics would be a statistical consequence of a more profound discrete deterministic theory in the same sense that thermodynamics is a consequence of the kinetic gas theory.
Classical field theories are described in terms of quantities which vary continuously over space and time according to certain wave equations. For example, electromagnetism has an electric field and a magnetic field each of which is described by three real numbers for each event of space-time. The equations which determine how they evolve are Maxwell's equations. The equations have derivatives in them which only make sense on continuous space and time, so if space-time is really a discrete lattice the equations will have to be replaced by some alternative which avoids the derivatives and approximates the original equations at large scales.
To make things simpler we will look at how this could be done for a simpler wave equation. The massless Klein-Gordon equation in two dimensions has just one field value at each event. The value will be a complex number since the Klein-Gordon equation was first proposed as a relativistic generalisation of the single particle Schrödinger equation. Usually it is denoted by (x,t). The equation is as follows:
This has solutions which describe localised wave packets of energy like particles of mass m moving at less than a speed of 1 unit which is the speed of light.
In discrete space-time the values of are only defined on the sites of a lattice which are spaced regularly at a distance d apart in the space dimensions and also in the time dimension.
The derivatives which appeared in the wave equation can no longer be defined exactly but they can be approximated using finite differences. E.g.
If this and a similar approximation for the time derivative is substituted into the Klein-Gordon equation we get an equation which is well defined on the lattice.
This equation must now hold true for each value of x and t on the lattice. It describes a simple numerical relation between the field values at a the site and the four nearest neighbour sites. Equations like this can be used to numerically solve wave equations on a computer. The lattice solution is not exact but in the limit as d becomes very small it gives a better and better approximation to continuum solutions. It also has wave packet solutions which look like particles of mass m moving through space, but close up they are revealed as discrete fields at fixed sites.
If we believe in discrete space-time we might guess that the equation could be exact for some fixed value of d such as the Planck length. If it is correct we should be able to do experiments which detect the differences from continuum physics that the theory predicts. At least we should in principle. In practice the difference would be too small to find and it is impossible to rule out the lattice theory directly.
Philosophically such a hypothesis seems a little strange. It would mean that time is advancing in small discrete steps yet we experience time as a continuous flow. There is no contradiction in this, after all, when we watch television we see only a sequence of discrete pictures made up of discrete pixels on the screen, yet it appears to flow continuously. A similar illusion could apply to real life but on a much smaller scale. A sceptic might ask about what happens between the discrete time steps or what lies in the space between the sites of the lattice. The answer is simply that there is nothing between. The sites are the only events of space-time which exist and the fields interact directly with their neighbours. Particles are formed as wave packets which are spread over many sites of the lattice so we never need to think of them as travelling between sites.
Quantum field theory as expressed by Richard Feynman starts from the Lagrangian formalism. In the case of the Klein-Gordon equation a Lagrangian density is defined as follows:
The modulus squared of the complex numbers is used so that the Lagrangian is always real. The action is given by
By the principle of least action for the classical filed theory, this must be minimised subject to boundary conditions which fix the value of at any given start and end times. By an application of the calculus of variations the Klein-Gordon field equations can be derived from this principle. According to Feynman the quantum theory replaces the principle of least action with a path integral which defines a transition amplitude for going from each initial field configuration to a final one.
The path integral must be taken over all possible evolutions of the field between the start and end. Not only does this sound complicated, it is not even possible to define rigorously except when the field equations are linear. Ordinary integration has been around since Newton and Leibniz and was rigorously defined by Riemann in the eighteenth century. Path integrals only appeared in the latter half of the twentieth century and are still not well defined accept in restricted cases. Informally the path integral is a sum over all possible ways the field can vary over space and time but defining exactly what such an infinite-integral means is less simple to do.
By comparison the lattice version of the same thing is much easier to grasp. The lattice Lagrangian is just a discretised version of the continuum Lagrangian.
The action is a sum over the lattice sites.
The classical lattice field equations already given above can be derived from the action relatively easily by just requiring that the action is minimised with respect to variation of each field variable (x,t).
The lattice quantum field theory is then specified in a similar way as for the continuum field except that now the integral is a multi-variable integral over each field variable. This may still sound complicated but at least multi-variable integrals are well defined (when they converge) which is a big improvement over path integrals. If we believed that space-time was a lattice we would never have to worry about problems like renormalisation because the lattice spacing sets a cut-off scale which turns the divergences of field theory into well-defined finite answers. Such convenience does not make them right, of course, but it might count for something.
The action for two-dimensional Klein-Gordon theory can be written differently by expanding the squares and collecting together the square terms in the sum over lattice sites. Actually the square terms from the difference terms cancel in the sum and we are left with a sum over an alternative Lagrangian.
Recall the gauge symmetry for the electromagnetic field is invariance of the wave equation when the wave function is multiplied by a complex phase.
The Lagrangian for the lattice Klein Gordon equation is already invariant under this transformation when the phase 
(x,t) is a global constant, independent of x and t. The principles of gauge theory require us to introduce a gauge field in such a way that the Lagrangian is an invariant even when the phase is not a constant. As it stands the Lagrangian is not invariant because the field values at (x,t) are directly multiplied by field values at (x+d,t) and (x,t+d). Notice that the mass term does not suffer this problem and is already invariant.
Remember the analogy between gauge fields and economics. Multiplying field values together at different places is like trying to exchange money between different countries with different currencies. An exchange rate must be used. In the gauge theory the exchange rate is a phase factor U which is a unit complex number. Since the Lagrangian has products extending between any site and its nearest neighbours we must introduce such a factor on each link between sites of the lattice in both space and time directions. We will use Ui(x,t) for the variables linking site (x,t) to (x+d,t) and (x,t+d).
These phases are the field values of the gauge field. They represent the electromagnetic force on the lattice. When a local gauge transformation changes the matter field variables by a phase which can vary from one site to another, the gauge field must also be adjusted, just as exchange rates must be modified by a factor if the values of currencies change.
The gauge transformation is as follows.
With these fields the Lagrangian can be modified to be gauge invariant. It suffices to introduce the appropriate gauge field in between the product of matter field terms. For example
becomes
This term and all others in the Lagrangian are then invariant under the local gauge transformation. However, the Lagrangian is still incomplete because the gauge field itself must have some dynamics. The Lagrangian should include a term made purely from gauge fields and, of course, it must be gauge invariant and real. It turns out that a suitable form for this term is a product of four gauge fields round a square of links on the lattice (known as a plaquette).
- is just a coupling constant parameter which controls the strength of the electromagnetic force. When this term is added to the matter field it gives a lattice version of electromagnetics in two dimensions. In 1974 Ken Wilson discovered this elegant Lagrangian and generalised it to a form which even gives a discrete lattice approximation to Yang-Mills theory with other gauge groups in any number of dimensions. Using Wilson's formulation of lattice QCD has been an important part of a method for performing numerical calculations to study theoretically the structure of particles composed of quarks and held together by the strong nuclear force of quantum chromodynamics.
Lattice gauge theory is an approximation to Yang-Mills theory which may become exact when the lattice spacing tends to zero if the fields and coupling constants can be suitably renormalised. Here we are more interested in the possibility that lattice theories could be an exact description of physics at very small length scales. The simple form of the theory and its elegant discrete version of gauge invariance are points in its favour but what about space-time symmetry? Lattice theories on a regular lattice have discrete translational invariance because the equations can be displaced by any multiple of the lattice spacing along any of the spatial axis. The same applies in the time direction. The greater difficulty lies with rotational and Lorentz invariance or more generally with co-ordinate transformations. Only ninety degree rotations are a symmetry of the theory. If space-time was such a lattice there would be a preferred set of space axis and a preferred reference frame but such things contradict relativity and have never been observed.
If the continuum limit is not to be restored by taking the limit where the lattice spacing goes to zero then the issue of the loss of rotational invariance must be addressed. A space-time constructed as a discrete lattice is analogous to a crystal whose atoms are arranged on a regular array. At first sight the internal structure of a crystal solid appears isotropic but there its mechanical properties can be carefully measured to determine the directions in which the atoms are aligned. If space-time was a regular lattice its loss of rotational invariance would also be present even though it might not be detectable with present technology. Lorentz invariance would also be lost so relativity would be violated in a way which is hard for theorists to accept.
The fact is that lattice theories of space-time cannot easily be ruled out but they are just too plain ugly to be right! The laws of physics seem to be based on elegant principles such as symmetry which help determine the correct form the laws of physics must take. If we abandon those principles we have little hope of making progress. Lattice theories are arbitrary in their form. There is an infinity of ways to approximate any field theory on a lattice. How would we know which is right if experiment can never probe at sufficiently small length scales? This arbitrariness is the price you pay whenever you abandon a principle of symmetry.
Nevertheless, the fact that we can accommodate gauge invariance on the lattice may be telling us something. If we could represent diffeomorphism invariance in such a clean discrete form too, there would be some hope. The discrete version of diffeomorphism invariance is permutation invariance. Diffeomorphisms are one-to-one mappings of the set of space-time events to itself which preserve its continuum properties. Permutations are one-to-one mappings of a discrete set of events to itself. We call this event symmetry. The event-symmetric analogue of a lattice gauge theory is a gauge glass with events each linked to each other using gauge fields. The lattice structure is discarded. This gives a complete model of symmetries but how could such a structureless model be anything to do with physics?
Quantum indeterminacy, which was another motivation for looking to discrete space-time, has also become an acceptable aspect of continuum physics. In 1964 John Bell showed that most ideas for local hidden variable theories would violate an important inequality of quantum mechanics. This inequality was directly verified in a careful experiment by Alain Aspect in 1982. There are still those who try to get round this with new forms of quantum mechanics such as that of David Bohm, but now they are a minority pushed to the fringe of established physics. Hugh Everett's thesis which leads us to interpret quantum mechanics as the dynamics of a multiverse has been seen as a resolution of the measurement problem for much of the physics community. Others are simply content with the fact that quantum mechanics provides the same way of doing calculations no matter what interpretation is used.
Without the physical motivation discrete space-time has been disfavoured by many physicists but others have found reason to persist with the idea.
In 1947 Claude Shannon laid the foundations of information theory. The smallest unit of information used in computers is the binary digit or bit. Each bit can just have a value 0 or 1 but many bits can record vast amounts of information in the form of numbers or binary coded characters. Shannon's information theory turned out to be important in physics as well as computers. It seems that the entropy of a system may be a measure of the amount of information it contains but it is difficult to make sense of such an idea unless the amount of information in a physical system is finite. If the positions and orientations of molecules can be specified to any degree of precision then there is no limit to the number of bits needed to describe the state of a gas in a box so entropy from information may only make sense if there is some minimum distance which can be measured.
Such reasoning has created a school of thought about the role of information processing in the fundamental laws of physics. John Wheeler has sought to extend this idea so that every physical quantity derives its ultimate significance from bits. He calls this idea "It from Bit." For Wheeler and his followers the continuum is a myth, but he goes further than just making space-time discrete.
Space-time itself, he argues, must be understood in terms of a more fundamental pregeometry. In the pregeometry there would be no direct concepts of dimension or causality. Such things would only appear as emergent properties in the space-time idealisation. All would be the consequence of complex interactions based on very simple basic elements, just as a complex piece of computer software is built from a simple set of instructions.
There are many different instruction sets which have been used to control computers. In RISC processors the number of different instructions is kept to a minimum. In the theory of computers, without the practical constraints of efficiency, it is possible to reduce the instruction set to very few elements indeed and still be able to use it to do any computation which is theoretically possible. Such a machine is called a universal computer. In 1979 while I was a student I attended an extra-circular course on logic given by the mathematics professor John Conway. He introduced the class to a hypothetical computer called a Minsky machine which had been devised by computer science theorist Marvin Minsky. The computer can store an unlimited number of non-negative integer values which are given variable names a, b, c, … etc. The computer obeys two fundamental instructions:
The branch with the double arrow is taken if b is zero on entering the circle. A program for a Minsky machine is a diagram made up of these two instructions. Here is an example of a simple program to add a to b.
If you want an interesting puzzle to solve try and work out what is the largest number which a Minsky machine can generate in a variable when it stops if it is only allowed to have k instructions where k is some small number of your choice.
In one lecture of the course, Conway showed us a program he had written for a Minsky machine which could calculate the nth prime number. It had only 16 instructions and he challenged us to do better. The next week I showed him how to do it with only 14 instructions. Can you do better still? Here is the program. Start with all variables set to zero except n. When you arrive at the end p will be the nth prime number. This Minsky machine program illustrates how the simplest of rules can be used to generate non-trivial systems. Perhaps some equally simple set of rules will account for physics.
The game of life is played on a two-dimensional array of square cells. Each cell at any given time step is either alive or dead. The state of the game at the next time step is determined by rules which are meant to mimic the life and death of living cells. If a living cell at one moment is isolated or it is accompanied by no more than one other living cell in the nearest neighbouring 8 cells, it will be dead the next moment through lack of support. If it is surrounded by two or three living cells in its neighbourhood it will continue to live but if there are more it will die from over competition. On the other hand, a cell which is dead will be revived if it is surrounded by exactly three living cells. Otherwise it remains dead. When these rules are applied iteratively to an initial picture of living and dead cells, the system evolves and patterns emerge. A computer can readily be made to simulate the game and display the progress.
Typically regions of cells will die out or stabilise into patterns which do not change such as an isolated square of four cells, or which repeat such as a line of three cells. From time to time a group of living cells will appear to separate from the activity and move away on its own. These are called gliders. The most common variety reflects about a diagonal axis after each second step and moves diagonally.
Despite its simple rules defined on a discrete lattice of cells the game has some features in common with the laws of physics. There is a maximum speed for causal propagation which plays a role similar to the speed of light in special relativity. Even more intriguing is the comparison of gliders with elementary particles. Cellular automata go a step further than lattice field theories. Even the continuous values of the field variables have been replaced with discrete quantities.
A great deal of research has been done to find out how cellular automata like this one behave on very large arrays. Numerical simulations suggest that stable regions develop but some activity can continue for a long time. It seems that self organised criticality is established. This means that the system stops evolving leaving steady or cyclic configurations of cells, but a small perturbation such as a glider wandering in from outside can set the thing off again like a spark lighting a fire. Little is known about how cellular automata might behave on very large arrays and over very large numbers of time steps. Recall that the smallest scales in physics seem to be around 10-35 m. To correspond in size to our universe, a cellular automaton would have to have an array of something like 10240 cells.
Despite its simple rules the game of life has sufficient complexity that we cannot imagine how an array that big would behave. On large scales some kind of physical laws may emerge from the statistical behaviour of the system. It is quite possible that complex organised structures would evolve. It is plausible that some cellular automata specified in 3-dimensions may be sufficiently interesting places for life to develop inside them. At present we have no idea if such things are likely. For those seeking to reduce physics to simple deterministic laws this was an inspiration to look for cellular automata as toy models of particle physics despite the obvious flaw that they broke space-time symmetries. Edward Fredkin is one of those people who suggests that the universe really does operate like a gigantic computer. Fredkin is a computer specialist with an interest in physics who has managed to influence a number of respected physicists to take the idea seriously.
In 1981 Fredkin was one of the organisers of a conference at MIT which he wanted to be called something like "On computational models of physics." Fredkin managed to persuade Richard Feynman to be the keynote speaker at the meeting, but when Feynman heard the title he said "Well if you have that as a name, and it implies that there are computational models of physics, then I am not coming." The title was changed to "Physics and computing" and so Feynman went. However, by time Feynman arrived to give his talk he had changed his mind and gave a talk about computational models of physics. He and many other speakers spoke about cellular automata which were very topical by then. Other speakers at the conference included Wheeler, Minsky and Fredkin himself. This conference and especially the presence of Feynman was very influential on the subject.
There has been some progress towards using cellular automata to study hydrodynamics and turbulence but there seems to be an impassable hurdle when we attempt to apply the automata to quantum physics. The evolution of automata is always based on what happens locally to any cell in the array, but Bell's inequality and the experiments of Aspect and others strongly suggest that quantum reality is not local in such a strong sense.
Another notable physicist who has been influenced by Fredkin is Gerard 't Hooft. He is not put off by locality arguments and suggests that the states of a cellular automaton could be seen as the basis of a Hilbert space on which quantum mechanics is formed. Although the idea is not popular, some interesting things may yet be learnt from such research.
Whichever approach to quantum gravity is taken the conclusion seems to be that the Planck length is a minimum size beyond which the Heisenberg Uncertainty Principle prevents measurement if applied to the metric field of Einstein Gravity. In ordinary quantum field theory the ability to measure small distances is limited only by the energy of the particles available and according to relativity there should be no theoretical limit to energy. When gravity is included, however, the metric itself becomes uncertain. At smaller distances the quantum fluctuations of the metric become more significant until, at the scale of the Planck length, it is impossible to do any reliable measurements.
Does this mean that space-time is discrete at such scales with only a finite number of degrees of freedom per unit volume? Recent theoretical results from string theories and the loop-representation of gravity do suggest that space-time has some discrete aspects at the Planck scale. These are akin to the discrete quantum numbers of the quantum mechanics of an atom which still also has a continuum description so the answer may be that space and time have a dual discrete and continuous nature.
The far reaching work of Bekenstein and Hawking on black hole thermodynamics has led to some of the most compelling evidence for discreteness at the Planck scale. The black hole information loss paradox which arises from semi-classical treatments of quantum gravity is the nearest thing physicists have to an experimental result in quantum gravity. Its resolution is likely to say something useful about a more complete quantum gravity theory. There are several proposed ways in which the paradox may be resolved most of which imply some problematical breakdown of quantum mechanics while others lead to seemingly bizarre conclusions.
One approach is to suppose that no more information goes in than can be displayed on the event horizon and that it comes back out as the black hole evaporates by Hawking radiation. Bekenstein has shown that if this is the case then very strict and counter-intuitive limits must be placed on the maximum amount of information held in a region of space. It has been argued by 't Hooft that this finiteness of entropy and information in a black-hole is also evidence for the discreteness of space-time. In fact the number of degrees of freedom must be given by the area in Planck units of a surface surrounding the region of space. This has led to some speculative ideas about how quantum gravity theories might work through a holographic mechanism, i.e. it is suggested that physics must be formulated with degrees of freedom distributed on a two-dimensional surface with the third spatial dimension being dynamically generated.
At this point it may be appropriate to discuss the prospects for experimental results in quantum gravity and small scale space-time structure. Over the past twenty years or more, experimental high energy physics has mostly served to verify the correctness of the standard model of particle physics as established theoretically between 1967 and 1973. We now have theories extending to energies way beyond current accelerator technology but it should not be forgotten that limits set by experiment have helped to narrow down the possibilities and will presumably continue to do so.
It may seem that there is very little hope of any experimental input into quantum gravity research because the Planck energy is so far beyond reach. However, a theory of quantum gravity would almost certainly have low energy consequences which may be in reach of future experiments. The discovery of supersymmetry, for example, would have significant consequences for theoretical research on space-time structure.
The concept is very much like a lattice except that it is not rigid. Instead of varying field values on sites the length of the links between the sites is allowed to be variable. It is sufficient to specify how the sites are connected and the lengths of all the links. Then the size and shape of all the simplexes can be determined. The curvature of the space-time surface can be derived from the angles of the simplexes around any site. It is possible to work out the equations which express the dynamics of the structure and which reduce to Einstein's field equations of general relativity in the limit where the size of the simplices becomes very small. The Regge calculus is therefore a discrete version of general relativity. Useful numerical simulations of either the classical or quantum dynamics can be done on a fast computer.
To Regge this discrete space-time was just an approximation scheme which would give ordinary general relativity in the fine limit. To us it could also be a pregeometric model of space-time, valid even while discrete. If space-time was a Regge skeleton we would have to find some rules about how it should be split into simplexes. Loss of space-time symmetry is also a problem just as it was with a regular lattice.
An alternative scheme which has proved to work better in numerical studies of quantum gravity is random triangulation. Instead of varying the lengths of the links joining sites the links are all the same length and the way space-time is divided into simplexes is varied. Space-time curvature varies with the number of simplexes which meet at each site. The path integral of quantum gravity is then effectively a sum over all the ways of triangulating a four-dimensional surface. The action can be given in terms of just the numbers of simplexes in the lattice. Discrete effects are averaged out so that rotational symmetry is exact in the quantum version. This is an interesting pregeometric model though it would be surprising if it was anything like reality.
Any pregeometric model can be characterised according to which of the highlighted properties in the previous paragraph it throws out and which it keeps. For example, lattice models discard continuity and symmetry but keep dimension, metric, events, etc. Cellular automata also discard quantum mechanics. Some physicists have played the game of building toy models which throw out all but a few of these concepts, the ones which they feel might be the most fundamental. They might try to keep causality, locality and quantum mechanics for example, because they think these things are of primary significance and must be part of the laws of physics at the most fundamental level. Another feature like topology, a metric or even information might be thrown in just to see what it led to.
Before about 1980 only a rare few physicists had made any serious attempts at this sort of thing. The best examples were Hartland Snyder with quantum space-time, David Finkelstein with his quantum net dynamics, Carl von Weizsäcker with Ur-theory and Roger Penrose with spin networks and twisters. Then in the 1980s and early 1990s there was a flurry of new speculative ideas. The time seemed right for bold ideas. Chris Isham and others looked at the quantum mechanics of spaces with just a distance metric between scattered points, or topologies of sets or even just random networks of links between space-time events.
Is there really any hope that such methods can tell us something about the real world? Physicists have succeeded before with theories they devised with little more than mathematics and insight. Dirac was a strong advocate of the power of mathematical beauty as an indicator of truth and successfully predicted the positron on such a basis. If you examine the pregeometries which have been studied up till now it is easy to dismiss them because none is complete.
However, rather than discarding each one because of some feature which does not correspond to reality, you can also look for features which seem promising. Better theories can then be produced by combining things from different models which might work well together. It seems improbable that someone is going to have complete success by such methods alone, but if clues from superstring theory and canonical quantum gravity are also considered there may be some hope.
Sadly, there is little encouragement or funding for such speculative research. Happily there are still a hand full of physicists and one or two journals which keep it alive.
How have modern physicists learnt to deal with these questions? The simplest answer is that they use mathematics to construct models of the universe from basic axioms. Mathematicians can define the system of real numbers from set theory and prove all the necessary theorems of calculus that physicists need. With the system of real numbers they can go on to define many different types of geometry. In this way it was possible to discover non-Euclidean geometries in the nineteenth century which were used to build the theory of general relativity in the twentieth.
The self consistency of general relativity can be proven mathematically from the fundamental axioms within known limitations. This does not make it correct, but it does make it a viable model whose accuracy can be tested against observation. In this way there are no paradoxes of the infinite or infinitesimal. The universe could be infinite or finite, with or without a boundary. There is no need to answer questions about what happened before the beginning of the universe because we can construct a self-consistent mathematical model of space-time in which time has a beginning with no before.
So long as we have a consistent mathematical model we know there is no paradox, but nobody yet has an exact model of the whole universe. Newton used a very simple model of space and time described by Euclidean geometry. In that model space and time are separate, continuous, infinite and absolute.
This is consistent with what we observe in ordinary experience. Clocks measure time and normally they can be made to keep the same time within the accuracy of their working mechanisms. It as if there were some universal absolute standard of time which flows constantly. It can be measured approximately with clocks but never directly.
So long as there is no complete theory of physics we know that any model of space-time is likely to be only an approximation to reality which applies in a certain restricted domain. A more accurate model may be found later and although the difference in predicted measurement may be small, the new and old model may be very different in nature. This means that our current models of space and time may be very unrealistic descriptions of what they really are even though they give very accurate predictions in any experiment we can perform.
Philosophers sometimes try to go beyond what physicists can do. Using reason alone they consider what space and time might be beyond what can be observed. Even at the time of Newton there was opposition to the notion of absolute space and time from his German rival Leibniz. He, and many other philosophers who came after, have argued that space and time do not exist in an absolute form as described by Newton.
If we start from the point of view of our experiences, we must recognise that our intuitive notions of space, time and motion are just models in our minds which correspond to what our senses find. This is a model which exists like a computer program in our head. It is one which has been created by evolution because it works. In that case there is no assurance that space and time really exist in any absolute sense.
The philosophical point of view developed by Gottfried Leibniz, the Bishop Berkeley and Ernst Mach is that space and time should be seen as formed from the relationships between objects. We experience objects through their relationships with our senses and infer space and time more indirectly. The mathematical models used by physicists turn this inside-out. They start with space and time, then they place objects in it, then they predict our experiences as a result of how the objects interact.
Mach believed that space and time do not exist in the absence of matter. The inertia of objects should be seen as being a result of their relation with other objects rather than their relation with space and time. Einstein was greatly influenced by Mach's principle and hoped that it would follow from his own postulates of relativity.
In the theory of special relativity he found that space and time do not exist as independent absolute entities but Minkowski showed that space-time exists as a combination of the two. In General Relativity Einstein found, ironically, that the correct description of his theory must use the mathematics of Riemannian geometry. Instead of confirming Mach's principle he found that space-time can have a dynamic structure in its own right. Not only could space-time exist independent of matter but it even had interesting behaviour on its own. One of the most startling predictions of general relativity; that there should exist gravitational waves, ripples in the fabric of space-time itself, may soon be directly confirmed by detection in gravitational wave observatories. In short, relativity succeeded in showing that all motion is relative but it failed to construct a complete relational model of physics.
Einstein's use of geometry was so elegant and compelling that physicists thereafter have always sought to extend the theory to a unified description of matter through geometry. Examples include the Kaluza-Klein models in which space-time is supposed to have more than four dimensions with all but four compacted into an undetectably small geometry. This is the opposite of what the philosophers prescribed. Thus physicists and philosophers have become alienated over the subject of space and time during the twentieth century.
Recent theories of particle physics have been so successful that it is now very difficult to find an experimental result which can help physicists go beyond their present theories. As a result they have themselves started to sound more philosophical and are slowly reviewing old ideas. The fundamental problem which faces them is the combination of general relativity and quantum theory into a consistent model.
According to quantum theory a vacuum is not empty. It is a sea of virtual particles. This is very different from the way that space and time were envisioned in the days of Mach. In a theory of quantum gravity there would be gravitons; particles of pure geometry. Surely such an idea would have been a complete anathema to Mach. But suppose gravitons could be placed on a par with other matter. Perhaps then Mach would be happy with gravitons after all. The theory could be turned on its head with space-time being a result of the interactions between gravitons.
Leibniz might also have been satisfied with such an answer. In his philosophy everything is constructed from monads. These could be packets of energy or more abstract entities. A discrete space-time would fit in well with the idea. Discrete elements of space-time can be put on a par with particles of matter suggesting the final unification of space-time and matter.
In string theory, the most promising hope for a complete unified theory of physics, we find that gravitons are indeed on an equal footing with other particles. All particles are believed to be different modes of vibration in loops of string. Even black holes, one of the ultimate manifestations of the geometry of space-time are thought to be examples of single loops of string in a very highly energised mode. There is no qualitative distinction between black holes and particles, or between matter and space-time.
The problem is that there is as yet no mathematical model which makes this identity evident. The equations we do have for strings are somewhat conventional. They describe strings moving in a background space-time. And yet, the mathematics holds strange symmetries which suggest that string theories in different background space-times and even different dimensions are really equivalent. To complete our understanding of string theory we must formulate it independently of space-time. The situation seems to be analogous to the status of electrodynamics at the end of the 19th century. Maxwell's equations were described as vibrations in some ether pervading space. The Michelson-Morley experiments failed to detect the hypothetical ether and signalled the start of a scientific revolution.
Just as Einstein banished the ether as a medium for electromagnetism we must now complete his work by banishing space-time as a medium for string theory. The result will be a model in which space-time is recovered as a result of the relationship between interacting strings. It will be the first step towards a reconciliation of physics and philosophy. Perhaps it will be quickly followed by a change of view, to a point from where all of our universe can be seen as a consequence of our possible experiences just as the old philosophers wanted us to see it. What other ways will we have to modify our understanding to accommodate such a theory? Not all can be foreseen.
At the same time, the mathematics of continuous manifolds seems to be increasingly important. Topological structures such as instantons and magnetic monopoles appear to play their part in field theory and string theory. Can such things be formulated on a discrete space?
Hawking says that he sees no reason to abandon the continuum theories that have been so successful. It is a valid point but it may be possible to satisfy everyone by invoking a discrete structure of space-time without abandoning the continuum theories if the discrete-continuum duality can be resolved as it was for light and matter.
The philosopher Immanuel Kant may have had some insight into this question. The human mind can pose questions about nature which have contradictory but perfectly logical answers. One such question is whether the world is made of elementary parts. The answer can be both yes and no. The riddle may be resolved through a dual theory of space-time which has both discrete and continuous aspects.