Theory of Theories
The Theory That Flies
As everybody knows, the job of a theoretical physicist is to invent theories of the universe. A layman might ask a physicist What is charge? or What is Time?. He will be disappointed when the physicist replies that his theories do not even try to explain what these things are. Theories are just mathematical models which make predictions about how they will behave in experiments.
When pressed the physicist will probably admit that he does physics because he too seeks deeper explanations for why things are the way they are in the universe. One day he hopes to understand the most basic laws of physics and he hopes that they will provide an answer to the most difficult question of all, Why do we exist?. Physicists can be justly proud of the fact that almost everything in physics can be accounted for with just a small number of basic equations embracing General Relativity and the Standard Model of particle physics. There remain many puzzles but those will probably be solved once a unified theory of quantum gravity and the other forces is found. Such a theory would be the final theory. It may be cast in other forms but they would always be exactly equivalent. There is no a priory reason why such a theory should exist but, as Steven Weinberg argues in Dreams of a Final Theory, the convergence of principles in modern physics seems to suggest that it does.
How many physicists have not wondered what principle of simplicity and beauty underlies that final theory? Could we not take an intellectual leap and figure it out from what we already know? John Wheeler wrote his thoughts thus, ... Paper in white the floor of the room, and rule it off in one-foot squares. Down on one's hands and knees, write in the first square a set of equations conceived as able to govern the physics of the universe. Think more overnight. Next day put a better set of equations into square two. Invite one's most respected colleagues to contribute to other squares. At the end of these labours, one has worked oneself out into the door way. Stand up, look back on all those equations, some perhaps more hopeful than others, raise one's finger commandingly, and give the order Fly! Not one of those equations will put on wings, take off, or fly. Yet the universe flies.
The Nature of Nature
If there is really a unique principle on which the laws of physics are founded then to understand it we should look for clues in the nature of nature, or as Feynman puts it the character of physical law. One thing is clear: Nature uses mathematics. If this were not the case, if nature was governed instead by a committee of demons who made nature follow their whims, then there would be little hope for us to understand physics and predict the outcome of experiments or invent new technology. Scientists would be replaced by sorcerers.
But the relationship between physics and mathematics seems to be much deeper than we yet understand. In Early history there was little distinction between a mathematician and a physicist but in modern times pure mathematicians have explored their subject independently of any potential application. Mathematics has an existence of its own. Those mathematicians have constructed a huge web of logical structures which have a remarkable inner beauty only visible to those who take the time to learn and explore it. They would usually say that they discovered new mathematics rather than invented it. It is almost certain that another intelligent race on another planet, or even in a different universe, would have mathematicians who discover the same theorems with just different notation.
What becomes so surprising is the extent to which mathematical structures are applicable to physics. Sometimes a physicist will discover a useful mathematical concept only to be told by mathematicians that they have been studying it for some time and can help out with a long list of useful theorems. Such was the case when Heisenberg formulated a theory of quantum mechanics which used matrix operations previously unfamiliar to physicists. Other examples abound, Einstein's application of non-Euclidean geometry to gravitation and, in particle physics, the extensive use of the classification of Lie groups. Recently the mathematical theory of knots has found a place in theories of quantum gravity. Before that, mathematicians had considered it an area of pure mathematics without application (except to tying up boats of course). Now the role played by knots in fundamental physics seems so important that we might even guess that the reason space has three dimensions is that it is the only number of dimensions within which you can tie knots in strings. Such is the extent to which mathematics is used in physics that physicists find new theories by looking for beautiful mathematics rather than by trying to fit functions to empirical data as you might expect. Dirac explained that it was this way that he found his famous equation for the electron. The laws of physics seem to share the mathematician's taste for what is beautiful. It is a deep mystery as to why this should be the case. It is what Wigner called The unreasonable effectiveness of mathematics in the natural sciences.
It has also been noted by Feynman that physical law seems to take on just such a form that it can be reformulated in several different ways. Quantum mechanics can be formulated in terms of Heisenberg's matrix mechanics, Schroedinger's wave mechanics or Feynman's path integrals. All three are mathematically equivalent but very different. It is impossible to say that one is more correct than the others.
Perhaps there is a unique principle which determines the laws of physics and which explains why there is such a tight relationship between mathematics and physics? If the laws of physics were merely some isolated piece of mathematics chosen for its simple beauty then there would be no explanation why so much of mathematics is incorporated into physics. There is no reason why one set of equations should fly. The fundamental principle of physics must be something more general. Something which embraces all of mathematics. It is the principle which explains the nature of nature. So what is it?
Many Anthropic Principles
There are other aspects of the universe which provide clues as to what principle determines the laws of physics. The universe is populated by an impressive menagerie of objects which exhibit organised complexity. They exist on all length scales from the atomic to the cosmological. Most impressive of all (that we know of) are living beings like ourselves.
Examination of the way that chemistry, nuclear physics, astrophysics, cosmology and other sciences are dependent on the details of the laws of physics suggests that the existence of so much complexity is no accident. The precise values of various constants of nature, such as the fine structure constant, seem to be just right to allow organised complexity to develop. Perhaps we might even say, to allow life to develop.
This observation has inspired much faith among physicists and philosophers in the Anthropic Principle. The anthropic principle supposes that the laws of physics are indeed selected so that intelligent life has a maximum chance of developing in the universe. Believers ask us to consider first why our planet Earth is so well suited to the evolution of life while other planets in the solar system seem to be more hostile. The answer is that we would not be on this planet to consider the question if it were not suitable for life to evolve here. The same principle can then be extended to the whole universe.
One way to understand the Anthropic Principle is to imagine that all possible universes exist with a validity which is equal to our own. When we say all possible universes we might mean any system which can be described by mathematics. Each such system has a set of physical laws which allow its structure to be determined in principle. Sometimes they will be simple and beautiful and often they will be complex and ugly. Sometimes the phenomenology of such a system will be dull or easily determined and nothing interesting will happen. Sometimes it will be so complicated that nothing can be determined, even a hypothetical computer simulation would reveal little of interest in the turmoil of those universes. Somewhere in between would exist our universe which has just the right balance of equations in its physical laws for intelligent life to exist and explore the nature of its environment.
Another interpretation of the anthropic principle, developed by Lee Smolin, is that there is one universe with a set of physical laws much as we know them. Those laws may have a number of variables which determine the physical constants but which can vary in certain extreme situations such as the collapse of massive stars into black holes. Universes governed by such laws might give birth to baby universes with different physical constants. Through a process of natural selection universes might evolve over many generations to have constants which are conducive to further procreation. This might mean they are optimised for the production of black holes and, from them, more baby universes. Within this population of worlds there will be some with laws conducive to life, indeed, the production of black holes may be linked to the existence of advanced life-forms which could have an interest in fabricating black holes as energy sources. This scenario makes a number of demands on the nature of physical laws. In particular it is essential that some physical parameters such as the fine structure constants should be able to vary rather than being determined by some equation. Future theories of quantum gravity may tell us if this is so.
Is the Anthropic Principle Enough?
The Anthropic Principle is compelling enough for us to wonder if it can determine the laws of physics on its own. I know of no convincing argument that it can. There is nothing in the anthropic principle which explains why so many of the most elegant discoveries of mathematics are so important in physics. There is nothing to explain why there is so much symmetry in physics, or why the elegant principle of least action is important or even why the laws of physics should be the same in one place as they are in another.
You might try to argue that the laws of physics have to take a certain form because otherwise they would be impossible to understand. I don't buy it! I am convinced that a suitable mathematical system, perhaps even something as simple as a cellular automata, can include sufficient complexity that intelligent life would evolve within it. There must be a huge variety of possible forms the laws of physics could have taken and there must be many in which life evolves. In the case of cellular automata, the cellular physicists living in it would probably be able to work out the rules of the automata because its discrete nature, and simple symmetry would be clear and easily uncovered. They would not need to know so much sophisticated mathematics as we do.
Whether or not the principle is valid as an explanation for some of the characteristics of nature and the values of its parameters I believe that there must be some other principle which explains those other things.
Universality
For centuries mathematicians confined themselves to looking at specific structures with simple definitions and interesting behaviour. With the advent of powerful computers they are now looking at general behaviour of complex systems. It was Feigenbaum who made the discovery that a large class of complex systems of chaotic non-linear equations exhibit a universal behaviour characterised by the Feigenbaum constants. This type of universality has an independent existence which transcends details of the specific equations which generate it. Other examples of universality can be identified in physics and mathematics. Statistical physics looks at the behaviour of systems with many degrees of freedom. Such systems exhibit a universal behaviour which can be described by the laws of thermodynamics. The microscopic details of the forces between particles are reduced to just a few macroscopic parameters which describe the thermodynamic characteristics.
A more mathematical example is the notion of computability. Computability of a sequence of integers can be defined in terms of a hypothetical programming language such as a Turing machine or a Minsky machine. Those languages and a large number of other possibilities turn out to give an equivalent definition of computability despite the fact that they look very different. There is no most natural or most simple way to define computability but computability itself is a natural and unambiguously defined concept. If we made contact with an alien intelligence we would probably find that they had an equivalent concept of computability but probably not quite the same definitions. Computability, then, can be seen as a universal characteristic of computing languages.
The message I wish to draw from this is that the laws of physics may themselves be a universal behaviour of some general class of systems. If this is the case then we should not expect the laws of physics to be given by one most natural formulation. Like computability there may be many ways to describe them. The universal behaviour of a class of complex systems would be likely to display organised complexity itself. Furthermore, there is evidence that thermodynamics runs deeper than just a behaviour of particle systems. It is also found to be a useful description of black hole dynamics. We can also remark that quantum mechanics and statistical physics are closely related through an exchange of real and imaginary time. All these things are intimately related and hint at the importance of universality in nature at its most fundamental level.
The Theory of Theories
At last we come to the main hypothesis of this essay. If the laws of physics are to be seen as a universal behaviour of some class of systems then it is necessary to ask what class to choose. We can regard any possible mathematical system as a theory of physics. I suggest that the laws of physics are a universal behaviour to be found in the class of all possible mathematical systems. This is known as The Theory of Theories
To understand the Theory of Theories we start from the same premise as we do with the anthropic principle, i.e. that all mathematically consistent models exist just as our own universe exists. We can simply take this to be our definition of existence.
We know from Feynman's Path Integral formulation of quantum mechanics that the evolution of the universe can be understood as a supposition of all possible histories that it can follow classically. The expectation values of observables are dominated by a small subset of possibilities whose contributions are reinforced by constructive interference. The same principle is at work in statistical physics where a vast state space is dominated by contributions at maximum entropy leading to thermodynamic behaviour. We might well ask if the same can be applied to mathematical systems in general to reveal the laws of physics as a universal behaviour which dominates the space of all possible theories and which transcends details of the construction of individual theories. If this was the case then we would expect the most fundamental laws of physics to have many independent formulations with no one of them standing out as the simplest. This might be able to explain why such a large subset of mathematics is so important in physics.
Can we use the Theory of all Theories to explain why symmetry is so important in physics? There is a partial answer to this question which derives from an understanding of critical behaviour in statistical physics. Consider a lattice approximation to a Yang-Mills quantum field theory in the Euclidean sector. The Wilson discretisation preserves a discrete form of the gauge symmetry but destroys the space-time rotational symmetry. If we had more carelessly picked a discretisation scheme we would expect to break all the symmetry. We can imagine a space of discrete theories around the Yang-Mills theory for which symmetry is lost at almost all points. The symmetric continuum theory exists at a critical point in this space. As the critical point is approached correlation lengths grow and details of the discretisation are lost. Symmetries are perfectly restored in the limit, and details of all the different discretisations are washed out.
If this is the case then it seems that the critical point is surrounded by a very high density of points in the space of theories. This is exactly what we would expect if universal behaviour dominating in theory space was to exhibit high symmetry. It also suggests that a dominant theory could be reformulated in many equivalent ways without any one particular formulation being evidently more fundamentally correct than another. Perhaps ultimately there is an explanation for the unreasonable effectiveness of mathematics in physics contained in this philosophy.
If physics springs in such a fashion from all of mathematics then it seems likely that discovery of these laws will answer many old mathematical puzzles. There is no a priori reason to believe that mathematical theories should have some universal behaviour, but if they did it might explain why there is so much cross-reference in mathematics. Perhaps mathematicians sense intuitively when they are near the hot spots in the space of theories. They notice the hightened beauty, the multitude of unexpected connections. Eventually, left to their own devices mathematicians might be capable of finding the central source of the heat, if physicists don't get there first.
I think therefore I am...
So, is it really possible to derive the laws of physics from pure mathematics without any reference to empirical observations? If the Theory of Theories is correct then the answer should be yes. At first it seems that it is rather hard to make progress with the theory of theories beyond the philosophical conception, since it is necessary to define an appropriate topology and measure in the space of all mathematical theories. Mathematics is just too large for this, or is it?
Perhaps we could search for a universal behaviour in the set of all possible computer programs. The set is sufficiently diverse to cover all mathematics because, in principle, we can write a computer program to explore any mathematical problem. John Wheeler proposed this as a place to start and called it It From Bit. Simple computer programs can be very complex to understand, but we are not interested in understanding the details of any one. We are concerned about the universal behaviour of very big programs randomly written in some (any) computer language.
The variables of a large program would evolve in some kind of statistical manner. Perhaps the details would fade into the background and the whole could be understood using the methods of statistical physics. So let's look at general systems of statistical physics. Suppose one system (one theory, one universe) had a number N of variables, its degrees of freedom.
a1, a2, ... aN
In addition there must be an energy function,
E(a1, a2, ... aN)
In the system, a possible set of values for these variables would appear with a weight given by
Z = e-E(a1, a2, ... aN)
I have not said much about the values of these variables. They could be discrete variables or real numbers, or points on a higher dimensional manifold. Somewhere in this complete set of systems you could find something close to any mathematical universe you thought of. For example, cellular automata would exist as limiting cases where the energy function forced discrete variables to follow rules.
But what did I mean when I said close? Two different systems would be isomorphic if there was a one to one mapping between them which mapped the weight function of one onto the weight function of the other. We could define a distance between two systems by finding the function mapping one to the other which minimised the correlations between them. This defines a metric space with the minimum correlation as metric.
A powerful property of metric spaces is that they can be completed by forming Cauchy sequences. Hence we can define a larger set of theories as the completed metric space of statistical systems. By means of this technique we include even renormalisable lattice gauge theories into the theory space. The renormalisation process can be defined as a Cauchy sequence of finite statistical systems. It remains to define a natural measure on this space and determine if it has a universal point where the total measure within any small radius of this point is larger than the measure on the rest of the space.
Needless to say, this is quite a difficult mathematical problem and I am not going to solve it. Perhaps I didn't really get much further than Descartes!
The Story Tellers Philosophy
On the off chance that someone has read this far, or skipped to the end, I will conclude with a lighter description of the theory of theories.
Suppose I were to tell you a story. It might be a story about wars between elves and goblins or a story about men exploring space or a story about creatures living in a two dimensional world or perhaps just the story of my life. A story defines a universe just as a mathematical theory does, but a story is incomplete. The same story might be played out in many different universes, or many times in one very large universe. When you listen to the story you add more details in your imagination. You picture the characters. Perhaps you see the hero wearing a blue coat which was not in the original tale.
Our universe is very large and will exist for a long time. It may even be infinite. There are probably many stories being told. Perhaps some are so close that they can be seen as accidentally telling different parts of the same story. If there are indeed many possible universes then stories will overlap sufficiently that complete universes can be built up from them. Details can be made consistent. The story of every possible universe will be told in the others. We could just as well regard ourselves as living in a story. There is no way to distinguish a real universe from a fictional one if all the details are told somewhere.
If a story teller ends his tale, there will always be another who continues from that point on. Indeed, there will be many who will all continue in different ways so we should not expect the future to be determined. But overall there would be some order to what was told. Some stories might get told more often than others. For each time a story is told where something too ridiculous and unexpected happens, there will be many others for which the story follows a more reasonable thread. Your story may never be told exactly but it can be pieced together many times over from those that do get told.