Lessons from Kepler and the theory of everything

Abstract

Johannes Kepler's successes and failures provide lessons for reductionists seeking the theory of everything as well as for those who have proclaimed the end of reductionism.

In trying to describe how fundamental discoveries are made, the work of Johannes Kepler provides a wonderful example. In contrast to present-day theoretical physics articles, which give only the conclusions, Kepler describes his process in agonizing detail. Furthermore, looking from the present, his failures, even more than his successes, may provide an important lesson.

The struggle of Kepler to discover his first laws is described in detail with all the false starts and the near successes in his New Astronomy (1, 2). The first two laws describe the orbits of the individual planets. The third law he found only 9 years later relates the different orbits giving T²/R³ = constant, where T is the period and R the semimajor axis. Kepler was motivated by some idea that the motion of every planet depended on the influence of the sun.

Kepler's three laws are the foundation stones of the Newtonian synthesis. All the planetary motions are explained in terms of two simple equations: Newton's second law and the universal law of gravitational force inversely proportional to the square of the distance. Given the initial position and velocity of each planet, these two laws predict their positions and velocities for the indefinite future. Of course, the further from the present time one goes, the more precisely those initial conditions must be specified and the greater the possibility of perturbations from comets, asteroids, etc., not originally considered.

The Newtonian synthesis is the model for the reductionist approach to physics to explain all phenomena as based on a fundamental set of interactions. In our present theoretical framework there are four interactions: besides gravity there are the electromagnetic, strong, and weak interactions. The latter three are described by the “standard model” gange theory. Many particle theorists are striving to find the “final theory” or the “theory of everything” that would define some single law that would unify all four interactions and perhaps even imply some interactions yet unknown. Some string theorists believe they are on the threshold of discovery, whereas other theorists may only “dream” of it (3).

There exist physicists, particularly those specializing in condensed matter physics, who are very skeptical of, or indeed very antagonistic toward, the reductionists. Their most vocal spokesman has been Phil Anderson (4). Their views are summarized in an article titled “Theory of Everything” (5) by David Pines and R. B. Laughlin, who proclaim “the end of reductionism.”

They believe that most everything of interest, with the exception of some esoteric particle physics experiments, involves systems of large numbers of particles. Starting with a fundamental theory involving elementary particles one cannot directly calculate properties of such systems. On the other hand, such complex systems display “emergent behavior” governed by “higher organizing principles” relatively independent of the fundamental theory. I will refer to these physicists as “emergentists.”

There was a problem that dominated Kepler much more than the problems solved by his three famous laws. He wanted to know why there were just six planets in those particular orbits. This required some symmetry principle that gave order to the universe following the Platonic–Pythagorean tradition. His first idea was that the six orbits were determined by the five regular solids of geometry. This he expounded in great detail in his first book, The Mysterium (6). His second idea was to add musical harmonies as expounded in the Harmony of the World (7), which just incidentally contains the third law.

From the reductionist point of view this question that so dominated Kepler was the wrong question. In the first place, of course, there are now nine planets if you include Pluto, not to mention the asteroid belt. More importantly, this is a problem that we believe has no simple answer. Our solar system was formed 4.5 billion years ago from some whirling mass of gas, much of which ejected from the explosion of earlier stars. The resulting sun and circling planets depend on the particular details of that original chaotic situation. In fact, we now believe there may be hundreds of millions of “solar systems” in our own galaxy, each one differing from the other. In the last decade, we have detected a large number of such planetary systems; those detectable thus far are very different from our own (see, for example, ref. 8).

However, we may consider the development of the solar system as an emergent phenomenon, and so it may be reasonable to look for some organizing principle. In fact, there exists stability criteria which limit the possibilities of a planetary system that could last for 4.5 billion years. However, for systems in which the planets are all much lighter than the central star, these limitations are not very restrictive (see, for example, ref. 9). Neither the reductionists nor the emergentists can solve Kepler's problem.

When we start analyzing the world around us, we don't know which aspects will have a beautiful “simple” explanation and which will remain very complicated. Thus, the motions of the planets and the moon and the moons of Jupiter are all explained by Newton's laws, but the “cast of characters” that is moving is not explained.

It is interesting to look at our present fundamental physics with the lessons from Kepler in mind. We believe that all physical phenomena ultimately depend on the four fundamental interactions; however, these interactions affect a strange cast of characters consisting of 12 fermions: 6 quarks, 3 charged leptons, and 3 neutrinos. They have a great range of masses from the t quark with a mass of 10¹¹ eV (1 eV = 1.602 × 10⁻¹⁹ J) to the neutrinos with masses <1 eV. Many theorists are fascinated with explaining this using symmetry principles (see, for example, ref. 10 and references therein). These are usually taken from group theory rather than geometry or music. Is it possible that there is no simple solution to this problem?

Both in the case of Newton and present-day particle physics we have a set of laws that allows us to make predictions given initial conditions. (Our predictions today are probability statements, but they became very accurate for large numbers of particles or repeated observations.) We can even use these laws to go backward in time. Thus, Newton could tell us quite accurately the positions of the planets 1 million years ago. However, it is impossible in this way to track our planetary system 4.6 billion years back in time, because then we are trying to recreate the very complex formation of the solar system.

In modern cosmology, we also go backward in time. The trick here is first to skip the complex stage of structure formation and go back ≈14 billion years to an early stage of a plasma of neutrons, protons, electrons, and neutrinos. Then it is possible to use the reductionist approach to calculate the consequences of the elementary particle and atomic processes going on in the early universe. Using this approach Alpher, Herman, and Gamow (for a first-hand account, see ref. 11) predicted the existence of a background of black-body radiation with a temperature of a few degrees. This seemed such an extreme form of reductionism that no one searched for this radiation; 15 years later it was discovered by accident by Penzias and Wilson (12). Over the last decade the cosmic microwave background radiation has been explored in wonderful detail. The results are understood in terms of fundamental physics (13).

One can go still farther back in time when the temperature of the universe was a few megaelectronvolts and, by analyzing elementary particle and nuclear reactions, successfully calculate (14) the relative abundances of hydrogen, helium, deuterium, and possibly even ⁷Li in the universe. Thus modern cosmology allows us to trace the evolution of the entire universe from the formation of the first elements to the present day in terms of a few initial parameters. This exciting success of reductionism resembles that of Kepler and Newton in the 17th century. The end of reductionism has been proclaimed prematurely.

The formation of structure in the universe emerged from the small initial matter fluctuations in the period between the formation of hydrogen atoms and the present time. As we have noted above, it is probably impossible to calculate the details of structures such as individual planetary systems. On the other hand, it is hoped that some general features of the large-scale structure, such as the distribution and sizes of galaxy clusters, could be understood. The reductionists have engaged in huge simulations (see, for example, ref. 15) starting with the fluctuation spectrum revealed by microwave background studies and applying fundamental physical laws. Nevertheless, much important physics must be left out. Pines and Laughlin suggest that this is an emergent system subject to organizing principles independent of the particle physics so that the large-scale structure ends up similar to “the structure of Styrofoam, popcorn or puffed cereal” (5).

In fact the simulations, despite their limitations, have demonstrated an important dependence on the particle physics. The large-scale structure cannot be fit if the dark matter consists of very light particles similar to neutrinos, referred to as hot dark matter, because the particles would be moving with large velocities when structure formation began (16). What is required instead seems to be cold dark matter consisting of some still-undiscovered heavy particles.

The reductionists will continue to increase the size of their simulations with the latest supercomputers, and yet many details of the physics must be omitted. The hope clearly is that once the major ingredients such as cold dark matter are determined, a pattern will be seen to emerge that is independent of further details. The solution may require a collaboration of reductionists and emergentists, if they can be persuaded to talk with one another.

If one tries to go still farther back in time to an era when the energies are measured in thousands of megaelectrovolts, there arises the problem that we might not know all the relevant fundamental laws. This troublesome but intriguing possibility often arises in the reductionist approach to astrophysics and cosmology. A recent example concerns the study of solar neutrinos (17). The theory explaining the energy generation near the center of the sun in terms of a set of nuclear reactions predicted that ≈3% of the energy would leave in the form of neutrinos that could penetrate the sun and arrive at the Earth in 8 min. A set of experiments pioneered by Raymond Davis (18), who won the Nobel Prize in Physics in 2002, discovered these neutrinos, demonstrating the existence of the reactions, but the flux of neutrinos was a factor 3 below the prediction. The proposed explanation was a new feature of the fundamental physics that resulted in the oscillation of two-thirds of the electron neutrinos (the type emitted by the sun and the type the experiments were measuring) into other types of neutrinos (muon and tau type, that were known to exist). During the last year the Sudbury Neutrino Observatory experiment actually measured the flux of these other types (19) and indeed found that it was twice that of the electron type. Thus our theory of the source of solar energy received confirmation, and we also discovered some new aspects of the fundamental physics.

One of the great problems of cosmology today is that it requires that most of the matter in the universe consists of some new, as-yet-undiscovered kind of particles (ref. 20 and references therein). The most popular theory is that these are the lightest of a new class of particles, additions to the cast of characters, called supersymmetric particles. If this is true, these particles will be discovered by experiments at the Large Hadron Collider, a large new accelerator to commence operation in 2007 at European Center for Nuclear Research in Geneva.

Many new aspects of fundamental physics have been proposed that affect the very early stages in the history of the universe. One example is the origin of the baryon–antibaryon asymmetry: the visible universe seems to contain matter made of protons and neutrons with practically no antiparticles, which would annihilate them. If one assumes that at some initial stage the universe was symmetric, there must have been some process that produced this asymmetry. After the discovery of the violation of charge–conjugation–parity invariance, Sakharov (21) proposed that a reaction or a decay that violated this invariance and baryon number in the early universe could have produced this asymmetry. A large number of models involving hypothetical new interactions and new particles have been proposed (22). However, most of them can never be tested directly in terrestrial experiments, because the new physics is restricted to a very high energy scale.

Thus we come back to our lesson from Kepler. Does this asymmetry have an explanation in terms of reductionist physics, or is it just part of the cast of characters that emerged from some chaotic state of the early universe? Indeed, it has been proposed that from the initial chaos many universes evolved, each presumably different from ours (see, for example, ref. 23). In some of these perhaps antibaryons predominate, although presumably inhabitants of such a universe will label our baryons as the “anti” ones. In contrast to the case of other solar systems, it seems very unlikely that we will ever detect any other universes even if they exist.

There is, however, at lease one selection criterion among the many possible planetary systems and the many possible universes. The planet must in its temperature and composition be hospitable to human life and the fundamental physics of the universe must allow for biological evolution. This is referred to as the “anthropic principle” and has been the subject of much debate and considerable scorn. Rees (23) refers to it as “anthropic reasoning.” From the particle physicist's reductionist point of view it provides no answer at all to the question, but from an earlier “teleological” tradition it might be considered the answer. If ours is just one of a multitude of universes, that may be the best we can do.

The lesson from Kepler is not that we must refrain from asking what seem to be fundamental questions; the lesson is that we cannot know whether there is any simple answer or where it may come from. For the reductionist the lesson is that there may be much that will not be encompassed by their final theory: many classes of emergent phenomena. For the emergentist the lessons are that (i) reductionism has been surprisingly successful in understanding the largest of many-body systems, our universe, and (ii) there may be emerging systems that cannot be understood simply by the use of their organizing principles.