The Tao of Physics

Encouraged by the brilliant success of Newtonian mechanics in astronomy, physicists extended it to the continuous motion of fluids and to the vibrations of elastic bodies, and again it worked. Finally, even the theory of heat could be reduced to mechanics when it was realized that heat was the energy created by a complicated ‘jiggling’ motion of the molecules. When the temperature of, say, water is increased the motion of the water molecules increases until they overcome the forces holding them together and fly apart. In this way, water turns into steam. On the other hand, when the thermal motion is slowed down by cooling the water, the molecules finally lock into a new, more rigid pattern which is ice. In a similar way, many other thermal phenomena can be understood quite well from a purely mechanistic point of view.

The enormous success of the mechanistic model made physicists of the early nineteenth century believe that the universe was indeed a huge mechanical system running according to the Newtonian laws of motion. These laws were seen as the basic laws of nature and Newton’s mechanics was considered to be the ultimate theory of natural phenomena. And yet, it was less than a hundred years later that a new physical reality was discovered which made the limitations of the Newtonian model apparent and showed that none of its features had absolute validity.

This realization did not come abruptly, but was initiated by developments that had already started in the nineteenth century and prepared the way for the scientific revolutions of our time. The first of these developments was the discovery and investigation of electric and magnetic phenomena which could not be described appropriately by the mechanistic model and involved a new type of force. The important step was made by Michael Faraday and Clerk Maxwell—the first, one of the greatest experimenters in the history of science, the second, a brilliant theorist. When Faraday produced an electric current in a coil of copper by moving a magnet near it, and thus converted the mechanical work of moving the magnet into electric energy, he brought science and technology to a turning point. His fundamental experiment gave birth, on the one hand, to the vast technology of electrical engineering; on the other hand, it formed the basis of his and Maxwell’s theoretical speculations which, eventually, resulted in a complete theory of electromagnetism. Faraday and Maxwell did not only study the effects of the electric and magnetic forces, but made the forces themselves the primary object of their investigation. They replaced the concept of a force by that of a force field, and in doing so they were the first to go beyond Newtonian physics.

Instead of interpreting the interaction between a positive and a negative charge simply by saying that the two charges attract each other like two masses in Newtonian mechanics, Faraday and Maxwell found it more appropriate to say that each charge creates a ‘disturbance’, or a ‘condition’, in the space around it so that the other charge, when it is present, feels a force. This condition in space which has the potential of producing a force is called a field. It is created by a single charge and it exists whether or not another charge is brought in to feel its effect.

This was a most profound change in our conception of physical reality. In the Newtonian view, the forces were rigidly connected with the bodies they act upon. Now the force concept was replaced by the much subtler concept, of a field which had its own reality and could be studied without any reference to material bodies. The culmination of this theory, called electrodynamics, was the realization that light is nothing but a rapidly alternating electromagnetic field travelling through space in the form of waves. Today we know that radio waves, light waves or X-rays, are all electromagnetic waves, oscillating electric and magnetic fields differing only in the frequency of their oscillation, and that visible light is only a tiny fraction of the electromagnetic spectrum.

In spite of these far-reaching changes, Newtonian mechanics at first held its position as the basis of all physics. Maxwell himself tried to explain his results in mechanical terms, interpreting the fields as states of mechanical stress in a very light space-filling medium, called ether, and the electromagnetic waves as elastic waves of this ether. This was only natural as waves are usually experienced as vibrations of something; water waves as vibrations of water, sound waves as vibrations of air. Maxwell, however, used several mechanical interpretations of his theory at the same time and apparently took none of them really seriously. He must have realized intuitively, even if he did not say so explicitly, that the fundamental entities in his theory were the fields and not the mechanical models. It was Einstein who clearly recognized this fact fifty years later when he declared that no ether existed and that the electromagnetic fields were physical entities in their own right which could travel through empty space and could not be explained mechanically.

At the beginning of the twentieth century, then, physicists had two successful theories which applied to different phenomena: Newton’s mechanics and Maxwell’s electrodynamics. Thus the Newtonian model had ceased to be the basis of all physics.

MODERN PHYSICS

The first three decades of our century changed the whole situation in physics radically. Two separate developments—that of relativity theory and of atomic physics—shattered all the principal concepts of the Newtonian world view: the notion of absolute space and time, the elementary solid particles, the strictly causal nature of physical phenomena, and the ideal of an objective description of nature. None of these concepts could be extended to the new domains into which physics was now penetrating.

At the beginning of modern physics stands the extraordinary intellectual feat of one man: Albert Einstein. In two articles, both published in 1905, Einstein initiated two revolutionary trends of thought. One was his special theory of relativity, the other was a new way of looking at electromagnetic radiation which was to become characteristic of quantum theory, the theory of atomic phenomena. The complete quantum theory was worked out twenty years later by a whole team of physicists. Relativity theory, however, was constructed in its complete form almost entirely by Einstein himself. Einstein’s scientific papers stand at the beginning of the twentieth century as imposing intellectual monuments—the pyramids of modern civilization.

Einstein strongly believed in nature’s inherent harmony and his deepest concern throughout his scientific life was to find a unified foundation of physics. He began to move towards this goal by constructing a common framework for electrodynamics and mechanics, the two separate theories of classical physics. This framework is known as the special theory of relativity. It unified and completed the structure of classical physics, but at the same time it involved drastic changes in the traditional concepts of space and time and undermined one of the foundations of the Newtonian world view.

According to relativity theory, space is not three-dimensional and time is not a separate entity. Both are intimately connected and form a four-dimensional continuum, ‘space-time’. In relativity theory, therefore, we can never talk about space without talking about time and vice versa. Furthermore, there is no universal flow of time as in the Newtonian model. Different observers will order events differently in time if they move with different velocities relative to the observed events. In such a case, two events which are seen as occurring simultaneously by one observer may occur in different temporal sequences for other observers. All measurements involving space and time thus lose their absolute significance. In relativity theory, the Newtonian concept of an absolute space as the stage of physical phenomena is abandoned and so is the concept of an absolute time. Both space and time become merely elements of the language a particular observer uses for describing the observed phenomena.

The concepts of space and time are so basic for the description of natural phenomena that their modification entails a modification of the whole framework that we use to describe nature. The most important consequence of this modification is the realization that mass is nothing but a form of energy. Even an object at rest has energy stored in its mass, and the relation between the two is given by the famous equation E= mc2, c being the speed of light.

This constant c, the speed of light, is of fundamental importance for the theory of relativity. Whenever we describe physical phenomena involving velocities which approach the speed of light, our description has to take relativity theory into account. This applies in particular to electromagnetic phenomena, of which light is just one example and which led Einstein to the formulation of his theory.

In 1915, Einstein proposed his general theory of relativity in which the framework of the special theory is extended to include gravity, i.e. the mutual attraction of all massive bodies. Whereas the special theory has been confirmed by innumerable experiments, the general theory has not yet been confirmed conclusively. However, it is so far the most accepted, consistent and elegant theory of gravity and is widely used in astrophysics and cosmology for the description of the universe at large.

The force of gravity, according to Einstein’s theory, has the effect of ‘curving’ space and time. This means that ordinary Euclidean geometry is no longer valid in such a curved space, just as the two-dimensional geometry of a plane cannot be applied on the surface of a sphere. On a plane, we can draw, for example, a square by marking off one metre on a straight line, making a right angle and marking off another metre, then making another right angle and marking off another metre, and finally making a third right angle and marking off one metre again, after which we are back at the starting point and the square is completed. On a sphere, however, this procedure does not work because the rules of Euclidean geometry do not hold on curved surfaces. In the same way, we can define a three-dimensional curved space to be one in which Euclidean geometry is no longer valid. Einstein’s theory, now, says that three-dimensional space is actually curved, and that the curvature is caused by the gravitational field of massive bodies.

Wherever there is a massive object, e.g. a star or a planet, the space around it is curved and the degree of curvature depends on the mass of the object. And as space can never be separated from time in relativity theory, time as well is affected by the presence of matter, flowing at different rates in different parts of the universe. Einstein’s general theory of relativity thus completely abolishes the concepts of absolute space and time. Not only are all measurements involving space and time relative; the whole structure of space-time depends on the distribution of matter in the universe, and the concept of ‘empty space’ loses its meaning.

The mechanistic world view of classical physics was based on the notion of solid bodies moving in empty space. This notion is still valid in the region that has been called the ‘zone of middle dimensions’, that is, in the realm of our daily experience where classical physics continues to be a useful theory. Both concepts—that of empty space and that of solid material bodies—are deeply ingrained in our habits of thought, so it is extremely difficult for us to imagine a physical reality where they do not apply. And yet, this is precisely what modern physics forces us to do when we go beyond the middle dimensions. ‘Empty space’ has lost its meaning in astrophysics and cosmology, the sciences of the universe at large, and the concept of solid objects was shattered by atomic physics, the science of the infinitely small.

At the turn of the century, several phenomena connected with the structure of atoms and inexplicable in terms of classical physics were discovered. The first indication that atoms had some structure came from the discovery of X-rays; a new radiation which rapidly found its now Well known application in medicine. X-rays, however, are not the only radiation emitted by atoms. Soon after their discovery, other kinds of radiation were discovered which are emitted by the atoms of so-called radioactive substances. The phenomenon of radioactivity gave definite proof of the composite nature of atoms, showing that the atoms of radioactive substances not only emit various types of radiation, but also transform themselves into atoms of completely different substances.

Besides being objects of intense study, these phenomena were also used, in most ingenious ways, as new tools to probe deeper into matter than had ever been possible before. Thus Max von Laue used X-rays to study the arrangements of atoms in crystals, and Ernest Rutherford realized that the so-called alpha particles emanating from radioactive substances were high-speed projectiles of subatomic size which could be used to explore the interior of the atom. They could be fired at atoms, and from the way they were deflected one could draw conclusions about the atoms’ structure.

When Rutherford bombarded atoms with these alpha particles, he obtained sensational and totally unexpected results. Far from being the hard and solid particles they were believed to be since antiquity, the atoms turned out to consist of vast regions of space in which extremely small particles—the electrons—moved around the nucleus, bound to it by electric forces. It is not easy to get a feeling for the order of magnitude of atoms, so far is it removed from our macroscopic scale. The diameter of an atom is about one hundred millionth of a centimetre. In order to visualize this diminutive size, imagine an orange blown up to the size of the Earth. The atoms of the orange will then have the size of cherries. Myriads of cherries, tightly packed into a globe of the size of the Earth—that’s a magnified picture of the atoms in an orange.

An atom, therefore, is extremely small compared to macroscopic objects, but it is huge compared to the nucleus in its centre. In our picture of cherry-sized atoms, the nucleus of an atom will be so small that we will not be able to see it. If we blew up the atom to the size of a football, or even to room size, the nucleus would still be too small to be seen by the naked eye. To see the nucleus, we would have to blow up the atom to the size of the biggest dome in the world, the dome of St Peter’s Cathedral in Rome. In an atom of that size, the nucleus would have the size of a grain of salt! A grain of salt in the middle of the dome of St Peter’s, and specks of dust whirling around it in the vast space of the dome—this is how we can picture the nucleus and electrons of an atom.

Soon after the emergence of this ‘planetary’ model of the atom, it was discovered that the number of electrons in the atoms of an element determine the element’s chemical properties, and today we know that the whole periodic table of elements can be built up by successively adding protons and neutrons to the nucleus of the lightest atom—hydrogen*—and the corresponding number of electrons to its atomic ‘shell’. The interactions between the atoms give rise to the various chemical processes, so that all of chemistry can now in principle be understood on the basis of the laws of atomic physics.

These laws, however, were not easy to recognize. They were discovered in the 1920s by an international group of physicists including Niels Bohr from Denmark, Louis De Broglie from France, Erwin Schrödinger and Wolfgang Pauli from Austria, Werner Heisenberg from Germany, and Paul Dirac from England. These men joined their forces across all national borders and shaped one of the most exciting periods in modern science, which brought them, for the first time, into contact with the strange and unexpected reality of the subatomic world. Every time the physicists asked nature a question in an atomic experiment, nature answered with a paradox, and the more they tried to clarify the situation, the sharper the paradoxes became. It took them a long time to accept the fact that these paradoxes belong to the intrinsic structure of atomic physics, and to realize that they arise whenever one attempts to describe atomic events in the traditional terms of physics. Once this was perceived, the physicists began to learn to ask the right questions and to avoid contradictions. In the words of Heisenberg, ‘they somehow got into the spirit of the quantum theory’, and finally they found the precise and consistent mathematical formulation of this theory.

The concepts of quantum theory were not easy to accept even after their mathematical formulation had been completed. Their effect on the physicists’ imagination was truly shattering. Rutherford’s experiments had shown that atoms, instead of being hard and indestructible, consisted of vast regions of space in which extremely small particles moved, and now quantum theory made it clear that even these particles were nothing like the solid objects of classical physics. The subatomic units of matter are very abstract entities which have a dual aspect. Depending on how we look at them, they appear sometimes as particles, sometimes as waves; and this dual nature is also exhibited by light which can take the form of electromagnetic waves or of particles.

This property of matter and of light is very strange. It seems impossible to accept that something can be, at the same time, a particle—i.e. an entity confined to a very small volume—and a wave, which is spread out over a large region of space. This contradiction gave rise to most of the koan-like paradoxes which finally led to the formulation of quantum theory. The whole development started when Max Planck discovered that the energy of heat radiation is not emitted continuously, but appears in the form of ‘energy packets’. Einstein called these energy packets ‘quanta’ and recognized them as a fundamental aspect of nature. He was bold enough to postulate that light and every other form of electromagnetic radiation can appear not only as electromagnetic waves, but also in the form of these quanta. The light quanta, which gave quantum theory its name, have since been accepted as bona fide particles and are now called photons. They are particles of a special kind, however, massless and always travelling with the speed of light.

The apparent contradiction between the particle and the wave picture was solved in a completely unexpected way which called in question the very foundation of the mechanistic world view—the concept of the reality of matter. At the subatomic level, matter does not exist with certainty at definite places, but rather shows ‘tendencies to exist, and atomic events do not occur with certainty at definite times and in definite ways, but rather show ‘tendencies to occur. In the formalism of quantum theory, these tendencies are expressed as probabilities and are associated with mathematical quantities which take the form of waves. This is why particles can be waves at the same time. They are not ‘real’ three-dimensional waves like sound or water waves. They are ‘probability waves’, abstract mathematical quantities with all the characteristic properties of waves which are related to the probabilities of finding the particles at particular points in space and at particular times. All the laws of atomic physics are expressed in terms of these probabilities. We can never predict an atomic event with certainty; we can only say how likely it is to happen.

Quantum theory has thus demolished the classical concepts of solid objects and of strictly deterministic laws of nature. At the subatomic level, the solid material objects of classical physics dissolve into wave-like patterns of probabilities, and these patterns, ultimately, do not represent probabilities of things, but rather probabilities of interconnections. A careful analysis of the process of observation in atomic physics has shown that the subatomic particles have no meaning as isolated entities, but can only be understood as interconnections between the preparation of an experiment and the subsequent measurement. Quantum theory thus reveals a basic oneness of the universe. It shows that we cannot decompose the world into independently existing smallest units. As we penetrate into matter, nature does not show us any isolated ‘basic building blocks’, but rather appears as a complicated web of relations between the various parts of the whole. These relations always include the observer in an essential way. The human observer constitutes the final link in the chain of observational processes, and the properties of any atomic object can only be understood in terms of the object’s interaction with the observer. This means that the classical ideal of an objective description of nature is no longer valid. The Cartesian partition between the I and the world, between the observer and the observed, cannot be made when dealing with atomic matter. In atomic physics, we can never speak about nature without, at the same time, speaking about ourselves.

The new atomic theory could immediately solve several puzzles which had arisen in connection with the structure of atoms and could not be explained by Rutherford’s planetary model. First of all, Rutherford’s experiments had shown that the atoms making up solid matter consist almost entirely of empty space, as far as the distribution of mass is concerned. But if all the objects around us, and we ourselves, consist mostly of empty space, why can’t we walk through closed doors? In other words, what is it that gives matter its solid aspect?

A second puzzle was the extraordinary mechanical stability of atoms. In the air, for example, atoms collide millions of times every second and yet go back to their original form after each collision. No planetary system following the laws of classical mechanics would ever come out of these collisions unaltered. But an oxygen atom will always retain its characteristic configuration of electrons, no matter how often it collides with other atoms. This configuration, furthermore, is exactly the same in all atoms of a given kind. Two iron atoms, and consequently two pieces of pure iron, are completely identical, no matter where they come from or how they have been treated in the past.

Quantum theory has shown that all these astonishing properties of atoms arise from the wave nature of their electrons. To begin with, the solid aspect of matter is the consequence of a typical ‘quantum effect’ connected with the dual wave/particle aspect of matter, a feature of the subatomic world which has no macroscopic analogue. Whenever a particle is confined to a small region of space it reacts to this confinement by moving around, and the smaller the region of confinement is, the faster the particle moves around in it. In the atom, now, there are two competing forces. On the one hand, the electrons are bound to the nucleus by electric forces which try to keep them as close as possible. On the other hand, they respond to their confinement by whirling around, and the tighter they are bound to the nucleus, the higher their velocity will be; in fact, the confinement of electrons in an atom results in enormous velocities of about 600 miles per second! These high velocities make the atom appear as a rigid sphere, just as a fast rotating propeller appears as a disc. It is very difficult to compress atoms any further and thus they give matter its familiar solid aspect.

In the atom, then, the electrons settle in orbits in such a way that there is an optimal balance between the attraction of the nucleus and their reluctance to be confined. The atomic orbits, however, are very different from those of the planets in the solar system, the difference arising from the wave nature of the electrons. An atom cannot be pictured as a small planetary system. Rather than particles circling around the nucleus, we have to imagine probability waves arranged in different orbits. Whenever we make a measurement, we will find the electrons somewhere in these orbits, but we cannot say that they are ‘going around the nucleus’ in the sense of classical mechanics.