A Stubbornly Persistent Illusion

by Stephen Hawking

  It is hard necessity and not speculation or a desire for novelty which forces us to change the old classical view. The difficulties of applying the old view have been outlined for one instance only, that of diffraction phenomena. But many others, equally convincing, could be quoted. Changes of view are continually forced upon us by our attempts to understand reality. But it always remains for the future to decide whether we chose the only possible way out and whether or not a better solution of our difficulties could have been found.

  We have had to forsake the description of individual cases as objective happenings in space and time; we have had to introduce laws of a statistical nature. These are the chief characteristics of modern quantum physics.

  Previously, when introducing new physical realities, such as the electromagnetic and gravitational field, we tried to indicate in general terms the characteristic features of the equations through which the ideas have been mathematically formulated. We shall now do the same with quantum physics, referring only very briefly to the work of Bohr, De Broglie, Schrödinger, Heisenberg, Dirac and Born.

  Let us consider the case of one electron. The electron may be under the influence of an arbitrary foreign electromagnetic field, or free from all external influences. It may move, for instance, in the field of an atomic nucleus or it may diffract on a crystal. Quantum physics teaches us how to formulate the mathematical equations for any of these problems.

  We have already recognized the similarity between an oscillating cord, the membrane of a drum, a wind instrument, or any other acoustical instrument on the one hand, and a radiating atom on the other. There is also some similarity between the mathematical equations governing the acoustical problem and those governing the problem of quantum physics. But again the physical interpretation of the quantities determined in these two cases is quite different. The physical quantities describing the oscillating cord and the radiating atom have quite a different meaning, despite some formal likeness in the equations. In the case of the cord, we ask about the deviation of an arbitrary point from its normal position at an arbitrary moment. Knowing the form of the oscillating cord at a given instant, we know everything we wish. The deviation from the normal can thus be calculated for any other moment from the mathematical equations for the oscillating cord. The fact that some definite deviation from the normal position corresponds to every point of the cord is expressed more rigorously as follows: for any instant, the deviation from the normal value is a function of the co-ordinates of the cord. All points of the cord form a one-dimensional continuum, and the deviation is a function defined in this one-dimensional continuum, to be calculated from the equations of the oscillating cord.

  Analogously, in the case of an electron a certain function is determined for any point in space and for any moment. We shall call this function the probability wave. In our analogy the probability wave corresponds to the deviation from the normal position in the acoustical problem. The probability wave is, at a given instant, a function of a three-dimensional continuum, whereas, in the case of the cord the deviation was, at a given moment, a function of the one-dimensional continuum. The probability wave forms the catalogue of our knowledge of the quantum system under consideration and will enable us to answer all sensible statistical questions concerning this system. It does not tell us the position and velocity of the electron at any moment because such a question has no sense in quantum physics. But it will tell us the probability of meeting the electron on a particular spot, or where we have the greatest chance of meeting an electron. The result does not refer to one, but to many repeated measurements. Thus the equations of quantum physics determine the probability wave just as Maxwell’s equations determine the electromagnetic field and the gravitational equations determine the gravitational field. The laws of quantum physics are again structure laws. But the meaning of physical concepts determined by these equations of quantum physics is much more abstract than in the case of electromagnetic and gravitational fields; they provide only the mathematical means of answering questions of a statistical nature.

  So far we have considered the electron in some external field. If it were not the electron, the smallest possible charge, but some respectable charge containing billions of electrons, we could disregard the whole quantum theory and treat the problem according to our old pre-quantum physics. Speaking of currents in a wire, of charged conductors, of electromagnetic waves, we can apply our old simple physics contained in Maxwell’s equations. But we cannot do this when speaking of the photoelectric effect, intensity of spectral lines, radioactivity, diffraction of electronic waves and many other phenomena in which the quantum character of matter and energy is revealed. We must then, so to speak, go one floor higher. Whereas in classical physics we spoke of positions and velocities of one particle, we must now consider probability waves, in a three-dimensional continuum corresponding to this one-particle problem.

  Quantum physics gives its own prescription for treating a problem if we have previously been taught how to treat an analogous problem from the point of view of classical physics.

  For one elementary particle, electron or photon, we have probability waves in a three-dimensional continuum, characterizing the statistical behavior of the system if the experiments are often repeated. But what about the case of not one but two interacting particles, for instance, two electrons, electron and photon, or electron and nucleus? We cannot treat them separately and describe each of them through a probability wave in three dimensions, just because of their mutual interaction. Indeed, it is not very difficult to guess how to describe in quantum physics a system composed of two interacting particles. We have to descend one floor, to return for a moment to classical physics. The position of two material points in space, at any moment, is characterized by six numbers, three for each of the points. All possible positions of the two material points form a six-dimensional continuum and not a three-dimensional one as in the case of one point. If we now again ascend one floor, to quantum physics, we shall have probability waves in a six-dimensional continuum and not in a three-dimensional continuum as in the case of one particle. Similarly, for three, four, and more particles the probability waves will be functions in a continuum of nine, twelve, and more dimensions.

  This shows clearly that the probability waves are more abstract than the electromagnetic and gravitational field existing and spreading in our three-dimensional space. The continuum of many dimensions forms the background for the probability waves, and only for one particle does the number of dimensions equal that of physical space. The only physical significance of the probability wave is that it enables us to answer sensible statistical questions in the case of many particles as well as of one. Thus, for instance, for one electron we could ask about the probability of meeting an electron in some particular spot. For two particles our question could be: what is the probability of meeting the two particles at two definite spots at a given instant?

  Our first step away from classical physics was abandoning the description of individual cases as objective events in space and time. We were forced to apply the statistical method provided by the probability waves. Once having chosen this way, we are obliged to go further toward abstraction. Probability waves in many dimensions corresponding to the many-particle problem must be introduced.

  Let us, for the sake of briefness, call everything except quantum physics, classical physics. Classical and quantum physics differ radically. Classical physics aims at a description of objects existing in space, and the formulation of laws governing their changes in time. But the phenomena revealing the particle and wave nature of matter and radiation, the apparently statistical character of elementary events such as radioactive disintegration, diffraction, emission of spectral lines, and many others, forced us to give up this view. Quantum physics does not aim at the description of individual objects in space and their changes in time. There is no place in quantum physics for statements such as: “This object is so-and-so, has this-and-this property.” Instead we
have statements of this kind: “There is such-and-such a probability that the individual object is so-and-so and has this-and-this property.” There is no place in quantum physics for laws governing the changes in time of the individual object. Instead, we have laws governing the changes in time of the probability. Only this fundamental change, brought into physics by the quantum theory, made possible an adequate explanation of the apparently discontinuous and statistical character of events in the realm of phenomena in which the elementary quanta of matter and radiation reveal their existence.

  Yet new, still more difficult problems arise which have not been definitely settled as yet. We shall mention only some of these unsolved problems. Science is not and will never be a closed book. Every important advance brings new questions. Every development reveals, in the long run, new and deeper difficulties.

  We already know that in the simple case of one or many particles we can rise from the classical to the quantum description, from the objective description of events in space and time to probability waves. But we remember the all-important field concept in classical physics. How can we describe interaction between elementary quanta of matter and field? If a probability wave in thirty dimensions is needed for the quantum description of ten particles, then a probability wave with an infinite number of dimensions would be needed for the quantum description of a field. The transition from the classical field concept to the corresponding problem of probability waves in quantum physics is a very difficult step. Ascending one floor is here no easy task and all attempts so far made to solve the problem must be regarded as unsatisfactory. There is also one other fundamental problem. In all our arguments about the transition from classical physics to quantum physics we used the old prerelativistic description in which space and time are treated differently. If, however, we try to begin from the classical description as proposed by the relativity theory, then our ascent to the quantum problem seems much more complicated. This is another problem tackled by modern physics, but still far from a complete and satisfactory solution. There is still a further difficulty in forming a consistent physics for heavy particles, constituting the nuclei. In spite of the many experimental data and the many attempts to throw light on the nuclear problem, we are still in the dark about some of the most fundamental questions in this domain.

  There is no doubt that quantum physics explained a very rich variety of facts, achieving, for the most part, splendid agreement between theory and observation. The new quantum physics removes us still further from the old mechanical view, and a retreat to the former position seems, more than ever, unlikely. But there is also no doubt that quantum physics must still be based on the two concepts: matter and field. It is, in this sense, a dualistic theory and does not bring our old problem of reducing everything to the field concept even one step nearer realization.

  Will the further development be along the line chosen in quantum physics, or is it more likely that new revolutionary ideas will be introduced into physics? Will the road of advance again make a sharp turn, as it has so often done in the past?

  During the last few years all the difficulties of quantum physics have been concentrated around a few principal points. Physics awaits their solution impatiently. But there is no way of foreseeing when and where the clarification of these difficulties will be brought about.

  PHYSICS AND REALITY

  What are the general conclusions which can be drawn from the development of physics indicated here in a broad outline representing only the most fundamental ideas?

  Science is not just a collection of laws, a catalogue of unrelated facts. It is a creation of the human mind, with its freely invented ideas and concepts. Physical theories try to form a picture of reality and to establish its connection with the wide world of sense impressions. Thus the only justification for our mental structures is whether and in what way our theories form such a link.

  We have seen new realities created by the advance of physics. But this chain of creation can be traced back far beyond the starting point of physics. One of the most primitive concepts is that of an object. The concepts of a tree, a horse, any material body, are creations gained on the basis of experience, though the impressions from which they arise are primitive in comparison with the world of physical phenomena. A cat teasing a mouse also creates, by thought, its own primitive reality. The fact that the cat reacts in a similar way toward any mouse it meets shows that it forms concepts and theories which are its guide through its own world of sense impressions.

  “Three trees” is something different from “two trees.” Again “two trees” is different from “two stones.” The concepts of the pure numbers 2, 3, 4 . . ., freed from the objects from which they arose, are creations of the thinking mind which describe the reality of our world.

  The psychological subjective feeling of time enables us to order our impressions, to state that one event precedes another. But to connect every instant of time with a number, by the use of a clock, to regard time as a one-dimensional continuum, is already an invention. So also are the concepts of Euclidean and non-Euclidean geometry, and our space understood as a three-dimensional continuum.

  Physics really began with the invention of mass, force, and an inertial system. These concepts are all free inventions. They led to the formulation of the mechanical point of view. For the physicist of the early nineteenth century, the reality of our outer world consisted of particles with simple forces acting between them and depending only on the distance. He tried to retain as long as possible his belief that he would succeed in explaining all events in nature by these fundamental concepts of reality. The difficulties connected with the deflection of the magnetic needle, the difficulties connected with the structure of the ether, induced us to create a more subtle reality. The important invention of the electromagnetic field appears. A courageous scientific imagination was needed to realize fully that not the behavior of bodies, but the behavior of something between them, that is, the field, may be essential for ordering and understanding events.

  Later developments both destroyed old concepts and created new ones. Absolute time and the inertial co-ordinate system were abandoned by the relativity theory. The background for all events was no longer the one-dimensional time and the three-dimensional space continuum, but the four-dimensional time-space continuum, another free invention, with new transformation properties. The inertial co-ordinate system was no longer needed. Every co-ordinate system is equally suited for the description of events in nature.

  The quantum theory again created new and essential features of our reality. Discontinuity replaced continuity. Instead of laws governing individuals, probability laws appeared.

  The reality created by modern physics is, indeed, far removed from the reality of the early days. But the aim of every physical theory still remains the same.

  With the help of physical theories we try to find our way through the maze of observed facts, to order and understand the world of our sense impressions. We want the observed facts to follow logically from our concept of reality. Without the belief that it is possible to grasp the reality with our theoretical constructions, without the belief in the inner harmony of our world, there could be no science. This belief is and always will remain the fundamental motive for all scientific creation. Throughout all our efforts, in every dramatic struggle between old and new views, we recognize the eternal longing for understanding, the ever-firm belief in the harmony of our world, continually strengthened by the increasing obstacles to comprehension.

  WE SUMMARIZE

  Again the rich variety of facts in the realm of atomic phenomena forces us to invent new physical concepts. Matter has a granular structure; it is composed of elementary particles, the elementary quanta of matter. Thus, the electric charge has a granular structure and—most important from the point of view of the quantum theory—so has energy. Photons are the energy quanta of which light is composed.

  Is light a wave or a shower of photons? Is a beam of electrons a shower
of elementary particles or a wave? These fundamental questions are forced upon physics by experiment. In seeking to answer them we have to abandon the description of atomic events as happenings in space and time, we have to retreat still further from the old mechanical view. Quantum physics formulates laws governing crowds and not individuals. Not properties but probabilities are described, not laws disclosing the future of systems are formulated, but laws governing the changes in time of the probabilities and relating to great congregations of individuals.

  Autobiographical Notes

  Albert Einstein famously said of himself, “Do not worry about your difficulties in mathematics. I can assure you mine are still greater.” Though modest about his own abilities and often caricatured as a poor student (though in reality, simply a willful one), Einstein showed an unusually intense curiosity about the natural world, and a drive to learn as much of the scientific and mathematical cannon as possible. In his “Autobiographical Notes,” Einstein presents his own very unusual scientific history, unusual if for no other reason than that it is replete with equations.

  This work, perhaps more than any other in the volume, gives us insight into why Einstein is the icon that he is. In describing his own education, Einstein gives us a guided tour of the state of science in his youth. By gradually describing both his contributions and those of others in relativity and quantum mechanics, we begin to see how much the world of physics was revolutionized over his lifetime.