The Analysis of Matter

with events rather than particles or impenetrable material units. This involves a re-interpretation of motion as it occurs in physics, which will be considered in a later chapter. For the present, I am concerned with the materials which will be required for this purpose as well as for the interpretation of other physical phenomena. A "steady event," as I use the term, is anything which is devoid of physical structure and is compresent with events which are not compresent with each other, but are one earlier and the other later; in other words, the steady event is a member of at least two points which have a time-like interval. When a steady event is contrasted with a rhythm, it is assumed that the steady event is not part of a periodic process; but it cannot be taken as certain that there are any elementary events which are not parts of such processes. It may be that all non-periodic changes occur by way of transactions; but this must be an open question in the present state of knowledge. A "rhythm," as already explained, is a recurring cycle of events, in which there is a qualitative similarity between[Pg 357] corresponding members of different periods. A rhythm may have a period consisting of a finite number of events, or one consisting of an infinite number; it may be discrete or continuous. If it is discrete, the proper time of one period is measured by the number of events in the period, and the "frequency" of the process is the reciprocal of this number. But here we are speaking of the frequency as measured by the time proper to the period; by an extraneous time the frequency may be quite different. What is commonly called the frequency of a light-wave is its frequency with respect to axes fixed relatively to the emitting body. Its frequency relative to axes which travel with it is zero; this is only the extreme of the Doppler effect. There is perhaps a certain inconsistency in the practice of studying bodies by means of axes which move with them, while light is always treated with reference to material axes. If we want to understand light in itself, not in its relation to matter, we ought to let our axes travel with it. In that case, its periodicity is spatial, not temporal; it is like that of corrugated iron. From the standpoint of the light itself, each part of a light-wave is a steady event in the sense defined above. One of the most fundamental of rhythmic processes will be the revolution of an electron about a nucleus, unless we accept the view of the new quantum mechanics, according to which there is no reason to suppose that this really occurs. In the Bohr-Sommerfeld theory, this revolution goes on by itself until it is altered either by a quantum change or by some more conventional chemical or electrical action. The question arises: why should we suppose that there is a process at all? Why not suppose that there is a steady event, possessed of a certain amount of energy, which is replaced, in a quantum change, by another steady event, possessed of a different amount of energy, the balance being radiated or absorbed? There is a certain attraction about this hypothesis, since the[Pg 358] atom gives no external indication of its presence while the supposed process continues, and therefore there can be no direct evidence that changes are occurring, such as a steady motion supposes. In any case, if an electron is revolving round a proton in a circle, and both are spherically symmetrical, it is not easy to see, from a relativist point of view, exactly what is meant by saying that the electron is revolving. This difficulty is not diminished by the hypothesis of spinning electrons. We have the same difficulties as in the case of absolute rotation and Foucault's pendulum—the difficulties, namely, which Newton advanced to prove the necessity of absolute motion. Within the system consisting of the electron and proton alone, nothing is changing while the electron revolves in its circular orbit; the change is only with reference to other bodies. Why not regard the state of affairs as static, but possessed of a certain amount of energy? Energy may be altered in amount by a change of axes, and is not an invariant property of the system; but reference to the outside world here is less serious, since the only purpose served by the energy of the atom is to provide physics with something which can be radiated into the outer world or absorbed from it. That is to say, energy is required only as something whose changes govern the causal relations of the atom with the outer world. This point of view is essentially that of the Heisenberg theory. There are several apparent difficulties in such a view. In the first place, the formula for energy obtained on the assumption that the electron revolves gives exactly the changes of energy required to account for spectroscopic phenomena; the Bohr-Sommerfeld theory agrees with observation so minutely that its formula for energy must be accepted. Of course we could say that the energy just happens to be what it would be if the electron revolved in one of the quantum orbits; but this would seem an almost miraculous coincidence. This, however, is not the strongest argument, which is that[Pg 359] derived from the quantum principle. The quantum principle in its older form can only be applied to periodic processes; if it is to apply, as we find that it does, to the interchange of energy between light and the atom, we must assume, if we adhere to the older theory, that within the atom there is something that can be called a "frequency," i.e. something which is periodic, which compels us to admit that, within an atom in a steady state, there is a recurring process whose formal properties are those which would be exhibited by a revolution of the electron, and perhaps also by a rotation. If we adhere to the Bohr theory, what can be supposed to be really occurring? If relative motion were all that was taking place, we should have either to find an interpretation for the spinning electron, or else to say that, taking axes fixed relatively to any large body, the line joining the electron to the proton rotates rapidly; any large body will do, since none rotates with an angular velocity comparable to that of the electron. But why should the electron be interested in this fact? Why should its capacity for emitting light be connected with it? There must be something happening where the electron is, if the process is to be intelligible. This brings us back to Maxwell's equations, as governing what is occurring in the medium. And there must be a rhythmic character in the events occurring where the electron is, if we are to avoid all the troubles of action at a distance. We suppose, therefore, that throughout an electromagnetic field there are events whose formal properties we know more or less, and that they, not the change of spatial configuration, are the immediate causes of what takes place. This brings us back to the cycle of events which we used in the preceding chapter to define a rhythm. The point is that a rhythm can never consist merely in periodic changes of spatial relation between two or more bodies, but must consist of qualitative cycles of events. We have experience of such cycles when[Pg 360] we watch a large-scale periodic event, such as the swing of a pendulum. All that happens to us during the cycle happens in us, not in a number of different places; and any effect upon us depends upon what happens to us. I am suggesting that this is a proper analogy when we wish to understand how a periodic motion affects an electron. I come now to what I shall call "transactions," by which I mean quantum changes. I call them "transactions" because energy is exchanged between different processes. The processes concerned must be periodic, since otherwise the quantum principle is unnecessary. In the simplest case, that of emission of light by a hydrogen atom, we have as antecedent, speaking the language of the older quantum theory, one periodic process (the revolution of the electron in an orbit other than the minimum orbit) and as consequent two processes, namely: (1) The revolution of the electron in a smaller orbit, (2) a light-wave. The latter, as already explained, is only periodic in a certain sense. The energy of the antecedent is the sum of the energies of the consequents. The amount of action during one period of the antecedent is a multiple of , and so are the amounts of action of the consequents during one period of each. Exactly the converse occurs when light is absorbed by a hydrogen atom. In other cases, both the antecedent and the consequent may consist of two or more rhythms; but always there will be conservation of energy, and each rhythm will contain an amount of action which is a multiple of . As yet, everything concerned with quanta is more or less mysterious, although Heisenberg's theory has somewhat diminished the mystery. We do not know whether quantum changes are really sudden or not; we do not know whether the space concerned in atomic structure is continuous or discrete. If electrons always moved in circles, as in the first form of Bohr's theory, we could be content with a granular[Pg 361] discrete space, and suppose that the intermediate orbits are geometrically non-existent. But the existence of elliptic orbits in Sommerfeld's development of the theory makes this difficult. And in atoms with many planetary electrons, the paths of some are supposed to cross those of others. In spite of these difficulties, however, I do not despair of the hypothesis that space-time is discrete. The older quantum theory uses the traditional conceptions of physics, and thinks of geometrical orbits in a constant space. The Heisenberg theory, on the contrary, has a completely new kinematics, according to which unquantized orbits (if we may still speak of orbits) are geometrically impossible. It is difficult, as yet, to translate this theory out of its technical form. But even according to the older theory, one can see that a discrete space-time is possible. For when we think of the matter in terms of space-time, we realize that the geometry of the neighbourhood of the atom may be different at different times. If an electron moves in one sort of orbit at one time and in another at another, it does not follow that each sort of orbit was geometrically possible at the time when the other was being described. Perhaps it is not superfluous to explain what is meant by saying that an orbit is "geometrically possible" though not physically actual. What is meant is this: there is a series of groups of events, each group being a point, and the series being one in which all the intervals of points are time-like, and in which, if a constant value is assigned to one of the co-ordinates, the remaining three give a curve in a three-dimensional space having the geometrical properties of the orbit in question. Whenever we speak of an orbit geometrically, we are assuming that we can distinguish one of the co-ordinates as "time," give it a constant value, and consider the relations of the remaining three co-ordinates. Now it is always possible that there may be a fallacy in this procedure, since it may be that such geometrical relations as we are considering are[Pg 362] impossible among "simultaneous" points. Moreover, in the general theory of relativity, it may be impossible to distinguish one co-ordinate as more representative of time than the others. When, from a traditional point of view, two orbits cross each other, this no longer happens from a relativity standpoint. We cannot assume, that is to say, that