The Analysis of Matter

part of the history of the other. In another equally legitimate system of co-ordinates, these two points would not have three co-ordinates identical. And the fact that a certain orbit passes from () in a certain direction does not imply that there is an orbit passing from () in a direction which is the same so far as , , are concerned. Therefore the apparent difficulties in the way of a discrete space are not necessarily insuperable. From our point of view, it is a difficulty in the quantum principle that it is stated in a form involving energy, which, from a relativity standpoint, requires re-interpretation. It is also a difficulty that we do not know any laws determining when a transaction will take place, and that we do not know whether it is really sudden or not. For all these reasons, we are compelled to be very tentative in philosophizing. I will, however, repeat the outcome of this chapter, such as it is. In one sense, the theory of space-time points as groups of events requires that all change should be discontinuous. An event e is a member of a certain set of space-time points, and of no others: the boundaries of the region constituted by this set are the boundaries of , so that it comes into existence suddenly and ceases to exist suddenly. Nevertheless, we can, if necessary, provide for continuity within this scheme.[Pg 363] Suppose a continuous series of qualities, like the colours of the rainbow; suppose that, in some process, each of these is compresent with its neighbours up to a certain distance in either direction, but not with more distant members of the series. Then the group of qualities existing at a point will change continuously, although each separate quality changes discontinuously. We may suppose this to be the nature of change between transactions, and in particular during a rhythm. There is no proof that change is ever continuous, but there is also no proof that it is not. We will assume, for the moment, that change between transactions is continuous in the above sense, but that transactions are discontinuous. This assumption is made only for the sake of brevity of statement; it is not asserted to be true, or even more probable than the opposite assumption. If we take the above view, there will be three kinds of things to consider in physics: transactions, steady events, and rhythms. Transactions are dominated by quantum laws. Steady events continue, without internal change, from one transaction to the next, or throughout a certain portion of a continuous change; percepts are steady events, or rather systems of steady events. The relation of a steady event to a rhythm I conceive according to a musical analogy: that of a long note on the violin while a series of chords occurs repeatedly on the piano. All our life is lived to the accompaniment of a rhythm of breathing and heart-beating, which provides us with a physiological clock by which we can roughly estimate times. I imagine, perhaps fancifully, something faintly analogous as an accompaniment to every steady event. There are laws connecting the steady event with the rhythm; these are the laws of harmony. There are laws regulating transactions; these are the laws of counterpoint. We must assume periodicity as a feature of the state of affairs where there are steady events, since we cannot state[Pg 364] the quantum principle without it. We have to find a meaning for "frequency" in order to connect energy with . It is not altogether easy to see how one frequency is to be compared with another. In the case of light, we can estimate the distance between the crest of one wave and the crest of the next. Knowing the velocity of light, this tells us how many waves pass a given place in a second. But here the periodicity exists for the outside observer; for an observer travelling on the crest of a given wave, there is no process and no periodicity. For an outside observer, there is a process in the motion of the light-wave; but our observer on the wave considers himself to be at rest, and presumably does not see objects flying past him. Thus for him the periodicity of a light-wave is spatial rather than temporal. One light-wave will consist of a series , , ... , ... of steady events, the intervals between which are space-like; the next will consist of a series , , ... , ..., again having space-like intervals from each other and from the previous series; and will have a similarity of quality which neither has to or (where is different from ). Each of these events is supposed to continue as long as the light-wave continues, i.e. until there is a transaction. Given any event which is connected with matter, may be compresent with , ... , ... , ... , ... successively, but not with all at once. This is what happens when a light-wave passes an observer or any other piece of matter. A series of events forming one light-wave are inseparably associated, in the sense that when there is one of them there will be others throughout the space covered by the wave. Similarly the series of events (if any) involved in the revolution of an electron are inseparably associated; but there is this difference, that these events form a temporal series from the standpoint of the electron, whereas the events constituting a light-wave form a spatial series from the point of view of the light-wave. [Pg 365] There are difficulties in the above which might be resolved in various ways, but we do not know which to choose. What, for example, shall we say about the transaction which consists in the absorption of energy by an atom from a light-wave? The correct view is supposed to be that, in such a case, a planetary electron passes suddenly from a smaller to a larger orbit. But if we imagine a light-wave to consist of a number of events , , ... , ..., one might expect that at least one whole wave would be required to produce one definite effect, and that a part of the wave would produce only part of the effect, if any. But a whole wave takes a finite time to reach the atom. This difficulty exists for any view which regards light as consisting of waves and quantum transitions as sudden, but would be obviated if either of these suppositions were dropped. We may therefore take it as part of the general unsolved problem of the relation between radiant energy and energy associated with matter. This problem, though it interests the philosopher, belongs to the domain of physics, and can only be profitably considered by a physicist. I am therefore content to await the discoveries of others. As regards quanta, let us examine once more what is implied by the fact that there is an important constant . In the first place, only exists, or at any rate is only important, in the case of periodic processes, and it is a characteristic of one complete period. In the second place, only integral multiples of occur. In the third place, when a transaction involves the loss by one system of a certain multiple of , another system may acquire another multiple of : what is transferred always unaltered in amount is energy. These seem to be the most significant facts about . It seems impossible to resist the view that represents something of fundamental importance in the physical world, which, in turn, involves the conclusion that periodicity is an element in physical laws, and that one period of a periodic[Pg 366] process must be treated as, in some sense, a unit. This follows from the fact that processes arrange themselves so as to secure that a period shall have an important property. This property is simplest in the case of a light-wave: the energy of one light-wave multiplied by the time it takes to pass a given material point is . If we take the velocity of light as unity, the time a light-wave takes to pass a given point is equal to the spatial distance between the beginning and end of the wave; therefore this distance multiplied by the energy is . This form might seem preferable for our purposes, since it does not involve reference to an extraneous material point. At least, it does not obviously involve such reference; but perhaps the reference is concealed in the process of estimating spatial distance. We have seen that this process must be indirect; one part of a light-wave cannot catch up another, so that the space-like interval between them can only be estimated by means of some process taking place in matter. If it should be found that quantum phenomena are not physically fundamental, much of what has been said in this chapter will become unnecessary. It should be said, however, that relativity should prepare our minds for the oddest feature of the quantum theory, namely the existence of causal laws involving whole periods. The causal unit, on relativity principles, should be expected to occupy a small region of space-time, not only of space; it should not therefore be instantaneous, as in pre-relativity dynamics. If we combine this with the hypothesis of a discrete space-time, we can imagine a theoretical physics which would make the existence of the quantum no longer seem surprising. I have to confess, reluctantly, that the theory developed in the present chapter, inadequate as it is, is the best that I know how to suggest on the topic of quanta. Perhaps the progress of physics will make a better philosophy of the subject possible before long. Meanwhile I commend the matter to the consideration of the reader. [Pg 367] CHAPTER XXXV CAUSALITY AND INTERVAL THE conception of "interval," upon which the mathematical theory of relativity depends, is very hard to translate, even approximately, into non-technical terms. Yet it is difficult to resist the conviction that it has some connection with causality. Perhaps a discontinuous theory of interval might diminish the obstacles to such an interpretation. Let us try to discover whether this is the case. The view which naturally suggests itself as a point of departure is something like this: Given two groups of co-punctual events, it may happen that at least one member of one group has a causal relation to at least one member of the other group; in that case, the interval between the two groups is time-like. If causality is a matter of discontinuous transitions, one might expect that the magnitude of the interval would be measured by the number of intermediate transitions. Again, it may happen that no member of one group has a causal relation to any member of the other, but that both contain members having causal relations to a member of a third group. In that case, the interval will be space-like, and again one might suppose that the number of intermediate links would determine the magnitude of the interval. This represents what might be hoped, but as it stands it is unduly simple, and open to obvious objections. Let us see, therefore, whether it is possible to answer the objections, or to introduce such modifications as will obviate them. First, let us be clear as to what we mean by a causal relation. There is a causal relation whenever two events, or two groups of events of which one at least is co-punctual, are related[Pg 368] by a law which allows something to be inferred about the one from the other. Formerly, one would have supposed that everything about the later event could be inferred from a sufficient number of antecedents; but in view of the explosive and apparently spontaneous character of radio-activity and quantum changes, we must be content with a more modest definition so far as this point is concerned. In another respect, however, our definition is less modest than it would formerly have been. In classical dynamics, causal laws connect accelerations with configurations, so that from the present state of a small region we cannot accurately infer anything as to what will be happening there