The analysis of matter

THE periodic character of many physical occurrences has been obvious ever since men observed their own respiration and the alternation of night and day, but it has acquired a quite new importance with the discovery of the quantum. The quantum characterizes a whole period of a rapid periodic process, not any one moment of the period; it thus requires us to consider the period as a whole, and in some sense reverses what has hitherto been the trend of physical laws, namely to proceed away from integrals towards differentials. It will be remembered that the quantum principle, as enunciated by Wilson and Sommerfeld, states: Given a periodic or quasi-periodic process, the kinetic energy of which has been expressed by means of "separated" co-ordinates, if is any one of these co-ordinates and is the kinetic energy, then where the integration is to extend over one complete period of , and is a small integer which is the quantum number associated with . This law is essentially concerned with a whole period, and thus makes periodicity fundamental in physics in quite a new way.

Before going further, it will be well to consider how far periodicity retains this importance in the newer quantum mechanics inaugurated by Heisenberg. For this purpose, we may concentrate attention upon the one fundamental equation involving h in the new system. This equation takes the form:[70] where and are matrices, being a Hamiltonian co-ordinate in the new sense, and the corresponding "impulse," also in the new sense; while is the unit matrix. This equation is asserted to hold for all motions, not only for such as are periodic. But in the case of motions which are not periodic, it gives a result which approximates to that of classical mechanics. Thus it remains the case that the new mechanics is only necessitated by periodic motions, although it is technically possible to find a quantum principle which is also applicable to non-periodic motions. Hence the importance of periodicity remains intact from an empirical point of view, though somewhat diminished from the point of view of a statement of fundamental laws. In any case, it remains sufficiently important to demand a separate discussion.

Traditionally, periodicity in physics was a question of motion: a body described the same path in space over and over again. With the coming of relativity, it has become necessary to modify this account somewhat. In space-time, every point has a date, and cannot be occupied twice; neither the earth nor an electron can describe again the orbit it described on a former occasion. And periodicity will be relative to a given system of co-ordinates: if, in one system, a co-ordinate runs through a given range of values repeatedly, and always in equal times, it may happen that, in another system, even if there is an oscillating co-ordinate, its periods are not all equal. A change of axes may even take away all trace of periodic character from a process. Since, however, the quantum principle compels us to treat periodicity as physically important, it would seem that we must regard it as a character belonging to a process when referred to axes which move with it, since this would overcome the difficulties connected with relativity. If, in certain cases, this method is not open to us, some other must be found which equally avoids these difficulties. But where processes connected with matter (as opposed to electromagnetic processes) are concerned we shall, I think, find no other possibility except to take axes which move with the matter concerned. But this makes it impossible to treat periodicity as fundamentally a character exhibited in a motion, since we have reduced to rest the body in which the periodic process is taking place. The suggestion I have to make is that, fundamentally, periodicity is constituted by the recurrence of qualities.

In the present chapter, I wish to consider what can be meant by the "quality" of an event; I wish also to investigate the connection of quality with causality and motion and periodicity.

Physics traditionally ignores quality, and reduces the physical world to matter in motion. This view is no longer adequate. Energy turns out to be more important than matter, and light possesses many properties—e.g. gravitation—which were formerly regarded as characteristic of matter. The substitution of space-time for space and time has made it natural to regard events, rather than persistent substances, as the raw material of physics. Quantum phenomena have thrown doubt on continuity of motion. For these and other reasons, the old simplicities have disappeared.

When we start from perception instead of from mathematical physics, we find that the events with which we are best acquainted have "qualities," by means of which they can be arranged in classes and series. All colours have something in common which is not possessed by sounds. Two colours may be so similar as to be almost or quite indistinguishable, but they may also be very dissimilar. As Gestalt-psychologie has emphasized, shapes are perceived qualitatively, not analytically as a system of interrelated parts. But this whole conception of quality, which plays such a large part in our perceptual life, has been wholly absent from traditional physics. Colours, sounds, temperatures, etc., have all been regarded as caused by various kinds of motions. There was no objection to this so far as it succeeded, but, if and where it proves insufficient, there can also be no objection to re-introducing qualitative differences into the physical world.

There is, however, one essential limitation. We may find reasons for supposing qualitative differences, in order to be able to build up the kind of structure which we have inferred; but we cannot have any means of knowing what are the qualities which differ. This point was discussed in Part II., and need not now detain us.

The apparatus so far assumed, apart from qualities, has been: co-punctuality, cause-and-effect, and the quantum laws. I say "cause-and-effect" because it is necessary to be able to distinguish the earlier from the later event in a transaction, and this is a smaller assumption than that of a general time-order among events in one causal series. The above apparatus sufficed except for one purpose: that of defining "repetition." The possibility of repetition is at the bottom of the common-sense distinction between space and time; the substitution of space-time should, one might suppose, make repetition impossible, and yet the whole of what is distinctive in quantum physics, and the theories of light and sound, not to mention other matters, depend upon periodicity, which involves repetition. So long as we had billiard-balls moving in an unchanging space, we could be content with repetition of configuration. But now spatial distance, which is essential to configuration, has to be analyzed into an elaborate indirect relation depending upon the existence of common causal ancestors or descendants. We must, therefore, be able to distinguish among events by means additional to their space-time relations.

There is, however, a considerable difficulty in finding laws governing what we are calling "qualities." In a world of continuous processes, one would say that qualities must change gradually. But in a quantum process they apparently change suddenly. Perhaps, however, this suddenness does not exist in a steady rhythmic process; or perhaps, even if it does, it may involve small changes producing a serial character in the successive qualities. Take, for example, the revolution of an electron about a nucleus. In the newer quantum theory this does not really occur, but we may consider how it could be interpreted if it were necessary to assume it. Let us make a fantastic hypothesis, purely for illustrative purposes: let us suppose that the electron and the nucleus can see each other, and that neither rotates on its own axis. Then they will get pictures of each other which change during each revolution, and repeat the cycle of changes each time. Now let us turn this hypothesis round, and begin by assuming the recurrent series of pictures. From this we can infer the revolution of the electron, provided we are free to construct space as we like, subject to certain formal laws. Now in fact we have this freedom: the "space" in which the electron revolves need only have certain abstract mathematical properties, and, so long as it has them, it may be constructed out of any material available. So long as the electron continues in one orbit, we may conceive, at any rate as a schematic simplification, that there is a persistent event which may be taken as representative of it, and in like manner that there is a persistent event P representative of the proton. Now let us suppose that, compresent with E but not with each other, there are successive events , , , ... which may be regarded as "aspects" of the proton, and are related to each other more or less in the way in which the appearances of the proton from different places would be related if the electron could see. Similarly let us assume a series of events , , , ... compresent with but not with each other, analogous to what would be appearances of the electron to the proton if the proton could see. And let us further suppose that, after a certain set of such events, an exactly similar set recurs, or a very approximately similar set. This supposition provides us with the material required for a periodic relative motion. We shall say, therefore, not that perspectives differ because spatial relations change, but that change in spatial relations consists of systematic alteration in perspectives. Such a view is feasible, but it makes similarity and difference of quality essential. It ceases to be fantastic if we drop the analogy with vision except as regards purely formal characteristics.

Let us now set forth the analysis of a periodic process suggested by the above, bringing it into relation with the construction of points in Chapter XXVIII. Let us assume, to begin with, that the process is discrete; this hypothesis can be dropped later, but simplifies the initial statement. Suppose, for the sake of illustration, that there are ten qualities, , , ... , and that there exist events which are subject to the following conditions:

(1) , a, , ... have the quality : , , ... have the quality etc.

(2) Each of the 's is compresent with its immediate neighbour to left and right, but with none of the other 's;

(3) If , any point of space-time of which but not is a member has a time-like interval from any point of which but not is a member.

In that case, the series of 's constitutes a periodic process, having ten 's in each period. The last digit in the suffix of an indicates the quality of the —i.e. if the last digit is , the quality is —while the remaining digits indicate the number of the period.

If all the 's are events in the history of one piece of matter, that piece of matter is undergoing the periodic process. If there is a correlative series of 's in another piece of matter, the two periodic processes together make up one relative motion of a periodic character, such as the revolution of an electron about a proton.

Generalizing the above, while still assuming that the process is discrete, suppose we have qualities , ... , and a set of events where, as before, the last suffix indicates the quality, i.e. has the quality (). Suppose, also, that each is compresent with of its predecessors and of its successors, where ; but that no is compresent with any except these. The remaining assumptions are to be as before. Then again we obtain a rhythm which may be regarded as an analysis of periodic processes in physics.

If we suppose that the 's are not compresent with any events except the other specified 's, then the group of 's with which a given is compresent constitutes a point, which may be taken as the middle point in the duration of the in question. We can take this point as representative of the in question, since their relation is one-one. Thus the in question is associated with a point, in spite of the fact that it lasts for a finite time, i.e. is compresent with events not compresent with each other.

It is to be observed that, according to the theory of space-time in Chapters XXVIII. and XXIX., it is quite possible for some parts of space-time to be continuous and others discrete. I am supposing, at the moment, that we are considering a periodic process in a discrete part of space-time; this does not involve the hypothesis that all space-time is discrete.

If the 's in one periodic process, as we supposed a moment ago, are not compresent with any events except certain neighbouring 's (which must be fewer than the whole of one period), then the number of points in a period is the same as the number of 's, and either affords a measure of the duration of the period, measured by its proper time. It is obvious that, in a discrete part of space-time, the natural measure of distance will be number of intermediate points. We see also how the proper time of one process can differ from that of another. Let us suppose that our 's form an "isolated" process (i.e. are not compresent with anything except other 's), except at the beginning and end; the first and last 's are to be compresent with the first and last terms of another periodic process composed of 's, which also is to be isolated except at its ends. Then the proper time of the -process is measured by the number of 's between the two ends, which need not have any relation to the number of 's. This illustrates, what of course follows from relativity, that periodicity must be measured by standards intrinsic to the process concerned, not by standards appropriate to other periodic processes. Such remarks would hardly be necessary but for the fact that relativity and quantum theory at present stand apart from each other, and have not yet been brought into one whole by the physicists.

The above can be stated in the language of mathematical logic, thereby making the character of the assumptions clearer and the generalization to continuous processes easier. Let be the series of qualities, the series of events in the rhythmic process. Let us imagine the events arranged in rows and columns, so that each row consists of one period and each column consists of all the events having a given quality. We assume a one-many relation , whose domain is the field of and whose converse domain is the field of . When has the relation to , we say " has the quality ." If is any term in the field of , let be the term which has the relation to ; then the next term below a in the same column (i.e. the corresponding in the next period) is the first term in the series which is after and to which has the relation . The "row of " consists of all 's earlier than and not earlier than . The "column of " consists of all 's to which has the relation . We assume that with its converse domain limited to one row is one-one, so that each row (i.e. each period) is a series which is similar (in the technical sense) to the series .

There is no difficulty in adapting the above analysis of periodicity to continuous processes. Instead of an enumerated set of qualities , , ..., we shall have to take some continuous series of qualities, such as the colours of the rainbow, or the notes produced on a violin by running one's finger up and down the string. The number of events compresent with a given event must now be infinite, but must still be less than the whole of one period (ignoring events outside the process concerned). The number of points in one period, or in any finite portion of it, is now infinite, and cannot therefore be used as a measure of distance. Thus in regard to metrical properties there are important differences between continuous and discrete processes. However, I shall not enlarge upon these, as I propose to consider the analysis of "interval" in a later chapter.

Hitherto I have been considering processes which may be regarded as taking place in matter, or which, at any rate, do not move with the velocity of light. But light, also, is commonly regarded as consisting of a periodic process. Accepting the wave-theory of light, let us proceed to analyze its periodic character. We shall find that it differs in important respects from that of periodic processes in matter.

The periodic character of a light-wave cannot exist from its own point of view, but only from that of the matter which it encounters or from which it radiates. We may suppose that when light radiates from an atom at the time of a quantum change, there is, from the point of view of the atom, a temporal series of what we may call "luminous events," and that this series is periodic in the sense which we have been considering. One period of such luminous events constitutes the emission of one light-wave. If we suppose that the light is absorbed by another atom, we may suppose that each of the luminous events is compresent with certain events in the absorbing atom, as well as with certain events in the emitting atom. As measured by the proper times of the atoms, the time-order of the luminous events is the same for the two atoms. But from the point of view of the luminous events themselves, there is no periodicity. So long as the light does not encounter matter, it consists of separated events which at most "touch" one other event at each boundary; the traveller who accompanies one of the events can have no cognizance of any of the other events, since they cannot catch each other up. If we could imagine a homunculus floating on the crest of a light-wave, he would have no means of discovering that anything periodic was occurring, since he could not "see" the other parts of the wave. The different parts of a light-wave cannot, in a word, interact causally in any way, because no causal action can travel faster than light.

We cannot even properly speak of a periodicity in the light-wave for an observer who watches it pass. We can only see light by stopping it. This applies to such phenomena as interference, which is only made visible by allowing light to meet matter. It is true that interference gives us a ground for inferring structure: two processes can neutralize each other, but two "things" cannot. If owes a pound, and owes a like sum, the result is zero; but if has a pound in his hand to give to , and has a pound in his hand to give to , there are two pounds. Wherever the sum of two occurrences can be null, both occurrences must have a relational character. Thus we are justified, by such facts as interference patterns, in supposing that, when light falls on a body, the body experiences a series of events whose effects upon it are of opposite kinds, as if some pushed it one way and some another. But all this is from the point of view of the body, not of the light. Thus the frequency of light is a characteristic which exists for a body which emits light, and for a body which absorbs it (e.g. the body of a scientific observer), but not for the light itself while it is in vacuo.

When light is emitted and absorbed, we may therefore suppose that what happens is according to the following scheme. We have a temporal series of events in the emitting body, and, compresent with each of these, a luminous event. These luminous events, arranged in the time-order of the compresent events in the emitting body, form a periodic process in the previous sense. Each of the luminous events is also compresent with some event in the absorbing body. The time-order of the events in the absorbing body is the same as that of the events in the emitting body; i.e. if , are events in the emitting body, compresent respectively with , , two luminous events; and if , , are respectively compresent with , , two events in the absorbing body, then if is earlier than , is earlier than . What happens to light-waves which are emitted but not re-absorbed we cannot tell, since, by the nature of the case, there can never conceivably be any evidence on the point.

According to the above, the frequency of a light-wave is a characteristic which it has in relation to matter, not in relation to itself. In this it differs from, e.g., the periodicity in the revolution of an electron, which may be supposed to exist for the electron itself.

The chief point of the above hypothesis is the suggestion that single "luminous events" extend from the emitting to the absorbing body. I do not advance it as anything more than a possible hypothesis. One of its main purposes is to account for the fact that the interval between two parts of a light-ray is zero; but this part of the argument belongs to a later chapter.