Chapter III (4)

[Footnote 27: M. La Rosa, Scientia, July-August, 1924.]

Einstein's restricted relativity has made a great contribution in so grouping and coordinating the phenomena that they can all be embraced in a simple mathematical formula, but he does not seem to have presented them in such a light that they are simple or easy to grasp physically. The explanatory aspect is completely absent from Einstein's work.

In view of all our present difficulties it would seem that we ought at least to try to start over again from the beginning and devise concepts for the treatment of all optical phenomena which come closer to physical reality. No one realizes more vividly than I that this is a most difficult thing to do. If we are ever successful in carrying through such a modified treatment, it is evident that not only will the structure of most of our physics be altered, but in particular the formal approach to those phenomena now treated by relativity theory must be changed, and therefore the appearance of the entire theory altered. I believe that it is a very serious question whether we shall not ultimately see such a change, and whether Einstein's whole formal structure is not a more or less temporary affair.

Although it is exceedingly difficult to forsee what the treatment of the future will be like, it is easy to surmise certain of its features. In essence the elementary process of all radiation perceived as radiation is twofold. There is some process at the source and some accompanying process at the sink, and nothing else, as far as we have any physical evidence; furthermore, the elementary act is unsymmetrical, in that the source and the sink are physically differentiated from each other. This is the most complete expression of the physical facts; there is nowhere any physical evidence for the inclusion of a third element (the ether). Therefore all the phenomena apprehended by an observer (and this embraces all physical phenomena) can be determined only by the source and the sink and the relation to each other of source and sink, for there is nothing else that has physical meaning in terms of operations. This formula covers not only the possibility of such first order phenomena as aberration and the Döppler effect, but also shows that such second order effects as that looked for by Michelson and Morley must be non-existent. It will thus be seen that some of the consequences of relativity theory are implicitly contained in certain very broad points of view. One interesting question that must be answered before we can get very far with a new treatment is whether the elementary optical process is of _necessity_ twofold, or whether we may have emission without absorption, that is, radiation into empty space. Lewis seems to imply in recent papers that this is not possible.[28] The astronomers have already pointed out difficulties in explaining phenomena like the temperature equilibrium of the planets if we suppose this is the case.

[Footnote 28: For example, in the book: G. N. Lewis, The Anatomy of Science, Yale University Press, 1926, p. 129.]

OTHER RELATIVITY CONCEPTS

We now turn to some of the other concepts of relativity. One of the most important of these is the "event"; in fact this concept is made fundamental by Whitehead.[29] We have already discussed the concept of "event" under the "identity" concept with which it is closely involved. The event is usually thought of by Einstein as merely an aggregate of four coordinates, three of space and one of time. The principle of general relativity, namely, that the laws of nature shall be of invariant form, when formulated mathematically, involves the assumption that nature may be analyzed into events, and is expressed by the requirement that the mathematical relations between the coordinates of a chain of events shall be invariant. The same idea is also expressed by Einstein in another form, namely, that nature may be completely characterized in terms of space-time coincidences. In elaborating this idea, Einstein assumes that the results of all measurement may be given in terms of such coincidences.

Now it appears to me most questionable whether the analysis of nature into events is possible or sufficient. With regard to the coincidence point of view, it seems perfectly obvious that the world of our immediate _sensation_ cannot be described in terms of coincidences; how, for example, shall we describe in terms of space-time coincidence the photometric comparison of the intensity of two sources of illumination, or the comparison of the pitch of two sounds, or the location of a sound by the binaural effect?

[Footnote 29: A. N. Whitehead, An Enquiry Concerning the Principles of Natural Knowledge, Cambridge University Press, 1919, Chap. V.]

To justify the coincidence point of view we apparently have to analyze down to the colorless elements beyond our sense perception. It does not seem unreasonable, perhaps, to expect that the universe is completely determined in terms of the positions as a function of time of all the positive and negative electrons; but to introduce such a thesis now certainly goes beyond present experimental warrant, and is contrary to the general spirit of relativity, which nowhere else involves any reference to the small scale structure of things. Even if we were willing to overlook all these objections, we would still have the fact that the difference between a positive and negative electron is not contained in any specification of the mere coordinates.

A further very important doubt in principle as to the possibility of the analysis of nature into events is afforded by the character of the concept of event itself. We have seen that the idea of event involves the existence of discontinuities, and that this can correspond only approximately to the physical fact, because discontinuities apparently lose their abruptness as we make our measurements more refined. The thesis that nature can be described in terms of discontinuities of a very small scale seems much too special to be made a fundamental part of a theory of the general pretensions of that of relativity. In fact this, as well as a consideration to be mentioned later, suggests that the argument and result of general relativity may be intrinsically restricted to large scale phenomena.

We now pass from these somewhat special questions to ask why it is that Einstein was able in the general theory of relativity to obtain new and physically correct results from general reasoning of an apparently purely mathematical character. We are convinced that purely mathematical reasoning never can yield physical results--that if anything physical comes out of mathematics it must have been put in another form. Our problem is to find where the physics got into the general theory.

There are two questions to be disentangled here: we have to consider in the first place the significance of the fact that Einstein has been able to describe relations in nature in mathematical form, and in the second place of the fact that he was able to arrive at the mathematical formulation of these physical relations by reasoning of apparently a purely mathematical character, from postulates of merely formal mathematical content (invariance of natural laws in generalized coordinates). Now the theory of relativity does not seem to differ in the first respect from any other branch of mathematical physics, such as the classical mathematical theory of electricity and magnetism, for instance, and this matter has already been touched in an earlier chapter. It is a fact that the behavior of nature can in many cases be expressed to a high degree of precision in mathematical language, and relativity is not unique in this respect. In any event, we must not allow this possibility of mathematical formulation to obscure the essential fact that all physical knowledge is by its nature only approximate, so that we may expect at any time to find, when we have carried our measurements to a higher degree of precision, that our mathematical expression of the laws was not quite exact, as seems now to be the case with Newton's law of gravitation, for example. I do not suppose that Einstein would claim that the statements of relativity differ in this respect from any of our other statements about nature, although apparently some of his followers see something more here. (From the operational viewpoint the meaning to be attached to "something more" is somewhat obscure.)

With respect to the second question, we may stop to notice that the special theory stands in quite a different position from the general theory. The special theory is much more physical throughout; its postulates are physical in character, and it is obvious that the physics got into the results through the postulates. It seems to me without question that Einstein showed the intuitive insight of a great genius in recognizing that there are mutual relations between physical phenomena which can be described in very much simplified language in terms of concepts slightly modified from those already in common use. In view of the remarks made on the nature of light, it is legitimate to wonder, however, whether the formulations of even the special theory will always stand. It seems to be true that _all_ the facts of nature, even in the absence of a gravitational field, cannot be connected by the simple formulations of the special theory; that the physical relations are simple only in a sub-group; and that if we wish to deal with _all_ optical phenomena, we have carried our simplifications too far, for the emission of a light signal is not a simple event, and light is not in nature like a thing travelling. Just the sorts of physical thing which are ignored in treating light as the special theory does are coming to be more and more important in the minds of physicists, and this is a reason for wondering whether ultimately Einstein's special theory may not be regarded merely as a very convenient way of tying together a large group of important physical phenomena, but not as being by any means a full or complete statement of natural relations.

With respect to the general theory, however, I believe the situation is quite different. The fundamental postulate that the laws of nature are of invariant form in all coördinate systems is highly mathematical, and of an entirely man-made character. Of what concern of nature's is it how man may choose to describe her phenomena, and how can we expect the limitations of our descriptive process to limit the thing described? Furthermore, Einstein's method of connecting his mathematical formulation and nature by way of coincidences of 4-events (three space, one time coordinates) seems to be very far removed from reality, since it entirely leaves out the descriptive background in terms only of which the 4-event takes on physical significance. Nevertheless, three definite conclusions about the physical universe have been taken out of the hat by the conjuror Einstein (shift of the perihelion of Mercury, displacement of apparent position of stars at the edge of the sun's disk, and the shift toward the infra-red of spectrum lines from a source in a gravitational field), and the problem for us as physicists is to discover by what process these results were obtained.

An examination of what Einstein actually did in deriving his results will show, I believe, that the situation is really different from that suggested above. In the first place, the requirement that the laws of nature be of invariant form actually places no restriction, as any one can see by setting himself the task of expressing, for example, an inverse cube law for gravitation in terms of generalized coordinates. The work of expressing such a law can be attacked in a perfectly routine way. (The essential difference between the invariability requirements of the special and general theories is to be noted; the special theory requires that the velocity of light, for instance, have the same _numerical_ value in all allowed systems: the general theory merely that all laws have the same _literal_ form, but with variable numerical coefficients.) But, as Einstein says, if any one actually attempts to carry through the work of expressing an inverse cube law in generalized coordinates, he will find the task prohibitively complicated, and will seek for some simpler formulation. What Einstein actually did, therefore, was to require that the laws of nature be _simple_ in generalized form. Now we know that the law of gravitation as formerly expressed in ordinary coordinates as an inverse square law was approximately exact, and was also simple. Any deviations from this law are small, and all experience leads us to expect that to the first order of small quantities the deviations can be taken care of mathematically in the form of small correction terms to this law. This by itself gives nothing, however, because a small correction term can be added to our equations in an infinite number of ways. If, however, we know that the equation must be of a certain type after the correction terms have been added, the possibilities are so much restricted that the form of the correction term may be determined. In arguing as to the probable type of the equation, Einstein advanced the considerations by which physics gets into the situation.

In the first place, the special theory had prepared us for the possibility of finding that our measuring instruments might be modified in a gravitational field, analogously to the shortening of a meter stick when set into motion. In fact, special theory had indicated that in an accelerated system the modifications might be too complicated to be treated by that theory. In the absence, then, of specific information we must be prepared for the most general possible alteration in space-time in a gravitational field. In describing space-time we must therefore use coördinates adapted to handling the most general possible relations, and these are the generalized coördinates of Riemann, which had been already discussed by mathematicians. Going back now to Einstein's criterion that the equations are to be simple, we have the demand that the equations be simple in generalized coördinates, and of course they must also reduce to the ordinary equations (that is, the equations of special relativity) in space where there is no gravitational field. In deciding the further question as to what the type of equation probably is, we are influenced by considerations of convenience as well as by physical considerations. Practically the only type of equation that can be handled mathematically is linear, so that we shall certainly try first whether this type of equation may not continue to hold. Now the Newtonian law of the inverse square may be expressed in terms of a linear differential equation of the second order in the old Cartesian coördinates (Poisson's equation), so that our most immediate suggestion is that the equations remain linear and of the second order in generalized coördinates. As a matter of fact, this requirement turns out to be sufficient to determine the small correction term by which the ordinary equations can be generalized; Einstein's papers must be studied to see how this works out in detail.

All this looks pretty mathematical, but as a matter of fact there is much physical content, because systems which can be described by linear equations of the second order have definite physical properties. The requirement that the equations be linear corresponds to one of the most fundamental properties of our universe--the causality concept would not be possible or would be much modified in a universe governed by non-linear equations, for the joint effect of two causes acting together would not be the sum of their effects acting separately, so that the analysis of a situation into simple elements would be impossible and the causality concept probably would not have arisen. Furthermore, an equation of the type of Poisson of the second order means that there are propagation phenomena, and equations of mechanics of the second order involve the existence of a scalar energy function. If, then, the behavior of the universe can be described by differential equations at all, these equations must be linear of the second order if the universe is to have the broadest physical characteristics of our own universe. What Einstein really did, therefore, was to demand that even when space-time is warped by the presence of a gravitational field, those physical phenomena which can be described in terms of differential equations continue to be described by linear differential equations of the second order; that is, that nature continues to be describable in terms of a causality concept, with propagation phenomena, and a simple energy function. The consequences of a guess like this about the properties of nature appeal to our physical intuition as being worth following out, and of course we know the experimental justification.

Several general comments may now be made on the structure reached in this way. In the first place, the whole structure is only descriptive in character; we find certain correlations in nature which we describe with considerable completeness in mathematical equations, without introducing any new element of explanation or of mechanism. We have seen that as we increase our range from the realm of ordinary phenomena to phenomena of different character we arrive at a stage where for a time the process of explanation apparently halts, and we have to be satisfied with a statement of mere correlation between elements; later, however, these elements may be accepted as the ultimates in a broadened scheme of explanation, and the explanatory process resumed. Are we at such a stage now with the general relativity theory, and may later a new scheme of explanation be established based on the correlations of Einstein? This is of course a matter of individual judgment; I personally question whether the elements of Einstein's formulation, such as curvature of space-time, are closely enough connected with immediate physical experience ever to be accepted as an ultimate in a scheme of explanation, and I very much feel the need for a formulation in more intimate physical terms.

In the second place, we must repeat the comment already made in discussing time, namely, that there is still a very wide gap between the theory and its physical application, in that we have no way of identifying our physical clocks and our physical measures of time with the thing called time in the formulas. This gap must be filled by a specification of the physical structure of a clock.

It has always been very puzzling to understand why Einstein has so strenuously insisted that the shift toward the infra-red is an integral part of the general theory, and that if the shift is not found, the theory must fall. In other words, Einstein insists that the assumption that an atom is a clock is an integral part of his theory. I believe that this attitude may be due to a realization by Einstein of that very flaw in the logical structure which we are now emphasizing. In the absence of any method of specifying the details of construction of at least one clock, relativity becomes a purely academic affair, unless there exist in nature concrete things which may serve as clocks. Einstein _must_ either be able to tell how to construct a clock, or else be able to point to a specific example of a clock. He chose the atom as the specific thing. Doubtless the reason was the apparent simplicity of the vibrating mechanism of an atom, as shown by the precise equality of the frequencies emitted by all atoms of the same element. If the atom is not a clock, where in nature can one be found? But in the last few years we have come to appreciate the exceedingly complicated quantum structure of an atom, and Einstein's thesis loses much of its instinctive appeal.

Since Einstein created the theory of relativity, it is perhaps ungracious to question his right to stipulate that the assumption that the atom is a clock is an integral part of the theory. This, however, degenerates to a mere matter of language, and does not touch the arbitrary nature of the procedure. It does not prevent us from having a second brand of relativity theory, that of X instead of Einstein, exactly like that of Einstein except that perhaps now the "clock" is constructed in terms of the life period of a radioactively disintegrating element. The only way to eliminate the arbitrariness seems to be to postulate that _all_ natural processes, which run naturally of themselves independently of what we may do, may equally well serve as clocks and give the same results. But in answering the question of the operational meaning of "independently of what we may do" we shall effectively have to answer the question of what is a clock. This point of view may possibly, however, get us a little nearer to our goal of finding how to specify the structure of a clock.

Finally, the general theory is not completely general, but applies only to a certain range of phenomena, just as we saw that the special theory does not embrace all optical phenomena. The general theory applies only to those phenomena which can be described in terms of differential equations, that is, par excellence, to large scale phenomena. If quantum phenomena cannot be described by differential equations,[30] as apparently now they cannot, general relativity cannot by its very nature be applicable. General relativity does not give us a comprehensive formulation of the behavior of all nature, and as far as we can see, we are still as far as ever from such a general formulation.

[Footnote 30: This statement now takes on a very questionable aspect in view of the new quantum wave mechanics (March, 1927).]

ROTATIONAL MOTION AND RELATIVITY

Physically there is a great difference between the behavior of systems in uniform relative rectilinear motion and those in uniform relative rotation. The special theory of relativity states that there is a triply infinite number of systems with all possible uniform rectilinear velocities with respect to each other, in all of which physical phenomena have exactly the same mutual relations, that is, natural laws are the same. Now the mere formulation of the principle suggests the sense in which "system" is here used. It is obvious that "system" refers only to a part of the universe; we are not making a hopelessly academic statement about what would happen if we had an infinite number of universes to experiment with, but are talking about operations that may be approximately realized in our own universe. The "system" of the formulation we may think of as a completely equipped laboratory, out in empty space, so far from the heavenly bodies that they can have no effect. The different systems of the formulation are different laboratories, all built to exactly the same architectural blue prints. The phenomenon to which the postulates of relativity apply are phenomena which pertain entirely only to one or another of these laboratories. The meaning of this restriction is not completely definite and has, in any special case, to be judged partly by the context. Obviously, to see from the window of a laboratory another laboratory passing with a certain velocity cannot be counted as one of the allowed phenomena. Still less is it one of the allowed phenomena to observe that the center of gravity of the entire stellar universe has a certain velocity of translation with respect to the laboratory. The special principle of relativity contains by implication therefore the statement that certain very large and important classes of physical phenomena may be isolated and treated as taking place unaffected by the rest of the universe. Granted now the possibility of isolation, we have a second statement, which is usually treated as if it were the entire statement of the restricted principle, namely, that there is a triply infinite set of systems in which these phenomena run in the same way independent of the relative motion of the systems with respect to each other. When once the significance of the observation is grasped that absolute motion has no meaning in terms of operations, we see that this last statement takes immediately a most simple and satisfying aspect, in fact, so simple and inevitable that we are inclined to see in this the complete essence of the situation and regard the meaninglessness of absolute motion as affording peremptory proof of the restricted principle.

With this bias we now turn to examine the facts of rotary motion, and are disconcerted to find them quite different. No meaning in terms of measuring operations can be given to absolute rotary motion any more than to absolute translation, but nevertheless phenomena are obviously entirely different in different systems in relative rotary motion (phenomena of rupture, for example), so that apparently there are physical phenomena by which the concept of absolute rotary motion might be given a certain physical significance. Given two worlds like our own in empty space, but surrounded by impenetrable clouds, and each provided with a Foucault pendulum, then we believe that it is physically possible that we may find on one of these worlds the plane of rotation of the pendulum gradually changing in direction, while on the other it remains stationary. This difference we regard as possible without other accompanying physical phenomena which are causally related to the rotation of the pendulum (of course we have to make the two worlds of infinitely rigid material and eliminate other phenomena which we regard as purely incidental), so that we apparently have here a contradiction of our cardinal physical principle of essential connectivity. We are certainly not inclined to give up our principle, and we believe that as a physical fact, if the clouds could be evaporated, an observer in one world would find that he was rotating with respect to the system of the fixed stars, whereas the corresponding observer on the other world would find that he was stationary. Our principle of essential connectivity is therefore maintained, in that the rotation of the plane of the pendulum is connected with a rotation with respect to the rest of the universe of the entire world in which the pendulum is mounted. As far as I am aware, no other way of maintaining our principle has ever been suggested. But this demands that we give up our physical hypothesis of the possibility of isolating a system. There is here no question of limiting behavior; we believe that no matter how far our rotating world gets from the rest of the universe the Foucault pendulum would always behave in the same way; the system can never be isolated, but such local phenomena as the invariance of the plane of the pendulum are always essentially determined by the rest of the universe.

If now our system cannot be isolated, we must return to the phenomena of translational motion. In principle the act of isolation cannot be performed, the rest of the universe cannot be disregarded, and we should expect that different states of translational motion as well as different states of rotational motion with respect to the rest of the universe would have an effect on phenomena. We set ourselves the problem of understanding this apparent enormous difference between phenomena of translation and rotation. We remark that what apparently is a difference in principle may, in virtue of the approximate character of all measurement, be only a difference in magnitude, and that translational effects may exist too small to detect. A physical basis for such a difference may be found in the enormously different numerical values of translational and rotational velocities with respect to the rest of the universe attainable in practice. In describing phenomena of cosmic magnitude, we may plausibly measure the phenomena in units commensurable with the scale of the phenomena. Thus in measuring linear distances, we may perhaps choose as the unit of length the diameter of the stellar universe, and in measuring rotation, a complete reversal of direction with respect to the entire universe. This last means a change of angular orientation of 2 π, the first means a length of the order of 10^6 light years. Measured in such cosmic units the angular velocities attainable in practice are incomparably greater than linear velocities. We now see that it is possible that the real state of affairs is as follows: namely, phenomena in any system are affected by motion with respect to the entire universe, whether that motion is of translation or of rotation, and the magnitude of the effect is connected with the velocity of the motion by a factor which is of the general order of unity when velocity is measured in cosmic units. This last is merely an application of the argument so often made in physics as to the order of magnitude of unknown numerical factors, and will be found expanded on page 88 of my book on _Dimensional Analysis_. The linear velocities attainable in practice are now so exceedingly low that their effect has not yet been detected experimentally, but angular velocities are high, and the effect is easily demonstrable. In this light the special principle of relativity is no different in character from any other physical law; it is only approximate, and some day our measurements may become refined enough to detect its limitations.

We have made a hypothesis here, which we may call the hypothesis of the immanence of the entire universe, namely, that isolation is impossible, or that the rest of the universe, no matter how distant, always has a local effect on at least some phenomena. This is essentially the hypothesis of Mach,[31] and leads to a situation which can, I think, be contemplated with logical equanimity, although it has always seemed to many physicists most highly antiphysical in character.

[Footnote 31: E. Mach, The Science of Mechanics, translated by McCormack, The Open Court Publishing Co., Chicago, 1893. See especially p. 235.]

It must certainly be admitted that most physical experience justifies us in thinking that effects may be made as small as we please by getting far enough away from the cause of the effect. But if we accept the considerations of the preceding pages, we must be prepared to admit that as phenomena change in range their character may change, and that in these new realms we must, at first at least, be satisfied with a mere statement of correlations. Certainly we have very strong physical evidence of a formal correlation between the Foucault pendulum and the rest of the universe. But a correlation of this sort may be without significance because of its very breadth; we never can prove the significance of the correlation by performing an experiment with the rest of the universe absent. Have we really done anything more than merely get things into such a formal situation that they cannot be assailed, a possibility which the mere laws of our thinking seem always to leave open, as has been suggested, or is there any physical content to what we have done? We have seen that if our correlation is also suggested by other phenomena, then we may accept it as having physical content. Now there is just a glimmer of a suggestion that our hypothesis of the immanence of the universe may be needed in other ways. The gravitational constant and the velocity of light are always treated as arbitrary magnitudes thrust on the universe from outside with no connection with other phenomena. Nevertheless, I suppose that no one regards this situation as ultimately satisfactory and does not entertain the hope that some day we may be able to give some sort of account of the numerical magnitude of these constants. We have not hitherto succeeded in finding any connection between these constants and small scale phenomena such as the charge on the electron, its mass, etc., so that there is some plausibility in expecting that a connection may be sometime found with cosmic things; indeed general relativity theory already prepares us for exactly this possibility. Now the velocity of light and the gravitational constant control small scale experiments, for of course these two constants can be measured by local experiments, so that if the cosmic connection is found, we should have a control of local behavior by cosmic things, and therefore another example of the immanence of the entire universe. There is no need for me to waste time in apologizing for the highly speculative character of all this. It is worth while to emphasize, however, that our general considerations on the meaning of "explanation" have prepared us to admit as reasonable just the sort of explanation contained in the hypothesis of the immanence of the universe, and therefore to reserve a place in our physical thinking for possibilities of this sort, in spite of the fact that such considerations are not usually entertained, and may seem to many opposed to the spirit of physics.

QUANTUM CONCEPTS[32]

The history of quantum theory up to the present is a repetition in many respects of that of the early theories of electricity, in that all our thinking has been in mechanical terms. As far as we now know, quantum phenomena are always associated with atoms. We make for the atom a mental model with all the properties of the mechanisms of the ordinary scale of magnitude and with a few impressed properties in addition which represent the new quantum relations. As we now think of it, the atom has a massive core about which electrons revolve under an inverse square law, the connection between the mass of the electron, its acceleration, and the force acting on it being that usual in Newtonian mechanics.

[Footnote 32: This section was written early in 1926 without access to recent literature. Our attitude toward quantum phenomena has been so much changed since then by the "new" quantum mechanics, that a number of the following statements are superseded as a statement of present opinion. However it has seemed worth while to let the section stand as written, because many of the developments actually taken in the new mechanics follow the lines that it is here urged they ought to take, and in so far afford interesting confirmation of the point of view of this essay.]

The space in which the electron circulates is thought of as Euclidean, and the motion is described in time, which may be measured with clocks in the usual way. The general equations of electrodynamics do not apply; there are no propagation effects inside the atom, the motion of the electrons does not produce a magnetic field, and there is no radiation when the electron is in one of its possible stable states, in spite of the acceleration. We may, if we please, in working out the character of the motion, entirely neglect the electrical origin of the inverse square law, and treat this merely as an impressed force without further implications. Superposed on the ordinary spatial, temporal, and mechanical characteristics of the model are additional quantum properties, one which determines the particular orbit in which the electron moves [∫ pdq = nh], and another which determines the frequency of the radiation emitted when the electron passes from one allowed orbit to another. No mechanism is suggested to account for these quantum conditions, although the conditions are formulated in mechanical terms.

We now have to ask what is the meaning in terms of operations of our usual concepts of space-time and mechanics when applied to phenomena of this order. It is of course evident, as has already been emphasized, that the concepts have entirely changed in character, because we do not measure an electron orbit, for example, by stepping off the diameter with meter sticks, or by measuring the time required for light to travel across the diameter. The particular feature of immediate interest in this changed situation is the change in number of our concepts on the atomic level. I shall not attempt to find by an exact analysis the number of independent concepts at this level; probably such an analysis is not possible. We may, however, make an approximate suggestion. Apparently the most important concept in describing relations inside a quantum system corresponds to that of energy on the ordinary scale. Changes of energy determine the frequency of emitted radiation, as well as the relations during collisions of atoms and electrons; these collisional relations make direct connection with experiment through the voltages applied to electrons in collision experiments. The analogue of the momentum concept also seems to have independent significance, as shown by the Compton effect. The frequency of emitted radiation is also something with independent experimental significance. I believe that these three things are all that have direct significance for quantum experiments made up to the present time. In any event, it is perfectly evident that on the quantum level the concepts which at present have operational significance are considerably fewer than on the level of ordinary experience.

Apart from the question of convenience, there may be justification in continuing to use our old mechanical forms of thought if new experimental relations are thereby suggested. That a very large number of such as yet undiscovered relations may be suggested in some such way is at once evident. Thus we have no present knowledge of any phenomenon associated with what the electron does when passing from one energy level to another. How long does it take to make the passage? What is its path during passage? Is it subject to the ordinary laws of electrodynamics during passage? When and where is the radiation emitted that corresponds to passage? When the electron leaves one stable orbit is the orbit on which it will eventually land already determined? Does the radiation train emitted during a change from one energy level to another have a definite length in space, or may it have a variable length and correspondingly something that corresponds to variable amplitude? What happens to the radiation when the electron passages are interfered with before the emission of a quantum has been completed? What is the mechanism by which the quantum conditions are imposed? Is it not possible that part of the clew to the riddle of the manner of transition from purely quantum behavior to the behavior of classical mechanics may be found in the behavior of the electron during passage from one energy level to another? Certainly we have a tendency to the classical behavior under those conditions, such as at high temperature or in strongly condensed systems, in which the time occupied in passage might be expected to become a more important part of the total time.

Corresponding to these questions there should be many as yet undiscovered phenomena, and the mechanical point of view therefore has its value in suggesting experiments to detect such effects. It is of course too early to see what the final result will be here; we cannot tell whether eventually enough new experimental kinds of behavior will be found to restore the number of independent concepts to that of the level of ordinary experience or not, or whether indeed it will turn out that a greater number of concepts is required. It is contrary to our instincts to expect a greater number, and a smaller number now seems to us not unnatural, but the considerations of this essay should prepare us for either possibility.

It is often said that quantum phenomena are inconsistent with ordinary mechanics, and proofs of this assertion are often offered. I believe that no such proof, in the spirit in which the attempt is usually made, can be correct, for it seems to me that the remark of Poincaré applies, namely, that any sort of behavior can be imitated by a mechanical system, provided it is only complicated enough. A peremptory proof of this can be given to any one who is not a believer in vitalism. If a sentient being can be regarded as a mechanical system, we merely have to station inside each atom a Maxwell demon, with instructions to make the atom react according to quantum rules. Opposed to the spirit of this sort of reduction of quantum phenomena to mechanical terms, we have to remember that it makes sense to talk about the character of our conceptual structure only when the number of concepts is reduced to the number that have independent operational significance, that is, to the minimum number.

In the meantime let us examine what may be the significance in the light of present experiment of statements like those ascribed to Bohr that our usual concepts of space and time may be inapplicable in dealing with quantum phenomena. This idea is often given the more explicit form that space and time may be essentially discontinuous at the quantum level. From the operational point of view, it is most difficult to see exactly what this more explicit statement means, at least in terms of those operations by which length and time were originally defined. Thus if space were discontinuous, it might mean that a point exists which may be reached by laying off a meter stick fourteen times, for example, and another point by laying off sixteen times, but that no point can be found with fifteen applications. Such a state of affairs seems to be inconsistent with our definition of the counting operation and to have no concern with any properties of space; for what shall we mean by laying off a meter stick sixteen times if it cannot be laid off fifteen times? It is conceivable that space might end, in the sense that beyond a certain limit there might be some irremovable physical hindrance to the continued laying off of distances with a meter stick (although I think that we should be inclined to describe such a state of affairs in terms of matter enclosing empty space rather than as the end of space), but to say that space may be discontinuous seems to be meaningless. In the same way, I believe it meaningless to speak of discontinuous time. We may have phenomena discontinuous in space and time, but not discontinuous space or time.

It seems then that we must give up the idea that in the quantum domain the usual concepts of space and time may fail, in the specific sense that they may become discontinuous. What may we understand by the failure of these concepts in a more general sense? No one of course would expect that even eventually the concepts will have the same operational significance for the inside of an atom that they have on the ordinary scale; it must be a modified sort of concept with which we are concerned, such as we have already seen is given by the field equations of electrodynamics. If now the number of operationally independent concepts on the quantum level turns out to be the same as on the level of ordinary experience, and if there is also the possibility of continuous transition from the operations of the quantum domain to those of ordinary experience, then it seems to me that we should say that our usual concepts of space and time still apply in the quantum domain. But if the number of operationally independent concepts is either greater or less than on the ordinary level, then I believe we must say that the ordinary concepts of space and time cannot apply. One might still look for the possibility of separating out from the complex of concepts on the quantum level a group which might change continuously to those of space and time on the ordinary level, but I think that such a possibility is very remote when one considers that the total number of concepts changes, and that in the zone where the number changes the definitions are not unique by which one extrapolates a concept from one domain to another.

If Bohr's idea is true that space and time cannot be used in describing ultimate quantum phenomena, one of the most immediate implications in terms of experiment might be that phenomena corresponding to intermediate positions of the electron between stable orbits do not exist.

Finally, we must comment on the general tactics of the quantum situation. It would seem that there have already been a sufficient number of unsuccessful attempts to formulate quantum behavior in terms of ordinary mechanics to justify the expectation that ultimately something quite different must evolve. The difficulties of an unmodified carrying over of ordinary mechanical notions to quantum phenomena may be illustrated by a simple example. Consider a particle of mass _m_ rotating in a frictionless circular track of radius _r_. Then according to quantum conditions it can move stably on this track only with certain definite velocities, such that ∫ pdq = mv 2πr = nh. Suppose now the particle rotating with one of the allowed velocities, and a tangential force applied. If the usual mechanical notions of force are still valid, the particle must respond by moving in its track with continually increasing velocity. After the velocity has been increased by a small amount, we remove the force. The motion is now no longer one of the allowed ones, and the particle must in some way change its velocity; it must either slow down or speed up. In the first case it must either radiate energy, which a system of the simple mechanical properties we have supposed is not capable of doing, or else the law of conservation of energy fails, and also Newton's first law of motion during the process of acquiring the steady condition. If, on the other hand, the particle speeds up, it must increase its energy from nowhere, and again ordinary mechanics does not apply.

It seems then a mistake to attempt to formulate the quantum conditions in terms of the notions of ordinary mechanics (momentum, and position coördinates in either the ordinary or the generalized Lagrangean sense). It would seem, on the other hand, plausible to expect that mechanics is not a fundamental thing, but is in some way an effect produced by the aggregate action of a great many elementary quantum processes. Amplitude of radiational vibration, for example, may be such a statistical aspect of a great many processes, in some such way as on the ordinary level of experience temperature is a statistical aspect of the average kinetic energy of the atoms. One possibility of this kind has already been more explicitly indicated; in the elementary process of emission of radiation, frequency and energy are not two independently assignable variables, but are connected [E = hν]. That is, on the quantum level radiation has only a single property, which is properly neither energy or frequency. [We are now neglecting the polarization aspect of radiation.] On a higher level, that of ordinary radiation, the single elementary property has expanded itself into two (energy and frequency) through the additional variable of the number of elementary quantum processes in the complex radiation.

The program of the immediate future should be an extension of something of this sort, namely, to invent new concepts corresponding to the experimentally independent things on the quantum level (such perhaps as the resultant of the fusion of the energy and frequency concepts for radiation), and then to show how the ordinary concepts of mechanics (and very likely those also of space and time) are generated by statistical effects in aggregates of great numbers. Perhaps it is yet too early for an attempt of this sort, because it may seem that there are still too many possibilities of new experimental discoveries which might upset the results of elaborate theoretical speculation. If this should really be felt to be the case, I believe that physics ought for the present to hold in partial abeyance its theoretical activities in this field, and devote itself to acquiring as rapidly as possible the necessary experimental facts. We may emphasize again that the possibility of carrying out this plausible program can be proved only by experiment; it may be that more concepts will be required on the quantum level than for ordinary experience.

The invention of new concepts is certainly not an easy thing, and is something which physics has always deliberately, and perhaps justifiably, shirked, as shown by the persistent attempts to carry the notions of mechanics down into the finest structure of things. This shirking has not had bad results, but on the contrary good results, as long as physics has been primarily concerned with phenomena near the range of ordinary experience, but I believe that as we get farther and farther away from ordinary experience, the invention of new concepts will become an increasing necessity.