The Analysis of Matter

diagonal consists of 's, and whose other terms are all zero. The above is the sole fundamental equation containing (Planck's constant), and it is true for all motions. Heisenberg does not claim that the new theory solves all difficulties. On the contrary, he says (5, p. 705): "The theory here described must be regarded as still incomplete. The real geometrical or kinematical meaning of the fundamental assumption (5)[19] has not yet been made completely clear. In particular, there is a serious difficulty in the fact that the time apparently has a different rôle from the space co-ordinates, and is formally differently treated. The formal character of the time co-ordinate in the mathematical structure of the theory is made particularly evident by the fact that in the theory hitherto the question of the temporal course of a process has no immediate meaning, and that the concept of earlier and later can hardly be defined exactly. Nevertheless, we need not consider these difficulties as an objection to the theory, since the appearance of just such difficulties was to be expected from the nature of the space-time relations that hold for atomic systems." In a more or less popular exposition (6), Heisenberg has set forth some of the consequences of his theory. Electrons and atoms, he says, do not have "the degree of immediate reality of objects of sense," but only the sort of reality which one naturally ascribes to light quanta. The troubles of the quantum theory have come, he thinks, from trying to make models of atoms and picture them as in ordinary space. If we are to retain the corpuscular theory, we can only do it by not assigning a definite point of space at each time to the electron or atom. We substitute a well-defined physical group of quantities which represent what was the place of the electron.[Pg 46] They are the observable radiation quantities, each of which is associated with two "terms," so that we obtain a matrix. The distinction of inner and outer electrons in an atom becomes meaningless. "It is, moreover, in principle impossible to identify again a particular corpuscle among a series of similar corpuscles" (p. 993). The matrix theory of the electron is too new to be amenable, as yet, to the kind of logical analysis which it is our purpose to undertake in this Part. It is clear, however, that it affects a scientific economy by substituting for the merely hypothetical steady motions of Bohr's atoms a set of quantities representing what we really know—namely, the radiations that come out of the region in which the atom is supposed to be. It is clear, also, that there is an immense logical progress in the construction of a dynamic which destroys the distinction between quantized and unquantized motions, and treats all motions by means of a uniform set of principles. And the greater abstractness of the Heisenberg atom as compared with the Bohr atom makes it logically preferable, since the pictorial elements in a physical theory are those upon which least reliance can be placed. An apparently different quantum theory, due to de Broglie[20] and Schrödinger,[21] has been found to be formally the same as Heisinger's theory, although at first sight very different. This is described by de Broglie as "the new wave theory of matter," in which "the material point is conceived as a singularity in a wave."[22] Here, also, the radiations which we think of as coming out of the atom have more physical "reality" than the atom itself. One of the merits of the theory is that it diminishes the difficulties hitherto[Pg 47] existing in the way of a reconciliation of the facts of interference and dispersion with the facts which led to the hypothesis of light quanta. Meanwhile, there remains the possibility that all the quantum phenomena may be deducible from classical principles, and that the apparent discontinuities may be only a question of sharp maxima or minima. The most successful theory known to me on these lines is that of L. V. King.[23] He assumes that electrons rotate with a certain fixed angular velocity, the same for all; he makes a similar assumption as regards protons. Consequently there is a magnetic field which introduces conditions that are absent if electrons and protons have no spin. There will be electromagnetic radiation of frequency , where: being Planck's constant, the invariant mass of the electron, and its velocity. (The identity of with Planck's constant is obtained by adjusting the hypothetical constants.) From this formula he deduces many of the phenomena upon which the quantum theory is based, and promises to deduce others in a later paper. An article by Mr R. H. Fowler ("Spinning Electrons," Nature, Jan. 15, 1927) discusses Mr King's theory without arriving at a verdict for or against. Presumably it will not be long before a definite answer as to the adequacy of Mr King's theory is possible. If it is adequate, the quantum theory ceases to concern the philosopher, since what remains valid in it becomes a deduction from more fundamental laws and processes which are continuous and involve no atomicity of action. For the moment, until the physicists have arrived at a decision, the philosopher must be content to investigate both hypotheses impartially. FOOTNOTES: [7] The numerical value of is , and its dimensions are those of "action"—i.e. energy x time. [8] Report on Radiation and the Quantum Theory, Physical Society of London, 1914, p. 58. [9] Annalen der Physik, vol. XVII., p. 146. [10] See Jeans, loc. cit., chap. VI. [11] See Sommerfeld, Atomic Structure and Spectral Lines, pp. 547 ff. [12] W. Wilson, The Quantum Theory of Radiation and Line Spectra Phil. Mag., June, 1915. [13] What follows is taken from Note 7 (pp. 555 ff.) in Sommerfeld's Atomic Structure and Spectral Lines, translated from the third German edition by Henry L. Brose, M.A., 1923. See also Note 4 (pp. 541 ff.). [14] This, however, is probably a temporary state of affairs. Certain Pasons for quantum transitions are already known. See J. Franck and P. Jordan, Anregung von Quaniensprüngen durch Stösse, Berlin, 1926; also P. Jordan, Kausalität und Statistik in der modernen Physik, Naturwissenschaften, Feb. 4, 1927. [15] See Sommerfeld, op. cit., pp. 232 ff. [16] This is not the same phenomenon as in the case of the orbit of Mercury. The latter depends upon the general theory of relativity, the former upon the special theory. [17] Sommerfeld, op. cit., pp. 467 ff. [18] The principal papers setting forth this theory are: 1. W. Heisenberg, Ueber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Zeitschrift für Physik, 33, pp. 879-893, 1925. 2. M. Born and P. Jordan, Zur Quantenmechanik. Ibid. 34, pp. 858-888, 1925. 3. M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik II. Ibid. 35, pp. 557-615, 1926. 4. P. A. M. Dirac, The Fundamental Equations of Quantum Mechanics. Proc. Royal Soc., Series A, vol. 109, No. A752, pp. 642-653, 1925. 5. W. Heisenberg, Ueber quantentheoretische Kinematik und Mechanik. Mathematische Annalen, 95, pp. 683-705, 1926. 6. W. Heisenberg, Quantenmechanik. Naturwissenschaften, 14 Jahrgang. Heft 45, pp. 989-994. I shall quote these papers by the above numbers. I am much indebted in this matter to Mr R. H. Fowler, F.R.S. [19] This is the assumption, mentioned above, that an atom or electron at time can be represented by a collection of terms of the form: [20] Annales de Physique, 3, 22, 1925. [21] Annalen der Physik, 1926. Four papers, 79, pp. 361, 489, 734; 80, p. 437. [22] Nature, Sp. 25, 1926, p. 441. See also Fowler, "Matrix and Wave Mechanics." ib. Feb. 12, 1927. [23] Gyromagnetic Electrons and a Classical Theory of Atomic Structure and Radiation. By Louis Vessot King. F.R.S., Macdonald Professor of Physics, McGill University. Louis Carrier, Mercury Press, 1926. [Pg 48] CHAPTER V THE SPECIAL THEORY OF RELATIVITY THE theory of relativity has resulted from a combination of the three elements which were called for in a reconstruction of physics; first, delicate experiment; secondly, logical analysis; and thirdly, epistemological considerations. These last played a greater part in the early stages of the theory than in its finished form, and perhaps this is fortunate, since their scope and validity may be open to question, or at least would be but for the successes to which they have led. One may say, broadly, that relativity, like earlier physics, has assumed that when different observers are doing what is called "observing the same phenomenon," those respects in which their observations differ do not belong to the phenomenon, but only those respects in which their observations agree. This is a principle which common sense teaches at an early age. A young child, seeing a ship sailing away, thinks that the ship is continually growing smaller; but before long he comes to recognize that the diminution in size is only "apparent," and that the ship "really" remains of the same size throughout its voyage. In so far as relativity has been inspired by epistemological considerations, they have been of this common-sense kind, and the apparent paradoxes have resulted from the discovery of unexpected differences between our observations and those of other hypothetical observers. Relativity physics, like all physics, assumes the realistic hypothesis, that there are occurrences which different people can observe. For the present, we may ignore epistemology, and proceed to consider relativity simply as theoretical physics. We may also ignore the experimental evidence, and regard the whole theory[Pg 49] as a deductive system, since that is the point of view with which we are concerned in Part I. The most remarkable feature of the theory of relativity, from a philosopher's standpoint, was already present in the special theory: I mean the merging of space and time into space-time. The special theory has now become only an approximation, which is not exactly true in the neighbourhood of matter. But it remains worth understanding, as a stage towards the general theory. Moreover, it does not demand the abandonment of nearly such a large proportion of our common-sense notions as is discarded by the general theory. Technically, the whole of the special theory is contained in the Lorentz transformation. This transformation has the advantage that it makes the velocity of light the same with respect to any two bodies which are moving uniformly relatively to each other, and, more generally, that it makes the laws of electromagnetic phenomena (Maxwell's equations) the same with respect to any two such bodies. It was for the sake of this advantage that it was originally introduced; but it was afterwards found to have wider bearings and a more general justification. In fact, it may be said that, given sufficient logical acumen, it could have been discovered at any time after it was known that light is not propagated instantaneously. It has grown by this time very familiar—so familiar that I have even seen it quoted (quite correctly) in an advertisement of Fortnum and Mason's. Nevertheless, it is, I suppose, desirable to set it forth. In its simplest form it is as follows: Suppose two bodies, one of which () is moving relatively to the other () with velocity v parallel to the -axis. Suppose that an observer on observes an event which he judges to have taken place at time , by his clocks, and in the place whose co-ordinates, for him, are , , . (Each observer takes himself as origin.) Suppose that an observer on judges that the[Pg 50] event occurs at time and that its co-ordinates are , , . We suppose that at the time when the two observers are at the same place, and also . It would formerly have seemed axiomatic that we should have . Both observers are supposed to employ faultless chronometers, and, of course, to allow for the velocity of light in estimating the time when the event occurs. It would be thought, therefore, that they would arrive at the same estimate as to the time of the occurrence. It would also have been thought that we should have: Neither of these, however, is correct. To obtain the correct transformation, put: where is, as always, the velocity of light. Then: For the other co-ordinates , , we still have, as before: It is the formulæ for and that are peculiar. These formulæ contain, implicitly, the whole of the special theory of relativity. The formula for embodies the FitzGerald contraction. Lengths on either body, as estimated by an observer on the other, will be shorter than as estimated by an observer on