Empirical Formulas

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In 1952, Nambu published the remarkable observation that the masses of elementary particles expressed in terms of the electron mass are close to integer multiples of half the reciprocal of the fine structure constant. Since then we have received numerous Letters and Articles about attempts to refine or modify the Nambu relation and to give a theoretical basis for the proposed formulas. Our referees usually recommend rejection of such communications, since they believe that the theoretical arguments are unconvincing. About an article of this kind, Pauli once made the comment, "It isn't even wrong."Sometimes the author of a speculative paper claims that the excellent agreement between his proposed formula and experimental data is proof of the correctness of his theory. That this alone is not sufficiently convincing can best be illustrated by an old example. In 1922, the late Ludwik Silberstein, an outstanding physicist, proposed a formula for the spectral lines of neutral helium based on the assumption that its two electrons behave as if helium were merely composed of two, not interacting, neutral hydrogen atoms. The helium spectrum was represented by the sum of two Balmer formulas. The agreement was so perfect that there was less than one chance in ${10}^{12}$ for it to be accidental, "small enough to discard every suggestion of the play of blind chance." Silberstein believed therefore that the numerical agreement proved his model for helium to be correct. What it really showed was that his formula was a good representation for the helium spectral lines, which is not surprising since it could be considered as an approximation to the well-established Rydberg formula. The probability argument had no bearing at all on the meaning of Silberstein's theory, which was obviously wrong. When discussing this case with one of his students, the great teacher Ehrenfest remarked, "If it is essential to use probability to prove that you are right, you are usually wrong."Even excellent agreement with data is sometimes misleading. Often an author seems not to realize that his vague theoretical novelties contain almost as many assumptions, or his formula about as many parameters, as there are data to be fitted.Empirical rules, from Boyle's law to the Balmer formula and the Geiger-Nuttall diagram, have always played a key role in the development of physics. Such relations often lead towards the discovery of new theories or to their corroboration. It is perhaps a major misfortune for nuclear physics and particle physics that the data in these fields seem not to be as conducive to finding numerical laws as was the case, for example, in atomic and molecular spectroscopy. However, sooner or later one of the speculations discussed in the first paragraph may turn out to have been an inspired correct guess. We have therefore recommended that these authors present the main features of their work at one of the meetings of our Society. In this way they will receive credit for their contribution if it later proves to be right. Moreover, the published Abstract brings the subject into open discussion, and might lead someone else onto the right path. We will gladly publish any Letter announcing a new empirical formula, provided the evidence for its validity is compelling by its simplicity and its excellent agreement with the data, or if it is based on clearly stated, plausible theoretical suppositions.