# Henri Poincaré, great mathematician

Poincaré was possibly the world's leading mathematician at the time. He contributed to the theory of special relativity in 1904 (before it was fully created in Einstein's paper in 1905) by showing that Maxwell's equations with sources are covariant under the Lorentz group. We call the inhomogeneous Lorentz group the "Poincaré group", a terminology favoured by Wightman, in honour of his contributions. I have not found a mathematical paper by Poincaré in which he defines this group; I thank Emili Bifet for raising this question. In a book Science and Method, Poincaré did write out the whole idea in words, in a chapter called The Relativity of Space. There, he also advocated the idea of invariance of the world under scale transformations. The group thus generated in now known as the conformal group, This, of course, is not an invariance group for particles of non-zero mass. W. Voigt was the first to prove (1887) that Maxwell's equations without sources are invariant under a class of conformal transformations. The conformal factor was not 1, so they are not Lorentz transformations. Indeed, the conformal factor was chosen so that there is no Lorentz contraction; inertial frames are related by a Galilean transformation, and time suffers a dilation equal to the square of that in a Lorentz boost. The theory, Ueber das Doppler'sche Princip, Goett. Nach Vol 41, 41-, 1887, is described by Andreas Ernst and Jong-Ping Hsu in Chinese J. of Phys, 31, 211-230, 2001. The Doppler shift predicted by this theory is the same in all frames of reference, but is not the same as that of Einstein. Voigt did correctly predict that the speed of light is the same in all (his) frames of reference, but this was not given enough credit at the time.

Poincaré's article can be read here, translated into English. This site dates the book as 1897, in contrast with Ruelle's article, which puts it ten years later, 1908, as do all other authors. It could be that the marxists confuse the later work with Poincaré's La science et l'hypothese, 1902, in which similar remarks about relativity are made. That Poincaré almost had special relativity is seen in his article of 1905, Sur le dynamique de l'electron, CR hebd. des séances de l'acad. des sci., vol. 140, 1504-1508, (1905), submitted before Einstein's own article. Pais, in his book on Einstein, Subtle is the Lord, says that Poincaré and Lorentz did not really get relativity; he says that even after Einstein's work, they hoped to set up a mechanical model of the electron with the properties required by the Maxwell equations. Pais says that Lorentz and Poincaré did not regard Einstein's paper as the final word, since it just postulates what we would want to be true (the constancy of the speed of light) but did not construct a theory in which it was true. Pais's remark does not seem to be true of Poincaré, however, who believed in the impossibility of detecting absolute motion from at least as early as 1902. Einstein did not often give references to other people; however in this case I agree with Pais, that Einstein did introduce a shift of paradigm, from a mechanical model to invariance under a group. Poincaré and Lorentz wanted both at once.

In some sense both Lorentz and Poincaré are right; to this day we have not been able to construct a mathematical model of interacting particles consistent with quantum mechanics and special relativity, obeying Einstein causality. We have the axioms (of Wightman or Haag), but no models in 3+1 dimensions.

In 1964, L. Gross (in Jour Math Phys) proved that the Wigner representation [0,1] + [0,1] of mass 0 and spin 1, of the Poincaré group, which is carried by solutions of Maxwell's equations with a suitable norm, can be extended to a unitary representation of the conformal group in the same space ... another triumph for Wignerism.

Poincaré is sympathetically discussed by David Ruelle here. Also one should consult the McHistory.