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Poincare's work in quantum mechanics
Posted:
Aug 17, 2003 6:22 PM


Although it's rather well known that Henri Poincare anticipated a number of results in special relativity prior to Einstein's 1905 publication, it seems that fewer people are aware that Poincare also played a role in another revolution in physics at the beginning of the 20th century  namely, quantum mechanics (I guess there are some people who might also argue that he participated in another revolution for his pioneering work in dynamical systems). Recently, I had the good fortune to come across Russell McCormmach's excellent article "Henri Poincare and the Quantum Theory" (Isis; Volume 58 (191); 1967; pages 3755). Besides discussing Poincare's work, it offers some fascinating glimpses into the first Solvay Conference that took place in October and November of 1911. Below are a few (six) excerpts that I've selected from McCormmach's article, which I highly recommend for people interested in the historical development of physics during this time period:
 1. At the time, Maurice de Broglie remarked to F. A. Lindemann that of all those present Poincare and Einstein were in a class by themselves (p 40).
2. Lorentz recalled that in the discussions Poincare had shown "all the vivacity and penetration of his spirit, and that one had admired the facility with which he entered vigorously into even those questions of physics which were new to him" (p. 40).
3. ... it was Planck, however, who stimulated Poincare's most penetrating, questioning spirit .... He twice pressed Planck to give good grounds for deciding among the several possible ways of decomposing phase space into the finite elementary areas for the probability calculations. He wanted to know how the energy of a system of several degrees of freedom might be quantized, since the onedimensional quantization procedure was incompatible with transformations of axes in higherdimensional systems. Poincare regretted that as yet there had been no discussion of mechanisms for the interaction of fixed resonators; for in the absence of any definite mechanism there could be no exchange of energy between radiations of different frequencies, and therefore no final equilibrium. Planck had stressed quanta of action rather than quanta of energy . . . but he did not know what it means to speak of the conservation of action. Finally, he was skeptical of Planck's new formulation of the radiation theory, according to which the absorption of energy by the resonators varies continuously with time (p. 41).
4. In a descriptive essay he spelled out the essence of Planck's theory as it appeared to him: "A physical system is capable of only a finite number of distinct states; it jumps from one of those states to another without going through a continuous series of intermediate states." The image of a physical system jumping from one discrete state to another put him in a speculative frame of mind. He considered the possibility that a particle might trace only certain allowed paths in phase space, shifting discontinuously between them. And he supposed that the universe as well as an electron ought to experience quantum jumps. Since there would be no distinguishable instants within the motionless states between universal jumps, there should exist an "atom of time." Such were the kinds of ideas going through Poincare's mind shortly before he died; there was nothing timid or grudging about his late acquaintance with the quantum theory (p 50).
5. ... above all it was the unquestioned authority of Poincare in mathematical matters which secured him an attentive audience. Jeans undoubtedly voiced a majority sentiment when he said that "we shall probably feel inclined to trust to the accuracy of Poincare's mathematics." (pp 5152).
6. Whereas Jeans had strongly opposed the quantum theory in Brussels . . ., he came out vigorously in support of quanta at the Birmingham meeting of the British Association in September 1913, fourteen months after Poincare's death. There is no doubt about what caused him to change his mind. Jeans had read Poincare's paper and been converted by it. ... The French scientist's arguments had been so completely persuasive that from this time on every theory would have to "logically involve either the belief that Poincare is wrong, or the belief that he is right, together with all that this involves. . . . And Jeans himself felt compelled to accept the quantum hypothesis in its entirety." (p. 53). 
As an undergraduate, I had become aware that Poincare's paper "Sur la theorie des quanta" (one of the last he ever wrote  he died prematurely in 1912, while undergoing an operation) had been influential in gaining wider acceptance for Planck's thencontroversial quantum theory of blackbody radiation. The brief historical excerpt I had read at the time implied that Poincare proved (essentially) that Planck's theory required the existence of discrete energy quanta.
After reading McCormmach's article, however, I'm less certain that this is an accurate statement. Instead, it would appear that Poincare's proof (modulo some necessary refinements [see page 52 of McCormmach]) was based on some debatable assumptions. My main questions regard the legitimacy of Poincare's proposed mechanisms whereby pairs of "resonators" exchanged energy (page 45 of McCormmach). There are other questions that one can raise as well (as some of Poincare's contemporaries did). I haven't read Poincare's original paper (my French not being particularly strong), so I'm wondering if anyone who is familiar with Poincare's work in quantum mechanics can comment on whether Poincare's legacy in quantum mechanics is either "both A and B" or just "B" for the following statements:
(A) A legitimate proof that Planck's theory required the existence of discrete energy quanta (in spite of working off of a physical foundation that predated Heisenberg and Schroedinger's work in QM by 1314 years).
(B) Helped to gain further acceptance for the theory of quantum mechanics among physicists circa 1912.
 Mark
p.s.  While on the topic of Poincare, can anyone comment on whether there is general concurrence on whether Perelman's third paper
successfully finishes off the Geometrization conjecture (and thereby the Poincare conjecture)?



