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April 13, 2005

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Chad Orzel provides the pointer to Helge Kraghe, who writes in Physics Web http://physicsweb.org/articles/world/13/12/8 about how quantum theory existed in the equations of physics half a decade before the human brain of any physicist understood it:

It was 100 years ago when Max Planck published a paper that gave birth to quantum mechanics - or so the story goes.... According to the standard story... quantum theory emerged when it was realized that classical physics predicts an energy distribution for black-body radiation that disagrees violently with that found experimentally. In the late 1890s, so the story continues, the German physicist Wilhelm Wien developed an expression that corresponded reasonably well with experiment - but had no theoretical foundation. When Lord Rayleigh and James Jeans then analysed black-body radiation from the perspective of classical physics, the resulting spectrum differed drastically from both experiment and the Wien law. Faced with this grave anomaly, Max Planck looked for a solution, during the course of which he was forced to introduce the notion of "energy quanta". With the quantum hypothesis, a perfect match between theory and experiment was obtained. Voila! Quantum theory was born.

The story is a myth, closer to a fairytale than to historical truth...

The study of black-body radiation had begun in 1859, when Robert Kirchhoff, Planck's predecessor as professor of physics in Berlin, argued that such radiation was of a fundamental nature.... [I]n 1896... Wien found a radiation law that was in convincing agreement with the precise measurements being performed at the Physikalisch-Technische Reichsanstalt in Berlin... the spectral density, u, - the radiation energy density per unit frequency - depended on the frequency, f, and temperature, T.... Planck was... interested in... establishing a rigorous derivation of it.... To secure a more fundamental derivation he... reinterpreted Boltzmann's theory in his own non-probabilistic way. It was during this period that he stated for the first time what has since become known as the "Boltzmann equation" S = k log W, which relates the entropy, S, to the molecular disorder, W.

To find W, Planck had to be able to count the number of ways a given energy can be distributed among a set of oscillators. It was in order to find this counting procedure that Planck, inspired by Boltzmann, introduced what he called "energy elements", namely the assumption that the total energy of the black-body oscillators, E, is divided into finite portions of energy, epsilon, via a process known as "quantization". In his seminal paper published in late 1900 and presented to the German Physical Society on 14 December... Planck regarded the energy "as made up of a completely determinate number of finite equal parts, and for this purpose I use the constant of nature h = 6.55 x 10-27 (erg sec)"....

Quantum theory was born. Or was it? Surely Planck's constant had appeared, with the same symbol and roughly the same value as used today. But... [Planck] explained in a letter written in 1931, the introduction of energy quanta in 1900 was "a purely formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result."... Far more interesting [to Max Planck] than the quantum discontinuity (whatever it meant) was the impressive accuracy of the new radiation law and the constants of nature that appeared in it.

If a revolution occurred in physics in December 1900, nobody seemed to notice it.... Very few physicists expressed any interest in the justification of Planck's formula, and during the first few years of the 20th century no one considered his results to conflict with the foundations of classical physics.... As to the quantum discontinuity - the crucial feature that the energy does not vary continuously, but in "jumps" - [Planck] believed for a long time that it was a kind of mathematical hypothesis, an artefact that did not refer to real energy exchanges between matter and radiation....

[N]owhere in his papers of 1900 and 1901 did Planck clearly write that the energy of a single oscillator can only attain discrete energies.... If this is what he meant, why didn't he say so? And if he realized that he had introduced energy quantization - a strange, non-classical concept - why did he remain silent for more than four years?...

[I]t was Einstein who first recognized the essence of quantum theory. Einstein's remarkable contributions to the early phase of quantum theory are well known and beyond dispute. Most famous is his 1905 theory of light quanta (or photons), but he also made important contributions in 1907... Einstein's 1907 theory of specific heats was an important element in the process that established quantum theory as a major field of physics. The changed status of quantum theory was recognized institutionally with the first Solvay conference of 1911, on "radiation theory and the quanta", an event that heralded the take-off phase of quantum theory...

Max Planck comes up with an equation that works. In order to do so he has to make a "purely formal assumption." And it is only half a decade later that Einstein realizes that the little h that appears in Max Planck's equation is not a formal assumption or an "artefact" but instead tells us what is perhaps the most important thing about the guts of the universe.

For half a decade the first equation of quantum theory was there. But nobody knew how to read it.

It is this "what if we took this equation seriously?" factor that is, to my mind at least, the spookiest thing about the unreasonable effectiveness of mathematics in physics. Take the h in Max Planck's equation seriously, and you have the quantum principle--something that was not in Planck's brain when he wrote the equation down. Take seriously the symmetry in Maxwell's equations between the force generated when you move a magnet near a wire and the force and the force generated when you move a wire near a magnet, and you have Special Relativity--something that was not in Maxwell's brain when he wrote down the equation. Take Newton's gravitational force law's equivalence between inertial and gravitational mass seriously and you have General Relativity--something never in Newton's mind. And take the mathematical pathology at r = 2M in the Schwarzchild metric for the space-time metric around a point mass seriously, and you have black holes and event horizons.

Posted by DeLong at April 13, 2005 11:42 AM