## Thursday, March 04, 2021

### Gregor Wentzel and the path integral

Provoked by the previous blog post about the path integrals, Frank S. sent me two fascinating 1924 papers by Gregor Wentzel which "almost" defined the Feynman's path integral definition of quantum mechanics but two decades earlier (and a year before Heisenberg started quantum mechanics as we know it). If you find English somewhat more comprehensible than German, like I do, you may read a 1998 review of these papers by Salvatore Antoci and Dierck-E. Liebscher
which probably contains more than what we will actually ever need, unless we plan to write a detailed book about the history of science. First, Wentzel may sound like a forgotten name. But on this blog, you find it in two previous texts. He had something to do with dispersion and Kramers etc. After all, Wentzel's name starts with a W, like WKB does, and indeed, he is also the first guy remembered in the WKB acronym. But how many people know what (J)WKB stands for now?

Just a Czech comment. Wentzel sounds very German because it's spelled in a German way. But it's just a Germanization of the Czech, Slavic name Václav. Why? Because the original form of the name, used some 1000 years ago, was Věnceslav (like Víceslav) which really means "more fame" or "more famous". Unless it literally means "wrath/crown of fame", as some sources say.

Ironically enough, "Boleslav" had the same meaning ("bole" is still a Russian word for "more"). Boleslav was the brother who murdered saint Václav. They were brothers but their first names had the same meaning. English adopted the old Czech "Věnceslav" as its (Good King) Wenceslaus, German shortened it to Wenzl or Wentzel, and modern Czech shortened it to Václav (or, colloquially, Vašek or Venca, the latter is probably a Bohemization of the German name Wentzel; note that Venclovský was the first Czech who swam over the English Channel while David Vencl – the surname is just a Bohemian respelled Wentzel – recently improved the world record in swimming under the ice to 81 meters, a great Czech dude)...

Gregor Wentzel was born in 1898 and died in 1978. He was some 2-4 years older than the likes of Heisenberg, Pauli, Dirac... but surely close enough to call him the same generation. And I think it's fair to say that guys like him really created the mood for the proper birth and expansion of quantum mechanics – and their role is hugely understated by the contemporary caricature of the history of physics.

The 1998 paper explains that Wentzel's first two 1924 papers were overlooked because their titles made them sound like papers on "optics" and that field wasn't and isn't too hot for those who study "foundations of quantum mechanics" which is why they don't read Wentzel's papers at all. But quantum mechanics of atoms is ultimately inseparable from optics because the atoms emit light when they transition from one state to another.

He didn't have a complete, usable theory but he was close to it, perhaps closer than Louis de Broglie who published his papers on the de Broglie waves in the same year, 1924. Wentzel postulated that atoms don't radiate when they are in a "mechanical" state. This confusing adjective meant to represent an atom that obeyed the classical equations of "mechanics". On the other hand, all the radiation emitted by atoms is due to "non-mechanical" states when there is some deviation from the classical solutions, if I use the no-nonsense modern terminology (the fix isn't just terminological but it's close to it).

You may see it's clever because quantum mechanics requires that the classical trajectory isn't the only one that matters. Some trajectories that are away from the classical one are important for atoms and totally crucial for the radiation coming from atoms. Wentzel understood that to decide whether radiation is emitted, he absolutely needs to use the description in terms of probabilities. He was able to see that a "sum of probabilities" was the classical answer but he needed the interference so he needed to sum the probability amplitudes of some kind before the sum's absolute value is squared.

And unlike everyone else, he also found it "obvious" – so that it wasn't even discussed much – that the interfering things should be complex, i.e. complex probability amplitudes. To determine the right phase of the probability amplitudes, he used something that was close to the Dirac-Feynman usage of the action. Well, Wentzel only had a part of the action, something like (let me include the exponential)$\exp\zav{\frac{i}{\hbar}\int dt\,\sum_j p_j \dot q_j }$ while he used letters $$\alpha,\beta$$ for $$p,q$$ instead. This isn't quite the action and it's lacking the information about the formula for the potential and kinetic energy but it's a term that may be added to the action. In fact, it's the difference between the action and the time-integrated Hamiltonian. With these rules, he had almost the full picture. This allowed him to learn quantum mechanics quickly and properly. The WKB paper emerged in 1926, on the same year as Schrödinger's paper on his equation.

The usage of "something like the energy times time over the reduced Planck constant" as the phase is pretty much equivalent to de Broglie's vague idea. De Broglie was more popular because he avoided all "esoteric" (and conceptually new) things but among the two, Wentzel was arguably closer to a full quantum mechanical theory.

Wentzel had something that could be called "mathematical intuition of a profound physicist". I mean his ability to instantly see that we needed to switch to probabilities from a deterministic description; and the probabilities should be constructing as squared absolute values of complex building blocks. (De Broglie, Schrödinger, and other people who quickly became the early foes of quantum mechanics had a problem with the complexity of the fundamental amplitudes, too. They spent some time by trying to work with real waves. The hostility towards the interpretational novelties of quantum mechanics; and towards the new theory's "purely" mathematical traits such as the complexity are heavily correlated.) In these respects, he scooped Born, Dirac, and Feynman, among others. But the history as shaped by Realpolitik sees things differently. The history of science prefers the papers which write down a complete, usable theory (which is a fair highlighting); and then some incomplete papers that are usable enough for misleading popular presentations (which is not too fair). But people like Wentzel who really perform a paradigm shift and are even certain about the need for such a paradigm shift often end up overlooked.

Of course, the men who did notice Wentzel's bright ideas are often forgotten, too, like K.F. Herzfeld. To some extent, other visionaries in quantum mechanics and quantum field theory, like Oskar Klein, are also heavily suppressed these days despite the fact that building on the Kaluza theory with an extra dimensions and adding non-Abelian and quantum aspects, Oskar Klein almost had a Yang-Mills (non-Abelian gauge) theory (like the Standard Model) by the 1930s. Even when these theories weren't quite right, they helped to create the right spirit for the complete theories to arise with a higher probability, I believe. But it's of course possible that many of the required ideas were rediscovered independently.