Why is it that particles with half-integral spin are Fermi particles whose amplitudes add with the minus sign, whereas particles with integral spin are Bose particles whose amplitudes add with the positive sign? We apologize for the fact that we cannot give you an elementary explanation. An explanation has been worked out by Pauli from complicated arguments of quantum field theory and relativity. He has shown that the two must necessarily go together, but we have not been able to find a way of reproducing his arguments on an elementary level. It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is deep down in relativistic quantum mechanisms. This probably means that we do not have a complete understanding of the fundamental principle involved.
–Richard Feynman, The Feynman Lectures on Physics, Volume III
The Feynman Lectures on Physics is a three-volume collection of essays based on undergraduate courses Richard Feynman taught at Caltech in the early 1960s. In addition to being a renowned physicist, Feynman was a skilled professor, able to draw upon everyday intuitions to illustrate technical points and explain complicated ideas clearly. On the bookshelves of practicing physicists and undergraduate students alike you are likely to find his distinctive red-bound paperbacks. They are, among other things, masterpieces of lucid prose.
A non-physicist may find that last claim hard to swallow on the basis of the quote above. If I redact from its first sentence all the language that only someone with a few semesters of quantum mechanics could be expected to understand, you are left with the following: “Why is it that particles with ████-████████ ████ are █████ particles whose ██████████ add with the minus sign, whereas particles with ████████ ████ are ████ particles whose ██████████ add with the positive sign?” This could mean anything, and a reasonable response would be to skip the remainder of the paragraph, figuring that you won’t understand any of that either so why bother? If you had that response, I invite you to go back and reread the whole thing anyway. Only the first sentence is a doozy. The rest is entirely comprehensible and only demands an appreciation of humility.
This passage comes at the beginning of a chapter on quantum mechanics, the branch of physics that describes very small objects like atoms and the elementary particles of which they are composed. Some of those elementary particles are called Fermi particles and others are called Bose particles (named after two physicists, Enrico Fermi and Satyendra Nath Bose) and they behave in at least two distinct ways: one that can be characterized mathematically as “half-integral” versus “integral” “spin” and the other as having “amplitudes” that one either adds or subtracts. This jargon refers to sets of equations that, when solved, make very specific predictions about the particles’ interaction with laboratory equipment that may be very clearly confirmed or disconfirmed. The meat of the assertion is that there is no obvious reason why the “spin” behavior and the “amplitude” behavior should bear the particular relationship to each other that they do. Their correlation would appear at first glance to be a coincidence, but it is not, and Feynman is unable to give even a hint as to why it is not in terms of general concepts. The best he can do is point to some other guy’s formalism which in turn does not admit any explanation beyond itself.
Crucially, Feynman is apologetic about all this. He says, “We apologize for the fact that we cannot give you an elementary explanation.” (Like many scientific writers, Feynman uses the editorial “we”, but it is also reasonable here to take him as speaking on behalf of all physicists, including himself.) Furthermore the apology is not just for an inability to communicate–to dumb down sophisticated ideas to an appropriate level–but for a failure to sufficiently grasp those ideas in the first place. “This probably means that we do not have a complete understanding of the fundamental principle involved.” If Feynman can’t explain it, the fault lies with him, not you. And it is not a superficial flaw, but points to a deep inadequacy in his, Feynman’s, understanding.
Richard Feynman was not a modest man. He was one of the most acclaimed geniuses of 20th century physics, right up there in the pantheon with Albert Einstein. By all accounts he possessed an almost superhuman level of self-confidence and, though generally charming, could be intellectually fearsome when the situation called for it. Here, lecturing an audience of mere undergraduates on his personal bailiwick of quantum mechanics, it would have been easy for Feynman to bully his way though. He could have presented an inability to express the relationship between half-integral spin and symmetric waveform summation as indicative of the profundity of the idea, and of his own intellect. Pauli’s “relativistic quantum mechanisms” could have been held up as an acid test to separate the tyros from those who had a chance of making it into the inner circle. The fact that certain concepts could not even begin to be expressed in any form other than the hermetically sealed jargon of mathematics could have been taken as a sign of their oceanic significance. The parts that were unknowable could have been made to seem mystical, poetic and profound.
Could have been, but were not. Instead for something to be inexpressible on “an elementary level” is a source of annoyance and a problem to be solved. Feynman’s sentiments are not unusual: his veneration of clarity is the party line for 20th century natural science. Granted it’s easy to lose sight of this because clarity is relative. Plenty of natural science is awfully unclear to someone who hasn’t studied it, and the field is populated by scientists who make careers out of navigating the murk better than their colleagues, and who may have penchants for vanity and poetic flights of their own. But there is in the end a sizable consensus that what is inexpressible is also incomplete.
Not everybody has to be a scientist. Striving for clarity is a choice like everything else. But it is a choice that has proved fruitful in a wide variety of human endeavor, which is why, for me, the Feynman quote above is the golden standard of intellectual honesty. When I encounter someone who insists on obscurity, who makes a virtue out of inexpressibility, who posits my lack of understanding as evidence of their sublimity, I think, well, physicists like Feynman who do great work in an intellectually forbidding, jargon-heavy field don’t let themselves off so easy, so what’s your excuse?