The Quantum Labyrinth

by Andrew Crumey

Review of The Quantum Labyrinth by Paul Halpern. The Wall Street Journal, 8 December 2017.

What's the square root of four? If you said two you're half right: the full answer is plus or minus two. Every number has two square roots, and a similar doubling can occur in equations of physics. When Paul Dirac melded quantum mechanics with special relativity to obtain an expression for negatively charged electrons, he realised there must be an additional solution with positive charge, later found experimentally and dubbed the positron. For a certain young graduate student and his supervisor, an analogous trick suggested an even more remarkable reversal: that of time itself. Hence the subtitle of Paul Halpern's book: "How Richard Feynman and John Wheeler Revolutionized Time and Reality".

The labyrinth of the main title aptly signals the problem Professor Halpern faced as author. Should it be a dual biography of Wheeler and Feynman, or an exposition of their ideas? An exploration of their key early collaboration, or a wider survey of their long careers? In fact Professor Halpern attempts all of these, to differing degrees and with varying levels of success. He is at his best when explaining concepts of physics, the subject he teaches at the University of the Sciences in Philadelphia. Less satisfying is his effort to convey the personalities of the two geniuses, whom he generally renders by way of simple contrasts: Wheeler the staid philosopher, Feynman the impish bongo-drummer. Admittedly, much has already been written about both men, especially Feynman, whose complexities and flaws are not explored here. And inevitably it is the colourful Feynman who tends to steal the show.

As a student in the 1930s, learning quantum theory from Dirac's writings, Feynman asked: can an electron interact with itself? Standard arguments said yes: the particle should be affected by its own electromagnetic field, rather like someone finding difficulty running because of their own weight. Yet this led to "divergences": the calculated self-interaction was infinite. Curing the problem would eventually win Feynman the Nobel Prize, though only after spending years on an incorrect approach.

Suppose there was no field, instead only particles magically influencing each other at a distance without any intermediary substance. It would answer the question of self-interaction, but could it be possible? Wheeler, Feynman's thesis supervisor at Princeton, immediately saw both a problem and its possible solution. Shake an electron and another far off will shake in response: this is how our television and phone networks can function, and the signals are mathematically described by equations found by James Clerk Maxwell in the 1860s. If Feynman's proposal was correct, then surely the wiggling of the receiving electron should in turn influence the sender, potentially leading to the same infinite circularity Feynman sought to avoid. But like the square root of four, Maxwell's equations have an extra solution: a signal going backwards rather than forwards in time. It had always been dismissed as an artefact; now Wheeler suggested it as reality, a way of cancelling the surplus interaction. He went even further: Dirac's positron could likewise be an electron in reverse. Together they hatched what came to be called "absorber theory". A nervous young Feynman presented it before an audience that included none other than Princeton's most famous resident, the elderly Albert Einstein who was, says Professor Halpern, "friendly but neutral". Others such as Wolfgang Pauli were more hostile, but Wheeler and Feynman pressed on with their theory.

They were interrupted by global events: Feynman's involvement with the Manhattan Project and brief marriage to the tragically sickly Arline Greenbaum provide a compelling interlude in Professor Halpern's narrative. Then as war ended, Feynman returned to the absorber theory. Its key feature was a mixing up of times - past and future - in contrast to the more usual procedure of modelling systems at successive instants. This proved to be its strongest point, even when the theory itself fell foul of experimental results that contradicted it. As Professor Halpern ably explains, Feynman had developed a calculation technique well suited to the time-distorting effects of special relativity. Reversing his previous position, Feynman now insisted that electrons really do self-interact - and with his new technique, called a "path integral", he could recast quantum theory in a way that eliminated the previous infinite results, employing a cancelling procedure dubbed "renormalization". This was the work that eventually took him to Stockholm where, in his 1965 Nobel Lecture, Feynman said that as a young man he had fallen in love with absorber theory, and "like falling in love with a woman, it is only possible if you do not know much about her, so you cannot see her faults." He had readily abandoned that theory when it no longer served his purpose: the sign of an excellent theoretician, though not necessarily the best companion, as people such as the second of his three wives might have attested.

Feynman's path integral splits quantum events into a sum of all possible ways they might occur. Wheeler called it a "sum over histories", a notion subsequently likened to the alternate-history scenarios of Philip K. Dick or the branching realities of Jorge Luis Borges's famous story "The Garden of Forking Paths". Feynman opposed that interpretation, but it is the Borgesian labyrinth that Professor Halpern's title hints at; and as his narrative continues we encounter others among Wheeler's students, including Hugh Everett and Bryce DeWitt, whose "many worlds" interpretation of quantum mechanics has further stimulated scientific speculations about multiple histories and parallel universes. In fact, as the theorists and theories multiply - reflecting the broad interests of Wheeler, Feynman and the generation they mentored or inspired - we almost seem in danger of another kind of infinite divergence. Every book, says Professor Halpern, requires an "Ariadne's thread... a linear narrative that serves as a guide through the greater labyrinth of information." The lives and careers of Wheeler and Feynman are meant to be the main thread here, but one must sometimes look hard to see its trail through the tempting tangle of corridors. Their strange joint theory might have been thread enough, no less absorbing or amazing.