Richard Feynman’s path integral is both a powerful prediction machine and a philosophy about how the world is. But physicists are still struggling to figure out how to use it, and what it means.
BY CHARLIE WOOD
The most powerful formula in physics starts with a slender S, the symbol for a sort of sum known as an integral. Further along comes a second S, representing a quantity known as action. Together, these twin S’s form the essence of an equation that is arguably the most effective diviner of the future yet devised.
The oracular formula is known as the Feynman path integral. As far as physicists can tell, it precisely predicts the behavior of any quantum system—an electron, a light ray, or even a black hole. The path integral has racked up so many successes that many physicists believe it to be a direct window into the heart of reality.
“It’s how the world really is,” said Renate Loll, a theoretical physicist at Radboud University in the Netherlands.
But the equation, although it graces the pages of thousands of physics publications, is more of a philosophy than a rigorous recipe. It suggests that our reality is a sort of blending—a sum—of all imaginable possibilities. But it does not tell researchers exactly how to carry out the sum. So physicists have spent decades developing an arsenal of approximation schemes for constructing and computing the integral for different quantum systems.
The approximations work well enough that intrepid physicists like Loll are now pursuing the ultimate path integral: one that blends all conceivable shapes of space and time and produces a universe shaped like ours as the net result. But in this quest to show that reality is indeed the sum of all possible realities, they face deep confusion about which possibilities should enter the sum.
All Roads Lead to One
Quantum mechanics really got off the ground in 1926 when Erwin Schrödinger devised an equation describing how the wavelike states of particles evolve from moment to moment. The next decade, Paul Dirac advanced an alternative vision of the quantum world. His was based on the venerable notion that things take the path of “least action” to get from A to B—the route that, loosely speaking, takes the least time and energy. Richard Feynman later stumbled upon Dirac’s work and fleshed out the idea, unveiling the path integral in 1948.
The heart of the philosophy is on full display in the quintessential quantum mechanics demonstration: the double-slit experiment.
Physicists fire particles at a barrier with two slits in it and observe where the particles land on a wall behind the barrier. If particles were bullets, they’d form a cluster behind each slit. Instead, particles land along the back wall in repeating stripes. The experiment suggests that what moves through the slits is actually a wave representing the particle’s possible locations. The two emerging wavefronts interfere with each other, producing a series of peaks where the particle might end up being detected.
The interference pattern is a supremely strange result because it implies that both of the particle’s possible paths through the barrier have a physical reality.
The path integral assumes this is how particles behave even when there are no barriers or slits around. First, imagine cutting a third slit in the barrier. The interference pattern on the far wall will shift to reflect the new possible route. Now keep cutting slits until the barrier is nothing but slits. Finally, fill in the rest of space with all-slit “barriers.” A particle fired into this space takes, in some sense, all routes through all slits to the far wall—even bizarre routes with looping detours. And somehow, when summed correctly, all those options add up to what you’d expect if there are no barriers: a single bright spot on the far wall.
It’s a radical view of quantum behavior that many physicists take seriously. “I consider it completely real,” said Richard MacKenzie, a physicist at the University of Montreal.
But how can an infinite number of curving paths add up to a single straight line? Feynman’s scheme, roughly speaking, is to take each path, calculate its action (the time and energy required to traverse the path), and from that get a number called an amplitude, which tells you how likely a particle is to travel that path. Then you sum up all the amplitudes to get the total amplitude for a particle going from here to there—an integral of all paths…
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