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.
Naïvely, swerving paths look just as likely as straight ones, because the amplitude for any individual path has the same size. Crucially, though, amplitudes are complex numbers. While real numbers mark points on a line, complex numbers act like arrows. The arrows point in different directions for different paths. And two arrows pointing away from each other sum to zero.
The upshot is that, for a particle traveling through space, the amplitudes of more or less straight paths all point essentially in the same direction, amplifying each other. But the amplitudes of winding paths point every which way, so these paths work against each other. Only the straight-line path remains, demonstrating how the single classical path of least action emerges from unending quantum options.
Feynman showed that his path integral is equivalent to Schrödinger’s equation. The benefit of Feynman’s method is a more intuitive prescription for how to deal with the quantum world: Sum up all the possibilities.
Sum of All Ripples
Physicists soon came to understand particles as excitations in quantum fields — entities that fill space with values at every point. Where a particle might move from place to place along different paths, a field might ripple here and there in different ways.
Fortunately, the path integral works for quantum fields, too. “It’s obvious what to do,” said Gerald Dunne, a particle physicist at the University of Connecticut. “Instead of summing over all paths, you sum over all configurations of your fields.” You identify the field’s initial and final arrangements, then consider every possible history that links them.
Feynman himself leaned on the path integral to develop a quantum theory of the electromagnetic field in 1949. Others would work out how to calculate actions and amplitudes for fields representing other forces and particles. When modern physicists predict the outcome of a collision at the Large Hadron Collider in Europe, the path integral underlies many of their computations. The gift shop there even sells a coffee mug displaying an equation that can be used to calculate the path integral’s key ingredient: the action of the known quantum fields.
“It’s absolutely fundamental to quantum physics,” Dunne said.
Despite its triumph in physics, the path integral makes mathematicians queasy. Even a simple particle moving through space has infinitely many possible paths. Fields are worse, with values that can change in infinitely many ways in infinitely many places. Physicists have clever techniques for coping with the teetering tower of infinities, but mathematicians argue that the integral was never designed to operate in such an infinite environment.
“It’s like black magic,” said Yen Chin Ong, a theoretical physicist at Yangzhou University in China who has a background in mathematics. “Mathematicians are not comfortable working with things where it’s not clear what’s going on.”
Yet it gets results that are beyond dispute. Physicists have even managed to estimate the path integral for the strong force, the extraordinarily complex interaction that holds together particles in atomic nuclei. They used two main hacks to do this. First, they made time an imaginary number, a strange trick that turns amplitudes into real numbers. Then they approximated the infinite space-time continuum as a finite grid. Practitioners of this “lattice” quantum field theory approach can use the path integral to calculate properties of protons and other particles that feel the strong force, overcoming rickety mathematics to get solid answers that match experiments.
“To someone like me in particle physics,” Dunne said, “that’s the proof that the thing works.”
Space-Time = The Sum of What?
The greatest mystery in fundamental physics, however, sits beyond experimental reach. Physicists wish to understand the quantum origin of the force of gravity. In 1915, Albert Einstein recast gravity as the result of curves in the fabric of space and time. His theory revealed that the length of a measuring stick and the tick of a clock change from place to place — that space-time is a malleable field, in other words. Other fields have a quantum nature, so most physicists expect that space-time should too, and that the path integral should capture that behavior.