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.

Naively, 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.

The gift shop at CERN, which houses the Large Hadron Collider, sells a coffee mug featuring a formula that’s necessary to calculate the action of the known quantum fields — the key input to the path integral.Courtesy of CERN/Quanta Magazine

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.

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