It is a radical view of quantum behavior that many physicists take seriously. “I think it’s absolutely true,” says Richard McKenzie, a physicist at the University of Montreal.
But how can an infinite number of winding roads add up to a single straight line? Feynman’s plan, roughly, is to take each path, calculate its action (the time and energy it takes to traverse the path), and then find a number called the amplitude, which tells you how likely the particle is to travel that path. Then sum up all the magnitudes to get the total number of particles going from here to there – the integral of all the paths.
Naively, circular paths look just like straight ones, because the width of any individual path is the same. Crucially, however, amplitudes are complex numbers. While real numbers represent points on a line, complex numbers act as arrows. The arrows point in different directions for different paths. And two arrows away from each other to sum to zero.
From a higher perspective, for a particle traveling through space, the width of more or less straight paths all point essentially in the same direction, amplifying each other. But the width of the curved paths indicates each path, so these paths work on each other. Only the straight-line path remains, showing how a single classical minimal step emerges from the infinite quantum possibilities.
Feynman showed that the dominant path is equivalent to the Schrödinger equation. The benefits of Feynman’s method are a more intuitive prescription for how to interact with the quantum world: sum up all possibilities.
Sum of all Ripples
Physicists soon understood particles as catalysts in quantum fields – entities that fill space with values at each point. Where a particle can move from place to place in different ways, a field can explode here and there in different ways.
Fortunately, the main way works for quantum fields as well. “It’s clear what he has to do,” said Gerald Dunn, a particle physicist at the University of Connecticut. “Instead of summarizing all traces, they sum up all the configurations of your fields.” You identify the first and last events of the field, then consider each story that connects them.
Feynman in In 1949 he relied on a unified approach to develop the quantum theory of the electromagnetic field. Others were figuring out how to calculate functions and amplitudes representing other forces and particles. When modern physicists predicted the results of the Large Hadron Collider in Europe, the main path underpinned many of their calculations. The gift shop there sells coffee mugs depicting the equation used to calculate the main ingredient of the trail: a function of known quantum fields.
“It’s absolutely fundamental to quantum physics,” Dunn said.
Despite his triumphs in physics, the mainstream makes mathematicians suspicious. Even a simple particle moving through space has an infinite number of possible paths. Fields are extreme, and the values can change in an infinite number of ways. Physicists have clever techniques to deal with the tower of infinity, but mathematicians argue that integration was never designed to work in an infinite environment.