It’s a radical view of quantum behavior that many physicists take seriously. “I consider it absolutely real,” said Richard McKenzie, a physicist at the University of Montreal.
But how can an infinite number of paths of curvature form a single straight line? The Feynman diagram, roughly speaking, is to take each path, calculate its effect (the time and energy required to pass the path), and from it we get a number called the amplitude, which tells you how likely it is that the particle will go that path. Then you sum all the amplitudes to get the total amplitude of a particle traveling from here to there – which is an integral of all trajectories.
Naively, skew paths seem just as likely as straight paths, because the amplitude of any individual path is of the same magnitude. Crucially though, the amplitudes are complex numbers. While real numbers locate 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, their sum is zero.
The result is that for a particle traveling through space, the amplitudes of more or less straight trajectories all essentially point in the same direction, amplifying each other. But the widening of the winding paths points in every direction, so these paths run against each other. Only the straight-line path remains, showing how the single classical path of least action emerges from endless quantum choices.
Feynman showed that his path integral is equivalent to the Schrödinger equation. The benefit of the Feynman method is a more intuitive prescription for how to approach the quantum realm: sum up all possibilities.
The 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 may move from one place to another along different trajectories, the field may ripple here and there in different ways.
Fortunately, path integration works with quantum fields, too. “It’s obvious what to do,” said Gerald Dunn, a particle physicist at the University of Connecticut. “Instead of collecting all paths, you collect all of your field configurations.” You determine the field’s initial and final arrangements, and then consider each possible date that connects them.
The gift shop at CERN, which houses the Large Hadron Collider, sells a cup of coffee containing a formula needed to calculate the effect of known quantum fields — the main entrance to the integrated path.Courtesy of CERN/Quanta Magazine
Feynman himself relied on the integral path to develop a quantum theory of the electromagnetic field in 1949. Others would work out how to calculate the acts and amplitudes of fields representing forces and other particles. When modern physicists predicted the outcome of a collision at the Large Hadron Collider in Europe, the path integral is the basis for many of their calculations. The gift shop there even sells a coffee mug displaying an equation that can be used to calculate the key component of the path integral: the action of known quantum fields.
“It’s absolutely fundamental to quantum physics,” Dunn said.
Despite its triumph in physics, integration makes mathematicians restless. Even a simple particle moving through space has an infinite number of possible trajectories. Fields are even worse, with values that can change in countless many ways in many places. Physicists have clever techniques for dealing with the swinging tower of infinities, but mathematicians argue that the integral was never designed to work in such an infinite environment.