This is a radical view of quantum behavior that many physicists take seriously. “I find it completely real,” he said Richard MacKenziephysicist on the University of Montreal.
But how can an infinite variety of curved paths add as much as a single straight line? Roughly speaking, Feynman’s scheme involves choosing each path, calculating its motion (the time and energy it takes to travel the trail), and getting a number from it called the amplitude, which tells you the way likely the particle is to travel that path. Then you add up all of the amplitudes to get the whole amplitude of the particle moving from here to there, the integral of all paths.
Naively, twisting paths look just as likely as straight paths since the amplitude for every individual path is similar size. But most significantly, the amplitudes are complex numbers. While real numbers represent points on a line, complex numbers act as arrows. The arrows point to different directions for various paths. And two arrows pointing away from one another add as much as zero.
The result’s that for a particle moving through space, the amplitudes of kind of straight paths point in essentially the identical direction, reinforcing one another. But the amplitudes of the winding paths point in all directions, so these paths work against one another. What stays is a linear path, showing how a single classical path of least motion emerges from infinite quantum options.
Feynman showed that his path integral is reminiscent of the Schrödinger equation. The advantage of Feynman’s method is a more intuitive recipe for coping with the quantum world: add up all the chances.
Sum of all waves
Physicists soon began to know particles as excitation in quantum fields—entities that fill space with values at every point. Where a particle can travel from place to position along different paths, the sector can undulate here and there in alternative ways.
Fortunately, the trail integral also works for quantum fields. “It’s obvious what to do,” he said Gerald Dunne, a particle physicist on the University of Connecticut. “Instead of summing up all paths, you sum up all of your field configurations.” You discover the initial and final arrangements of the sector, then consider every possible story that connects them.
The gift shop at CERN, home to the Large Hadron Collider, sells a coffee mug with the formula crucial to calculate the consequences of known quantum fields – a key entry into the trail integral.Courtesy of CERN/Quanta Magazine
Feynman himself relied on the integral path to develop quantum theory of the electromagnetic field in 1949. Others wondered the right way to calculate the actions and amplitudes for fields representing other forces and particles. When modern physicists predict the final result of a collision on the Large Hadron Collider in Europe, the trail integral underlies a lot of their calculations. The gift shop there even sells a coffee mug with an equation that may be used to calculate the important thing term of the trail integral: the motion of known quantum fields.
“It’s absolutely fundamental to quantum physics,” said Dunne.
Despite its triumph in physics, trajectory integrals make mathematicians sick. Even an easy particle moving through space has infinitely many possible paths. Things are worse with fields whose values can change in infinitely some ways in infinitely many places. Physicists have clever techniques for coping with a wobbly infinity tower, but mathematicians say the integral was never designed to operate in such an infinite environment.
