How reality can be a sum of all possible realities

It’s a radical view of quantum behavior that many physicists take seriously. “I consider it completely real,” he said Richard MacKenziePhysicist from the University of Montreal.

But how can an infinite number of curved paths add up to a single straight line? Feynman’s scheme, roughly speaking, is to take each path, compute its action (the time and energy required to travel the path), and from there derive a number called the amplitude, which indicates the probability that a particle will walk that path. Then you add up all the amplitudes to get the total amplitude of a particle going from here to there: an integral of all paths.

Naively, deviating paths seem just as likely as straight ones, because the width of any individual path is the same size. Crucially, however, the 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 moving away from each other add up to zero.

The result is that, for a particle traveling through space, the amplitudes of more or less straight paths point in essentially the same direction, amplifying each other. But the widths of the winding paths point in all directions, so these paths work against each other. Only the straight-line path remains, demonstrating how the only classical path of least action emerges from endless quantum options.

Feynman proved that his path integral is equivalent to Schrödinger’s equation. The benefit of the Feynman method is a more intuitive recipe for how to deal with the quantum world: add up all the possibilities.

sum of all waves

Physicists soon came to understand particles as excitations in quantum fields—space-filling entities with values ​​at each point. Where a particle can move from one place to another along different paths, a field can undulate here and there in different ways.

Fortunately, the path integral also works for quantum fields. “It’s obvious what needs to be done,” he said. Gerald Dunne, a particle physicist at the University of Connecticut. “Instead of adding up all the paths, you add up all of your field settings.” You identify the initial and final arrangements of the field, then consider all the possible antecedents that link them.

Feynman himself leaned on the integral path to develop a quantum theory of the electromagnetic field in 1949. Others would discover 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 is the basis for many of their calculations. The gift shop even sells a coffee mug that shows an equation that can be used to calculate the key ingredient of the path integral: the action of known quantum fields.

“It’s absolutely fundamental to quantum physics,” Dunne said.

Despite its triumph in physics, the path integral worries mathematicians. Even a single particle moving through space has an infinite number of possible paths. Fields are worse, with values ​​that can change in infinite ways in infinite places. Physicists have clever techniques for dealing with the teetering tower of infinities, but mathematicians argue that the integral was never designed to operate in such an infinite environment.