Feynman's Infinite Quantum Paths

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Matt O'Dowd (1:02)

Quantum mechanics seems to imply that all possible properties, paths, or events that could reasonably occur between our measurements do occur. Whether or not this is true, a mathematical description of this crazy idea led to the most powerful expression of quantum mechanics ever devised. Richard Feynman's path integral formulation.

There is a fundamental limit to the knowability of our universe. The Heisenberg uncertainty principle tells us that the more precisely we try to define one property, the less definable is its counterpart. Knowing a particle's location perfectly means its velocity is unknowable. But unmeasured properties are not just uncertain, they are fundamentally undefined. One of the most profound illustrations of this is the famous double-slit experiment. If you aren't familiar, check out our previous episode that discusses it in detail. You might also want to catch up on the first two in our quantum field theory playlist, because we are going to be building on that. But the too-long-didn't-watch for the double-slit experiment is this. A particle, say a photon or an electron, travels through a barrier containing two slits to a screen. Its initial and final positions are known, but what path does it take in between? Which slit does it go through? The interference pattern produced by particles on the screen can only be explained if each of them travels through both slits, not as a particle, but as a wave that fills the intervening space, interacts with itself, and defines the probability of that particle actually showing up at a given point on the screen. There's a story about a quantum mechanics professor explaining the double-slit experiment to a class.

The prof showed how the locations of the particles on the screen can be calculated by adding the amplitude of a wave passing through one slit to the amplitude of a wave passing through the other slit. One impertinent student asked, What happens if you cut a third slit in the barrier? The professor replied, Obviously you have to add together the amplitudes of waves passing through all three slits.

The student pushed it. But what if you add four slits? What about five? To which the agitated professor repeated, Add the amplitudes of four slits, five slits, etc.

Feeling cocky, the student asked, What about if you cut infinite slits, so there's no more screen? And then what if you add a second screen with infinite slits?

As the probably apocryphal story goes, that student was Richard Feynman, and he just outlined the basics of what was to become the path integral formulation of quantum mechanics. It was a simple idea, but it led to the most elegant formulation of quantum mechanics ever devised, and became a key to quantum field theory, that in turn has provided us with our clearest understanding of the fundamental structure of the subatomic world.

The idea is essentially this. To know the likelihood of a particle traveling between two points, A to B, we need to take into account all of the conceivable ways that could happen. The double slit experiment is a special case where we only think about two possible paths. But when something travels through empty space, it's like it's traveling through infinitely packed barriers, each with infinite slits. There are infinite possible paths, and Feynman actually figured out a way to combine the infinite paths to give a very real, finite probability of a particle reaching its final destination. His trick was to slice the time taken for the journey into small intervals, and at each time step allow the particle to take any conceivable straight line step in space. That gives a set of paths from A to B, some of which look sensible, but most of which are ridiculous. For example, there are paths that loop in circles or take detours to the edge of the universe. There was absolutely no physics in this description so far, not even the limit of the speed of light. The amazing thing about the path integral formulation is that Feynman added one and only one piece of real physics. From that, it was possible for him and others to re-derive all of quantum mechanics.