Feynman's Infinite Quantum Paths, QFT Pt. 3 (Script) (Patreon)

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

Let’s talk about the double-slit experiment. This is one of the most profound experimental results in all of physics. If you aren’t familiar, check out our previous episode that discusses it in detail. But the too-long-didn’t-watch 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 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 any given point IF we were to try to localize it there.

There’s a story about a quantum mechanics professor explaining the double slit experiment  to a class. The prof showed how locations of 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. 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 about 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’d 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 travels 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. Feynman actually figured out a way to combine the infinite possible paths to give a very real, finite probability of a particle reaching a given 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.

That piece of physics was the “principle of least action”, and it was borrowed from old-school classical physics. It states that an object will always follow the path that minimizes this quantity called the action. The action is tricky to define; it’s proportional both to the change in energy over a path and the travel time. In relativity it’s proportional to the proper time – so the time measured by the clock on a given trajectory. For the large-scale, classical universe, minimizing proper time let’s you derive all equations of motion. Objects always take the path of least action. Basically, the universe is lazy.

However in the quantum universe, there is no single path. Feynman instead used quantum action to assign an importance – a weight – to each of the infinite paths that a single particle could take. Then, using the miracle of calculus, he was able to add up the contributions from all of those infinite possible paths to find the probability of a particle making that simple journey from A to B.

Feynman’s paths don’t each have some separate probability of occurring; instead each path contributes what we call a probability amplitude to the entire A-B journey. 

OK, a quick quantum 101 aside. Schrodinger’s wavefunction and Feynman’s path integral describe this probability amplitude thing. Regular probabilities have to be positive – there’s can’t be a negative chance of something happening. However probability amplitudes can be positive or negative. If you add two paths or wavefunctions with a positive and a negative amplitude they cancel each other out. The total probability of, say, finding an object in a certain location is the square of the probability amplitudes at that location. That probability is always positive, but it can end up very small if the probability amplitudes cancel. This relationship between probability and the amplitude of the wavefunction is the Born Rule, by the way, and no one has any idea why it works.

OK, so when Feynman used this action quantity to figure out the probability amplitudes of his infinite paths, something amazing happened. All of the really crazy paths cancelled each other out, leaving mostly just the sensible paths. That doesn’t mean that only one of these sensible paths actually happened - ALL paths happened, but some just don’t happen with much amplitude. Whatever that means.

Feynman’s path integral formulation allowed him to derive the Schrodinger equation from scratch. With a bit more work and help from others – like figuring out how to add particles with spin – the path integral approach is both mathematically equivalent AND more powerful to earlier derivations of quantum mechanics. That power comes from the principle of least action. This action quantity is a function of a particle’s path through spacetime. That means it treats space and time symmetrically and so works very naturally with Einstein’s theory of special relativity. On the other hand, Schrodinger’s equation gave time a special role so doesn’t work with relativity at all. We already saw how Paul Dirac managed to fix this two decades earlier with his Dirac equation. But Feynman’s solution produced a quantum mechanics that didn’t need fixing.

But perhaps the greatest power of the path integral is that it very naturally converts into a true quantum field theory. See, when I say there are lots of ways for a particle to travel from point A to B I mean LOTS. It’s not just that a particle can travel infinite physical paths. Also infinite things can happen to the particle on the way. And you have to account for ALL of them. For example a photon traveling between two points could spontaneously become an “virtual” electron-positron pair, before they annihilate back into the original photon. And a traveling electron could emit and re-absorb a photon, which itself could make its own particle-antiparticle pair, ad infinitum. And let’s not even get started with the complexity of two or more particles interacting.

The path integral method is able to deal with all of this weirdness because it’s able to describe a universe of oscillating fields just as well as it can describe a universe of moving particles. Instead of adding up all possible paths that particles can take, you can instead add up all possible histories of quantum fields. So a photon is an excitation – a vibration – in the electromagnetic field. Its motion can be described as changes in that excitation. The quantum action principle gives the probability amplitude of changes in the state of the field. That includes motion of the photon, but also the probability amplitude of the photon’s energy moving from the electromagnetic field into, say, the electron field, where it might become an electron-positron pair. In the quantum field version of path integrals, we can describe all possible paths AND all possible events for that simple journey between A and B.

This is pretty crazy. If we have to take those infinite simultaneous paths seriously then we also have to take those infinite intervening events seriously. One interpretation of the path integral formulation is that everything in between A and B that can happen does. However unlike the more ridiculous infinite trajectories a particle can take, those infinite events don’t cancel out nearly so neatly. In fact they lead to rampant, uncontrolled, infinite probabilities. It was a major challenge to fix these. One powerful tool in making sense of these infinite possible events also came from Richard Feynman. Specifically, Feynman diagrams. How can a bunch of doodles be used to tame the infinities? In part by describing anti-matter as regular matter traveling backwards in time. Next on Space Time.