[MUSIC PLAYING] 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 the most powerful expression of quantum mechanics ever devised-- Richard Feynman's path integral formulation.

[MUSIC PLAYING] 12 00:00:31,140 --> 00:00:34,730 There's 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.

It's 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 had four slits?

What about five?

To which the agitated professor repeated, add the amplitudes of four slits, five slits, et cetera.

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 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.

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 we call the action.

The action is tricky to define.

It's proportional to both the transfer between kinetic and potential 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 lets 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.

Schrodinger's wave function and Feynman's path integral describe this probability amplitude thing.

Now, regular probabilities are normal positive numbers between 0 and 1.

However, probability amplitudes are what we call complex numbers.

We can think of each path's probability amplitude as a vector or an arrow of a certain length and direction in an imaginary 2D space.

The total length of each arrow gives the probability of that path being taken.

But to get the total probability that a particle travels from A to B, you connect the probability amplitude arrows for all possible paths end to end.

The length of the arrow connecting the start and the end of this chain represents the total probability from all paths.

But if two parts contribute individual probability amplitudes that point in exactly opposite directions, they cancel each other out.

This is equivalent to the wave function along those paths being perfectly out of phase when they reach the destination.

In this case, they destructively interfere with each other.

Now, 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 cancel each other out.

Only the most sensible paths, those with the least action, added significantly to the probability.

The familiar paths of the large-scale classical world are just the small set of infinite possible paths that don't cancel each other out.

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 to and more powerful than earlier derivations of quantum mechanics.

That power comes from the principle of least action.

This action quantity is a function of the particle's path through space-time.

That means it treats space and time symmetrically and so works very naturally with Einstein's theory of special relativity.

On the other hand, the Schrodinger equation gave time a special role, so it doesn't work with relativity at all.

We already saw how Paul Dirac manage 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 a virtual electron-positron pair before they annihilate back into the original photon.

And a traveling electron could emit and reabsorb a photon, which itself could make its own particle-antiparticle pair ad infinitum.

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 instead add up all possible histories of quantum fields.

So a photon is an excitation of 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 it also includes the probability amplitude of a 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 from A to 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 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 things.

One powerful tool in making sense of these infinite possible events also came from Richard Feynman, specifically Feynman diagrams.

But how can a bunch of doodles be used to tame the infinities?

In part, by describing antimatter as regular matter traveling backwards in time.

Find out how on the next episode of "Space Time."