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'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. 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 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 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. Schrödinger's wave function, and Feynman's path integral describe this probability amplitude thing. A regular probability is a normal, positive number 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 paths 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 Schrödinger equation from scratch. With a bit more work, and the 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 particles 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 Schrödinger equation gave time a special role, so it 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 to 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, 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 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 these. 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. Thanks a tonne to all of our supporters on Patreon. You make this show possible. And a special big thanks to Max Levine for supporting us at the quasar level. The universe may obey the Principle of Least Action, but Mr. Levine is more of a maximum action guy. Thanks for your help. It means a lot. Actually, we want to show you all a little more than verbal thanks. We made something for you! On August 21st, for the first time in 40 years, the United States will experience a total solar eclipse. Uh, no, we didn't make the eclipse, but astrophysicists agree that the most stylish way to view this event is with the official Space Time eclipse glasses. That's why we're sending eclipse glasses to all of our patrons who are contributing at the $5 level or above. If you're not currently a patron, that's okay. Anyone who joins us at the $5 level, or increases their pledge to the $5 level, during the month of July will also receive glasses. We have 2,400 glasses to give away. We're hoping that will cover everyone, but if we do reach our limit then glasses will be sent out on a "first come, first served" basis, and this is a really great way to simultaneously support the show and to not burn your eyes out staring at the Sun. Okay, now let's get to your comments on our episode on quantum electrodynamics, "The First Quantum Field Theory". Jakub asks, 'What is the difference between the electromagnetic field of quantum field theory and the æther?' That's a great question. As a reminder, the æther, or more precisely the luminiferous æther, was proposed in the late 19th century as the medium for the propagation of light waves. It was imagined to be very closely analogous to air as the medium for propagation of sound waves. Just like air, the medium was imagined to have some preferred reference frame in which it was motionless. If true, then it should be possible to be moving relative to that frame. That should cause different observers to measure a different speed of light. The Michelson-Morley experiment disproved that idea. So, the æther as imagined in the 19th century doesn't exist, but the EM field does. The crucial difference is that this field has no preferred reference frame. No matter what speed you're traveling, it's as though the field is stationary with respect to you. Satya points out that if matter and antimatter particles are always created in pairs, shouldn't they be just as much antimatter as matter in the universe? Well, good point. There should. Up to around a millionth of a second after the big bang, the universe was hot enough for photons to be continuously forming matter-antimatter pairs. At some point, it became cool enough for this process to stop. Almost all the matter and antimatter annihilated each other, leaving only one in a billion particles of matter. This tiny imbalance of matter over antimatter is a deep mystery. It indicates a break in what we once thought to be a fundamental symmetry of the universe. This so-called charge parity, or CP, violation has been seen in experiment, implying the universe does treat antimatter differently to matter. The details might also make for a future episode. A few of you asked whether quantum field theory and string theory are the same thing. Well, the answer is no. Quantum field theory describes particles as a field vibration in 4D space-time, and each elementary particle has its own field. String theory states that all particles are different vibrational modes in one-dimensional objects called strings. These in turn may exist in a space of 11 or even 26 dimensions, most of which are compactified. While QFT has been tested with stunning success and is the foundation of the standard model, string theory is untested and may have nothing to do with reality. But whatever, it's fun stuff, so we're going to get to it anyway.
Love me some PBS Space Time.
This channel is so, so good. It's an example of the perfect intermediate-level science educational programming that is seriously lacking. It's essentially a general relativity/QM course without a focus on HOW the mathematics works, but WHY it works.
Actually explaining physics by highlighting the mathematical concepts is honestly the only true way to explain physics to a general audience. I mean these videos are 100 times better than any Neil degrasse Tyson video
Where would be a good place to learn about the mathematical formulation of the concepts mentioned in this video?
Great video.
Does anybody know someone of similar video style that actually dives into the math a little deeper? This guy is great but I'd like to see more of the math.
Key take-away point :
Quantum mechanics is not "opposed" to classical mechanics. Rather classical mechanics is a limiting case of the larger and more abstract quantum mechanics. When you take the limit as Planck's constant approaches zero, then "at the bottom of the chalkboard" after some algebra, the Principle of Least Action pops out.
Large objects like basketballs and cars are so large compared to atoms , that Planck's Constant is "essentially" zero for them. This is why the approximation works.