The Closest We Have to a Theory of Everything

in english they talk about a theory of everything in german we talk about the belt formula the world equation i've always disliked the german expression that's because equations in by themselves don't tell you anything take for example the equation x equals y that may well be the world equation the question is just what's x and what is y however in physics we do have an equation that's pretty damn close to a world equation it's remarkably simple looks like this and it's called the principle of least action but what's s and what's this squiggle that's what we'll talk about today the principle of least action is an example of optimization where the solution you're looking for is optimal in some quantifiable way optimization principles are everywhere for example equilibrium economics optimizes the distribution of resources at least that's the idea natural selection optimizes the survival of offspring if you shift around on your couch until you're comfortable you're optimizing your comfort what these examples have in common is that the optimization requires trial and error the optimization we see in physics is different it seems that nature doesn't need trial and error what happens is optimal right away without trying out different options and we can quantify just in which way it's optimal i'll start with a super simple example suppose a lonely rock flies through outer space far away from any stars or planets so there are no forces acting on the rock no air friction no gravity nothing let's say you know the rock goes through point a at a time we'll call ta and later through point b at the time t b what path did the rock take to get from a to b well if no force is acting on the rock it must travel in a straight line with constant velocity and there is only one straight line connecting the two dots and only one constant velocity that will fit to the duration it's easy to describe this particular path between the two points it's the shortest possible path so the path which the rock takes is optimal in that it's the shortest this is also the case for rays of light that bounce off a mirror suppose you know the ray goes from a to b and you want to know which path it takes you find the position of point b in the mirror draw the shortest path from a to b and reflect the segment behind the mirror back because that doesn't change the length of the path the result is that the angle of incidence equals the angle of reflection which you probably remember from middle school this principle of the shortest path goes back to the greek mathematician hero of alexandria in the first century so not exactly cutting-edge science and it doesn't work for a fraction in a medium like for example water because the angle at which array of light travels changes when it enters the medium this means using the length to quantify how optimal a path is can't be quite right in 1657 pier de fomar figured out that in both cases the path which the ray of light takes from a to b is that which requires the least amount of time if there's no change of medium then the speed of light doesn't change and taking the least time means the same as taking the shortest path so reflection works as previously but if you have a change of medium then the speed of light changes too let's just use the previous example with a tank of water and let us call the speed of light in rc1 and the speed of light in water c2 we already know that in either medium the light ray has to take a straight line because that's the fastest you can get from one point to another at constant speed but you don't know what's the best point for the ray to enter the water so that the time to get from a to b is the shortest but that's pretty straightforward to calculate we give names to these distances calculate the length of the paths as a function of the point where it enters multiply each path with the speed in the medium and add them up to get the total time now we want to know which is the smallest possible time if we change the point where the ray enters the medium so we treat this time as a function of x and calculate where it has a minimum so where the first derivative with respect to x vanishes the result you get is this and then you remember that those ratios with square roots here are the signs of the angles et voila verma may have said this is the correct law of refraction this is known as the principle of least time or as fema's principle and it works for both reflection and refraction let us pause here for a moment and appreciate how odd this is the ray of light takes the path that requires the least amount of time but how does the light know it will enter a medium before it gets there so that it can pick the right place to change direction it seems like the light needs to know something about the future crazy it gets crazier let us go back to the rock but now we do something a little more interesting namely throw the rock in the gravitational field for simplicity let's say the gravitational potential energy is just proportional to the height which is too good precision correct near the surface of earth again i tell you the particle goes from point a at time t a to point b at time t b in this case the principle of least time doesn't give the right result but in the early 18th century the french mathematician mo patra figured out that the path which the rock takes is still optimal in some other sense it's just that we have to calculate something a little more difficult we have to take the kinetic energy of the particle subtract the potential energy and integrate this over the path of the particle this expression the time integral over the kinetic minus potential energy is the action of the particle i have no idea why it's called that way and even less do i know why it's usually abbreviated s but that's how it is this action is the s in the equation that i showed at the very beginning the thing is now that the rock always takes the path for which the action has the smallest possible value you see to keep this integral small you can either try to make the kinetic energy small which means keeping the velocity small or you make the potential energy large because that enters with the minus but remember you have to get from a to b in a fixed time if you make the potential energy large this means the particle has to go high up but then it has a longer path to cover so the velocity needs to be high and that means the kinetic energy is high if on the other hand the kinetic energy is low then the potential energy doesn't subtract much so if you want to minimize the action you have to balance both against each other keep the kinetic energy small but make the potential energy large the path that minimizes the action turns out to be a parabola as you probably already knew but again note how weird this is it's not that the rock actually tries all possible paths it just gets on the way and takes the best one on first try like it knows what's coming before it gets there what's this squiggle in the principle of least action well if we want to calculate which path is the optimal path we do this similarly to how we calculate the optimum of a curve at the optimum of a curve the first derivative with respect to the variable of the function vanishes if we calculate the optimal path of the action we have to take the derivative with respect to the path and then again we ask where it vanishes and this is what the squiggle means it's a sloppy way to say take the derivative with respect to the path and that has to vanish which means the same as that the action is optimal and this is usually a minimum hence the principle of least action okay you may say but you don't care all that much about paths of rocks all right but here's the thing if we leave aside quantum mechanics for a moment there's an action for everything for point particles and rocks and arrows and that stuff the action is the integral over the kinetic energy minus potential energy but there's also an action that gives you electrodynamics and there is an action that gives you general relativity in each of these cases if you ask what the system must do to give you the least action then that's what actually happens in nature you can also get the principle of fleece time and of the shortest path back out of the least action in special cases and yes the principle of least action really uses an integral into the future how do we explain that well it turns out that there is another way to express the principle of least action one can mathematically show that the path which minimizes the action is that path which fulfills a set of differential equations which are called the euler lagrange equations for example the euler lagrange equations of the rocket sample just give you newton's second law the oil lagrange equations for electrodynamics are maxwell's equations the all lagrange equations for general relativity are einstein's field equations and in these equations you don't need to know anything about the future so you can make this future dependence go away what's with quantum mechanics in quantum mechanics the principle of least action works somewhat differently in this case a particle doesn't just go one optimal path it actually goes all paths each of these paths has its own action it's not only that the particle goes all paths it also goes to all possible end points but if you eventually measure the particle the wave function collapses and the particle is only in one point this means that these paths really only tell you probabilities for the particle to go one way or another you calculate the probability for the particle to go to one point by summing overall paths that go there this interpretation of quantum mechanics was introduced by richard feynman and is therefore now called the definement path integral what happens with the strange dependence on the future in the final and path integral well technically it's there in the mathematics but to do the calculation you don't need to know what happens in the future because the particle goes to all points anyway except it doesn't in reality it goes to only one point so maybe the reason we need the measurement postulate is that we don't take this dependence on the future which we have in the path integral seriously enough this video was sponsored by brilliant yes we use a lot of mathematics and physics in my mind mathematics is kind of a code that we decipher to understand nature but it isn't as difficult as it looks the principle of least action is basically a more elaborate version of finding the optimum of a curve and if you need to freshen up your knowledge on calculus or maybe on differential equations check out brilliant brilliant is an amazing tool for learning that offers interactive courses on a large variety of subjects in science and mathematics i like working with them because their courses fit so well to the topics i'm talking about for this video have a look for example at the course calculus in a nutshell and those on differential equations a big part of understanding the principle of least action is really just getting used to what the maths means and brilliant will really help you make sense of the symbols with brilliant you can learn at your own pace and whenever you can find the time and all their courses will challenge you with questions so you can check your understanding along the way if you want to try it out use our link brilliant.org sabine and sign up for free the first 200 subscribers using this link will get 20 off the annual premium subscription thanks for watching see you next week

Channel: Sabine Hossenfelder

Views: 459,509

Keywords: what is a theory of everything, physics, do we have a theory of everything, principle of least action, least action, least path, Fermat's principle, equation for everything, how close are we to a theory of everything, mathematics, mathematical physics, theory of everything, hossenfelder, science without the gobbledygook

Id: A0da8TEeaeE

Length: 13min 28sec (808 seconds)

Published: Sat May 21 2022