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I was reading the comments in an earlier video, when I saw a question from a viewer. They asked "Hey Dr. Don- what does that equation on your shirt mean?" That's actually pretty interesting, and that seems like something we want to do. Let's do it. (intro music) If you are one of those people who is interested in particle physics- and, seriously- how could you not be- you've encountered this equation. You've may have even heard that it is the equation of the standard model of particle physics. So, what, exactly does it mean? And is it useful? First of all, I should identify what this equation is called. It is called a Lagrangian.
The name Lagrangian is a generic one which is found throughout physics. It's a type of equation that allows scientists to predict the motion of both particles and fields. It's not a new idea. The Lagrangian was first introduced in 1788, when Joseph-Louis Lagrange published his book "Mécanique Analytique." Many of you are aware of how Isaac Newton formulated the laws of physics, which describe how objects move under the direction of a force. If you ever took a physics class, you're familiar with his second law of motion,
which says that the force equals mass times acceleration, and it is written as F = ma. There is nothing wrong with this formulation of physics, but it can be very difficult to use. Because forces and accelerations are vectors, this means that you often have many different components and the equations can get very thorny. However, the Lagrangian formulation is both equivalent to the Newtonian approach, and often much easier to write down. Furthermore, essentially all modern subatomic physics work is done using Lagrangians. So, if you're serious about doing physics, you have to learn about them. Rather than using forces and accelerations, a Lagrangian is built using energy. Energy, being a scalar- which means it has no
components- is much easier to work with. Okay- so just how do you make a Lagrangian? The full answer can be a bit tricky, but in the world of standard, classical physics, the Lagrangian is usually written as the Lagrangian equals kinetic energy minus potential energy. And, if one writes the equation in the most common way, using accepted symbols, one gets L equals T minus V. T stands for kinetic energy and V represents potential. Bizarre choice of symbols, I know, but whaddya do? Now there is so much you can say about how to use the Lagrangian to solve for the motion of particles and fields, but the fact is that the
math can be daunting. Not only does it require calculus and differential equations, but it also requires partial differential equations and what is called the calculus of variations. And I briefly toyed with the idea of digging into the math- showing all the gory details- but then my video editor threw cold water on that idea. Such a video would really be appropriate only for advanced physics students and not most viewers. So, I decided to compromise. If you're one of those physics students, I found a website that I very much like that explains the Lagrangian idea
in a fairly intuitive way. It's really very helpful and I wish I had known about it when I was a student. I put a link in the description below that points to this fantastic resource. But, for the rest of us, let's skip the deep dive and just think of the Lagrangian as a versatile and modern way to write down the laws of motion. So let's get back to the standard model Lagrangian. What does it mean? Well, let's start with this one here. It was originally written by John Ellis back in 2007 as part of a photo shoot for the CERN media department. This equation is pretty simple. The first term - the one with the F's - that line describes the behavior of the subatomic forces, specifically
the strong and weak forces and electromagnetism, and how the forces interact among themselves. The second line - the one with the D with a slash through it - describes the particles - specifically the fermions, like quarks and leptons, and how they interact with the subatomic forces. The third line covers how the Higgs boson interacts with quarks and leptons. The lefthand side of the fourth line represents how the Higgs interacts with the other forces, and the righthand side represents how the Higgs interacts with itself. Very, very, roughly, bits with an F or D involve the subatomic forces except for the Higgs. Those with a psi include quarks and leptons. And, finally, ones with a phi involve the Higgs field.
So that's the short answer. Now for some interesting asides. If you look in the second and third line, there is the term "plus h.c." "h.c." is shorthand for something called the "Hermitian conjugate," which is a complicated feature of complex numbers. Complex numbers involve the square root of minus one, which you might have been told can't be done, but turns out it's possible if you do advanced math. If you want to know more about it, feel free to look it up. The exact meaning of the term "Hermitian conjugate" isn't super important here unless you're a real math wonk. If we ignore the math meaning, the "plus h.c." basically means "do the same thing with antiparticles."
However, the reason I'm drawing your attention to those is that it turns out that the first "h.c." actually isn't necessary. In fact, it was a mistake to include it. That first bit- the I, psi with a line over it, D with a slash in it, followed by a final psi, already has all of the matter and antimatter in it. So that "h.c." is an oopsie and really should be erased. Okay- that's basically what the equation means. If you want more details, I put some links in the video description to articles by Flip Tanedo, a physics professor at University
of California Riverside. He does a good job at explaining complex theoretical ideas. So that could be the end of the road, but there's more. This is because this equation is written in a really ultra compact format. To give you an idea what I'm talking about, here is a more familiar type of math equation, what is called "summation notation." Summation notation is basically a way to write a very long equation compactly. In this case, like most equations- it has two sides a left and right, separated by an equal sign, and the two sides mean exactly the same thing. The same thing is true for the standard model Lagrangian. You can take the compact equation and write it out long hand. When you do that,
you get this one here, which is way scarier looking than the compact version. This version was written down back in 1999 or thereabouts by Thomas Gutierrez, who is a professor of physics at California Polytechnic State University. He used a formulation composed by Nobel Prize winner Martinus Veltman. Note that the symbols have changed from the compact version of the equation, for example where the Greek letter psi meant all fermions, here they are called out specifically, with e's meaning electrons, d's meaning down-type quarks,
u's meaning up-type quarks, etc. So don't try to make a one to one correspondence. But it's kind of fun to see what a mess the equation can be. Okay, how about we translate this equation? What terms mean what? This first bit- the one with all of the different kinds of g's- describes how gluons interact among themselves. This second section deals with the interaction between electroweak force particles, mostly W and Z bosons, with a smattering of photons and with the appearance of Higgs bosons.
This third section represents how the weak force interacts with quarks and leptons. This fourth section is kind of technical. It describes what are called ghost particles, which aren't true particles, but they are a mathematical construct that fixes some unwanted predictions of the theory. And this fifth section is more of that ghost cleanup- this time what are called Faddeev-Popov ghosts. Again, these are introduced to maintain the integrity of the calculation. If you want to dig more into these ghost particles, make sure to buy some aspirin first. They always gave me headaches when I had to calculate them.
So that's kind of it. I've shown you two versions of the standard model Lagrangian and given you a sense of what they mean. If you want to dig more deeply into it, well, you have a fair bit of studying in front of you. I mean- I suppose I could do it for you, but I think this would be a great time to duck that responsibility. I know what I'll do. I'll do what my professors used to tell me when I asked these kind of questions and say "the solution is left as an exercise for the student." Good luck.
(phasing sound) Okay, so that was a fun dive into some crazy looking equations. They're fascinating and they represent all of what we know about the subatomic world. On the other hand, I kind of think of them as being like dragons- fascinating to look at, but not something you want to get too close to. If you liked getting a peek at the theoretical world, please subscribe to the channel and share on social media. The Standard Model Lagrangian does a pretty complete job of describing the subatomic world, which makes it very important, because of what I always like to say, which is that physics is everything.
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