The birth of gauge theory — Yang–Mills

42:09politics, there's certainly a difference between >> certain there's certainly a difference, right? >> Maybe that's why we shouldn't be labeling them in these directions because in physics there's there's >> it seems it seems there's no difference. >> Or maybe this is where horseshoe theory comes from. [laughter] It's the same. >> It's the same. Yeah. Exactly. Yeah. >> Exactly. So, so we've been discussing this question, right? Is parody conservation a law of the universe? And so far it seems yes. Okay. Meaning that if I have some experiment and I do some experiment and then I do the mirror image of that experiment, the results of the experiment are exactly going to be the same. Okay. So let's review a few a few things. Okay. Certain vectors flip sign. >> Okay. >> Okay. Like position for example, if I'm

42:51moving in this direction in the mirror image, it's going to flip sign. >> When you say this direction, how do you mean just flip? >> I mean like Yeah. For example, like let's say there's a mirror in front of me. If I'm moving towards the mirror, >> in the mirror world, that particle is going to be moving towards me, right? So, it's like it's like I'm >> I'm moving towards the wall. >> Let's say the mirror is on the wall and the particle is moving towards the wall. Well, in the mirror world, the particle is moving out of the wall, right? So, it's it's flipped its sign, right? That's what par means. We're flipping all of the x's. X becomes negative x, y becomes negative y, z becomes negative z. Okay? Now certain vectors these true vectors they flip their sign but then

43:32there's other vectors called axial vectors that do not. For example angular momentum suppose I have uh suppose I'm doing this with my hands >> uh rotating in a clockwise direction. >> Yeah. Rotating in a clockwise direction. So the spin is towards the wall. Okay. >> Right. It's towards the wall into the wall. >> Yeah. The righty tidy idea. >> Yeah. Righty tighty idea. Well, that rotation is going to look the same in the mirror in the mirror world, right? In the mirror world, if I'm doing my hand like this, the mirror's trajectory is also going to be going clockwise. So, in the mirror world, >> that the the the spin is going to be going into the wall, right? It's going

44:12to be going like the the the angular momentum actually doesn't change. >> Wait, this is actually really >> it's kind of trippy. I really need you to focus here and >> that's actually really crazy because what you're basically saying is in the mirror. >> Yeah. And and if Yeah. We pull it back up. Like on the left hand side we've got we've got a a wheel that's turning like this. And so the spin vector is to the right. >> Now I flipped it along the middle. >> Mhm. >> It's spinning, but the spin vector is still still right because it's like imagine I got I got a I got I got a mirror in front of me. >> I'm doing this in the mirror world. that thing is my hand is still doing the same kind of thing. >> It's because I've changed both x and y

44:53to be negative and negative and the spin is a product of my >> of those of those two coordinates. So the negative, you know, it's still going to be pointing in the same direction. >> So an electron that's spinning this way into the mirror is going to be still pointing like going the same direction in the mirror world. in the mirror world. >> It's this is this this is a crucial thing for you to understand. >> This is I've never >> And can you imagine it? >> No. So I I can in my head. Yeah. No, I can I can visually >> and it it's it's making me uncomfortable because it doesn't make sense. >> Yeah. like but not not even that it doesn't make sense but it's it's it's the the the distinction between the

45:35first use case we just talked about um of the um position vector like versus this axial vector and simply like we it's it seems like such a simple difference >> but the implications of that simple difference are are change everything >> and and and and the the real underlying substrate here is that axial vectors are not true vectors. They're kind of like okay >> they're they're products of two vectors, right? Angular momentum is R cross V. So So because you put a negative on one and the negative on the other, it's going to be the same, >> right? >> The position vector is literally like