1 00:00:00,450 --> 00:00:07,440 So here in this table, I have colored the different entries according to their symmetry behavior and 2 00:00:07,440 --> 00:00:12,270 also according to their scalar object, our character. 3 00:00:13,290 --> 00:00:20,100 So first we can consider only the and the entries that are symmetric on the both of these operations 4 00:00:20,100 --> 00:00:21,140 and that are scalar. 5 00:00:21,540 --> 00:00:25,440 And these are the charged density and the divergence of each. 6 00:00:26,580 --> 00:00:34,140 And then we can also take all the entries that are symmetric with respect to both of these operations, 7 00:00:34,410 --> 00:00:36,750 but are vectors of pseudo vectors. 8 00:00:37,290 --> 00:00:43,470 And these are, for example, here, the green ones, the time derivative of B and the rotation of E.. 9 00:00:44,880 --> 00:00:52,170 Now, I've summarized this categorization in these four tables, so here we have positive, positive, 10 00:00:52,170 --> 00:00:54,630 positive, positive, negative, negative, negative, negative. 11 00:00:55,050 --> 00:01:00,550 And here we have scalar, pseudo vector, vector and pseudo scalar. 12 00:01:01,890 --> 00:01:10,700 And now we must understand that when we have an equation with left and the right hand side, then all 13 00:01:10,710 --> 00:01:15,690 of the terms and both sides of this equation must have the same character. 14 00:01:15,700 --> 00:01:18,750 So they must either be vectors or scalars. 15 00:01:20,250 --> 00:01:24,670 And they must behave equally under the time and version and the space and version. 16 00:01:25,110 --> 00:01:33,450 So this means an equation, for example, can only consist these two or these two or these three or 17 00:01:33,450 --> 00:01:36,480 this one entry here, but they cannot mix. 18 00:01:37,530 --> 00:01:42,870 And maybe you have realized, as already, the four tables correspond to the four different Maxwell's 19 00:01:42,870 --> 00:01:43,410 equation. 20 00:01:44,400 --> 00:01:49,560 Let's start with the first two that are both symmetric under these operations here, we have to scale 21 00:01:49,590 --> 00:01:51,660 our ante, we have to pseudo vectors. 22 00:01:52,320 --> 00:02:01,230 And this means we have here the charge density plus or minus the divergence of E must be zero. 23 00:02:02,720 --> 00:02:08,640 So the sum of these two objects or let's say these two objects can be in the same equation. 24 00:02:08,840 --> 00:02:11,020 That's all that we developed here. 25 00:02:11,810 --> 00:02:16,460 And if you compare this to the Maxwell equation and we see, OK, this already looks pretty good, we 26 00:02:16,460 --> 00:02:21,890 have divergence of E, then we have this prefecture that we did not know yet about and we have to charge 27 00:02:21,890 --> 00:02:22,460 density. 28 00:02:24,380 --> 00:02:30,200 Now, the prefecture can, of course, be here because this does not affect any symmetry and it does 29 00:02:30,200 --> 00:02:32,990 also not change the scalar or vector behavior here. 30 00:02:33,800 --> 00:02:41,480 And only thing that we can do now is that we can determine the dimension or the unit of this prefecture. 31 00:02:42,230 --> 00:02:45,200 So we can do this by looking at the unit of the charged density. 32 00:02:45,740 --> 00:02:49,130 So this is a charge as the unit is Coulomb. 33 00:02:49,400 --> 00:02:51,200 So this is Amper second. 34 00:02:51,860 --> 00:02:54,290 And since it's a density, it's over the volume. 35 00:02:54,290 --> 00:02:56,990 So it's divided by meters to the power of three. 36 00:02:57,320 --> 00:02:59,450 So it's per second over meter three. 37 00:03:00,680 --> 00:03:07,300 Now, the divergence of the electric field is the electric field, which is volt of a meter. 38 00:03:07,880 --> 00:03:13,550 And this Nabala operator is essentially a vector of first order derivatives. 39 00:03:13,560 --> 00:03:20,250 So the derivative with respect to the coordinate, so the unit of this is one of a meter. 40 00:03:20,480 --> 00:03:22,570 So this is your volt of a meter square. 41 00:03:23,600 --> 00:03:29,640 So if you want to relate these to these two terms here, then they must also have the same unit. 42 00:03:30,320 --> 00:03:36,080 So there must be a prefecture which has the unit like this, what meter per second. 43 00:03:36,950 --> 00:03:40,270 So this is why this makes total sense that we have had this prefecture. 44 00:03:41,120 --> 00:03:47,210 However, we cannot tell anything about the numerical value of this prefecture and also not about to 45 00:03:47,210 --> 00:03:47,720 sign. 46 00:03:49,490 --> 00:03:55,970 Now we can do a very similar thing for the vectors that have the same symmetry behavior in both of these 47 00:03:55,970 --> 00:04:03,950 operations, and this means the time derivative of B and the rotation of E both can enter the same maximal 48 00:04:03,950 --> 00:04:04,520 equation. 49 00:04:05,740 --> 00:04:11,500 And you actually, when you look at the unit, we see that they already have the same unit, so there 50 00:04:11,500 --> 00:04:15,640 is not or there must not be a factor that can be put. 51 00:04:15,640 --> 00:04:17,130 There must not be a factor. 52 00:04:17,860 --> 00:04:22,590 As you can see in the Maxwell equation, the prefecture is actually a minus one. 53 00:04:23,590 --> 00:04:27,640 So this we cannot explain by the symmetry and analysis here. 54 00:04:28,030 --> 00:04:32,500 But I can tell you, if there's a minus sign, wouldn't it be there then all of the physics would be 55 00:04:32,500 --> 00:04:32,810 wrong. 56 00:04:32,830 --> 00:04:39,070 So that's unfortunately something that our method to you cannot explain. 57 00:04:39,070 --> 00:04:44,320 But OK, it is then related to experiments, you could say. 58 00:04:46,740 --> 00:04:53,100 Now, on this slide here, I show you all of the quantities that are anti symmetric with respect to 59 00:04:53,100 --> 00:04:54,970 time inversion and space inversion. 60 00:04:56,640 --> 00:05:01,320 So these are the current density, the time derivative of the electric field and the rotation of the 61 00:05:01,320 --> 00:05:02,180 magnetic field. 62 00:05:02,850 --> 00:05:08,190 And you can see here that these are all vectors, of course, and they have all different units. 63 00:05:09,000 --> 00:05:10,110 So there must be three. 64 00:05:11,340 --> 00:05:15,840 And when we look at the result, which is one of our Maxwell equations, we see that the rotation of 65 00:05:15,840 --> 00:05:25,110 B is equal to this prefecture here zero times Epsilon zero, which is second square meter square and 66 00:05:25,170 --> 00:05:27,210 the unit times. 67 00:05:27,450 --> 00:05:29,160 There's one here time derivative of it. 68 00:05:29,550 --> 00:05:33,840 And then we have another term, which is zero times, Jane. 69 00:05:35,480 --> 00:05:41,930 And then the very last one is actually the shortest one, we only have a single entry for our objects 70 00:05:41,930 --> 00:05:46,010 that are scalar and that aren't anti symmetric with respect to time and space. 71 00:05:46,700 --> 00:05:49,640 So this is just a divergence of the magnetic field. 72 00:05:50,270 --> 00:05:58,550 And so in terms of first order derivatives of B and E, there is not a single other term that has the 73 00:05:58,550 --> 00:06:02,450 same behavior under these transformations and there's a scalar. 74 00:06:03,110 --> 00:06:06,960 So this means just this divergence of B must be zero. 75 00:06:07,370 --> 00:06:10,970 So that's that's the reason why there cannot be any magnetic monopoles. 76 00:06:12,840 --> 00:06:19,590 So once again, let me admit that this lecture was probably kind of hard to understand because this 77 00:06:19,590 --> 00:06:26,040 is a very different approach, but this is really, I would say, theoretical physics approach to all 78 00:06:26,040 --> 00:06:26,730 of this topic.