1 00:00:00,120 --> 00:00:06,870 So because it's not really possible to straightforwardly drive to Maxwell's equation most of the time 2 00:00:06,870 --> 00:00:12,600 and discourse, we will just discuss these equations and in the end come up with the results that we 3 00:00:12,600 --> 00:00:18,450 compare to experiments so that we understand or that we see that everything makes sense and everything 4 00:00:18,450 --> 00:00:19,290 fits together. 5 00:00:20,100 --> 00:00:28,140 However, this very special lecture is a bit more theoretical because here we want to motivate how we 6 00:00:28,140 --> 00:00:32,680 can construct Maxwell's relations based on symmetry arguments. 7 00:00:33,570 --> 00:00:41,640 So if you cannot follow this lecture 100 percent, it's no big deal because this is somewhat an exotic 8 00:00:41,640 --> 00:00:47,150 lecture, because here we really go into theoretical physics and do everything a bit different. 9 00:00:47,790 --> 00:00:54,630 But I think it's really worth it to do this because when I was a student, I really, really like this 10 00:00:54,630 --> 00:00:55,150 lecture. 11 00:00:55,530 --> 00:01:01,650 So for this lecture, I really have to credit my professor, Ingrid Mantashe, who came up with this 12 00:01:01,650 --> 00:01:02,680 nice explanation. 13 00:01:03,690 --> 00:01:11,430 So that was first discussed several objects, for example, at the time, or the position vector or 14 00:01:11,430 --> 00:01:19,410 the velocity and understand how these objects relate or how they behave under certain symmetry operations. 15 00:01:20,310 --> 00:01:24,460 So you will later understand or realize why this is important. 16 00:01:25,560 --> 00:01:33,480 So, for example, we can consider the time what happens if we, for example, apply a space inversion 17 00:01:33,480 --> 00:01:34,210 operation? 18 00:01:34,800 --> 00:01:40,060 So this means we take the whole space and put a minus sign in front of it? 19 00:01:40,980 --> 00:01:48,060 Well, the time T doesn't care at all about the space because these are independent variables, at least 20 00:01:48,060 --> 00:01:50,310 in a non relativistic world. 21 00:01:51,180 --> 00:01:56,860 So in this sense, the time is symmetric under space inversion. 22 00:01:57,000 --> 00:02:01,800 So this just means it does not change when we change our minds. 23 00:02:01,800 --> 00:02:03,050 Are it? 24 00:02:03,060 --> 00:02:10,020 The same is also true for the for the derivative aspect to time, because this is something like one 25 00:02:10,020 --> 00:02:10,490 over T. 26 00:02:10,770 --> 00:02:12,060 So it doesn't really matter here. 27 00:02:13,230 --> 00:02:21,510 And if we apply a time inversion operation, this means that we change T to minus T and of course T 28 00:02:21,510 --> 00:02:22,590 changes to minus T. 29 00:02:23,480 --> 00:02:27,850 So this is not really an application of these concepts. 30 00:02:27,870 --> 00:02:33,870 This is really the definition of the of the time inversion operator and then here also for the space 31 00:02:33,870 --> 00:02:34,770 inversion operator. 32 00:02:35,310 --> 00:02:37,470 So time inversion means the. 33 00:02:38,130 --> 00:02:38,490 Yeah. 34 00:02:38,670 --> 00:02:39,780 The time is a.. 35 00:02:39,780 --> 00:02:43,560 Symmetric and space inversion means that the space is a.. 36 00:02:43,560 --> 00:02:44,260 Symmetric. 37 00:02:44,280 --> 00:02:50,820 So here when we take the position vector and we apply a space inversion operation, this does not matter 38 00:02:50,820 --> 00:02:56,430 at all because when we when we invert the time, the spaces unaffected. 39 00:02:56,440 --> 00:02:57,740 So it's symmetric. 40 00:02:58,350 --> 00:03:01,950 But when we apply the space inversion, then of course we get this minus sign here. 41 00:03:02,880 --> 00:03:07,050 And now the same thing is true for the derivative aspect to space. 42 00:03:07,370 --> 00:03:17,040 So this would be true for the derivative with respect to X, Y, Z, and also when we take this whole 43 00:03:17,040 --> 00:03:19,070 vector here, does KNOBLER operator. 44 00:03:19,860 --> 00:03:24,360 So this operator is symmetric at a time inversion and it's a.. 45 00:03:24,360 --> 00:03:27,450 Symmetric under space inversion. 46 00:03:29,240 --> 00:03:39,740 Now, we can also take the velocity, so here the velocity is basically the derivative of the position 47 00:03:39,740 --> 00:03:40,940 with respect to the time. 48 00:03:41,120 --> 00:03:44,390 So it's something like position divided by time. 49 00:03:45,800 --> 00:03:51,470 So now, since this is just the quotient of these two quantities, we can take these two rows here and 50 00:03:51,470 --> 00:03:51,990 divide them. 51 00:03:52,010 --> 00:03:53,780 So if we have a.. 52 00:03:53,780 --> 00:04:01,430 Symmetric divided by symmetric or if we have the other way around, in both cases, we will get an anti 53 00:04:01,430 --> 00:04:01,970 symmetry. 54 00:04:02,330 --> 00:04:07,460 So the velocities aren't symmetric with respect to time inversion and with respect to space inversion. 55 00:04:09,040 --> 00:04:17,400 Now, the acceleration is the second derivative of space time, so this is something like our over T-square. 56 00:04:18,580 --> 00:04:25,900 So here we have to take this one here squares and both of these will be symmetric and so on. 57 00:04:25,900 --> 00:04:27,940 The forward is space inversion. 58 00:04:27,940 --> 00:04:30,460 Just will give us an A. symmetric function like here. 59 00:04:30,880 --> 00:04:36,790 And for time inversion, this will give us a symmetric function because here we get the minus sign twice. 60 00:04:38,370 --> 00:04:44,520 Now, that force is exactly the same thing because it's just mass times acceleration and the mass is, 61 00:04:44,520 --> 00:04:48,090 of course, independent of time inversion and space inversion. 62 00:04:48,840 --> 00:04:53,120 And the charged density is also it's just a scalar. 63 00:04:53,160 --> 00:04:58,390 So it's also, yeah, really trivial, I would say, like the mass. 64 00:04:58,410 --> 00:05:00,180 So it's symmetric in both cases. 65 00:05:01,740 --> 00:05:08,010 However, if we take the current density, which is basically charged density multiplied by the velocity, 66 00:05:08,430 --> 00:05:14,670 then we get the same symmetric symmetry, transformation, behavior as for the velocity. 67 00:05:14,940 --> 00:05:16,140 So we get a.. 68 00:05:16,140 --> 00:05:20,310 Symmetric behavior for the time inversion and for the space inversion. 69 00:05:21,920 --> 00:05:31,500 This was just some examples where you can understand how these objects, for example, acceleration 70 00:05:31,500 --> 00:05:35,880 or the velocity, behave when we reverse the time or the space. 71 00:05:37,040 --> 00:05:43,130 And I hope that this makes sense to you, because, for example, if you think of sitting in a car going 72 00:05:43,130 --> 00:05:49,820 with a certain velocity, if you would film yourself and then rewind the film, you would see that you 73 00:05:49,820 --> 00:05:50,720 go backwards. 74 00:05:51,810 --> 00:05:57,210 And also, if you would look in the mirror, for example, you would also see that you go into the opposite 75 00:05:57,210 --> 00:05:57,730 direction. 76 00:05:58,530 --> 00:06:00,960 So in both cases, this is a.. 77 00:06:00,960 --> 00:06:03,150 Symmetric under these two operations.