So because it's not really possible to straightforwardly drive to Maxwell's equation most of the time
and discourse, we will just discuss these equations and in the end come up with the results that we
compare to experiments so that we understand or that we see that everything makes sense and everything
fits together.
However, this very special lecture is a bit more theoretical because here we want to motivate how we
can construct Maxwell's relations based on symmetry arguments.
So if you cannot follow this lecture 100 percent, it's no big deal because this is somewhat an exotic
lecture, because here we really go into theoretical physics and do everything a bit different.
But I think it's really worth it to do this because when I was a student, I really, really like this
lecture.
So for this lecture, I really have to credit my professor, Ingrid Mantashe, who came up with this
nice explanation.
So that was first discussed several objects, for example, at the time, or the position vector or
the velocity and understand how these objects relate or how they behave under certain symmetry operations.
So you will later understand or realize why this is important.
So, for example, we can consider the time what happens if we, for example, apply a space inversion
operation?
So this means we take the whole space and put a minus sign in front of it?
Well, the time T doesn't care at all about the space because these are independent variables, at least
in a non relativistic world.
So in this sense, the time is symmetric under space inversion.
So this just means it does not change when we change our minds.
Are it?
The same is also true for the for the derivative aspect to time, because this is something like one
over T.
So it doesn't really matter here.
And if we apply a time inversion operation, this means that we change T to minus T and of course T
changes to minus T.
So this is not really an application of these concepts.
This is really the definition of the of the time inversion operator and then here also for the space
inversion operator.
So time inversion means the.
Yeah.
The time is a..
Symmetric and space inversion means that the space is a..
Symmetric.
So here when we take the position vector and we apply a space inversion operation, this does not matter
at all because when we when we invert the time, the spaces unaffected.
So it's symmetric.
But when we apply the space inversion, then of course we get this minus sign here.
And now the same thing is true for the derivative aspect to space.
So this would be true for the derivative with respect to X, Y, Z, and also when we take this whole
vector here, does KNOBLER operator.
So this operator is symmetric at a time inversion and it's a..
Symmetric under space inversion.
Now, we can also take the velocity, so here the velocity is basically the derivative of the position
with respect to the time.
So it's something like position divided by time.
So now, since this is just the quotient of these two quantities, we can take these two rows here and
divide them.
So if we have a..
Symmetric divided by symmetric or if we have the other way around, in both cases, we will get an anti
symmetry.
So the velocities aren't symmetric with respect to time inversion and with respect to space inversion.
Now, the acceleration is the second derivative of space time, so this is something like our over T-square.
So here we have to take this one here squares and both of these will be symmetric and so on.
The forward is space inversion.
Just will give us an A. symmetric function like here.
And for time inversion, this will give us a symmetric function because here we get the minus sign twice.
Now, that force is exactly the same thing because it's just mass times acceleration and the mass is,
of course, independent of time inversion and space inversion.
And the charged density is also it's just a scalar.
So it's also, yeah, really trivial, I would say, like the mass.
So it's symmetric in both cases.
However, if we take the current density, which is basically charged density multiplied by the velocity,
then we get the same symmetric symmetry, transformation, behavior as for the velocity.
So we get a..
Symmetric behavior for the time inversion and for the space inversion.
This was just some examples where you can understand how these objects, for example, acceleration
or the velocity, behave when we reverse the time or the space.
And I hope that this makes sense to you, because, for example, if you think of sitting in a car going
with a certain velocity, if you would film yourself and then rewind the film, you would see that you
go backwards.
And also, if you would look in the mirror, for example, you would also see that you go into the opposite
direction.
So in both cases, this is a..
Symmetric under these two operations.