So here in this table, I have colored the different entries according to their symmetry behavior and
also according to their scalar object, our character.
So first we can consider only the and the entries that are symmetric on the both of these operations
and that are scalar.
And these are the charged density and the divergence of each.
And then we can also take all the entries that are symmetric with respect to both of these operations,
but are vectors of pseudo vectors.
And these are, for example, here, the green ones, the time derivative of B and the rotation of E..
Now, I've summarized this categorization in these four tables, so here we have positive, positive,
positive, positive, negative, negative, negative, negative.
And here we have scalar, pseudo vector, vector and pseudo scalar.
And now we must understand that when we have an equation with left and the right hand side, then all
of the terms and both sides of this equation must have the same character.
So they must either be vectors or scalars.
And they must behave equally under the time and version and the space and version.
So this means an equation, for example, can only consist these two or these two or these three or
this one entry here, but they cannot mix.
And maybe you have realized, as already, the four tables correspond to the four different Maxwell's
equation.
Let's start with the first two that are both symmetric under these operations here, we have to scale
our ante, we have to pseudo vectors.
And this means we have here the charge density plus or minus the divergence of E must be zero.
So the sum of these two objects or let's say these two objects can be in the same equation.
That's all that we developed here.
And if you compare this to the Maxwell equation and we see, OK, this already looks pretty good, we
have divergence of E, then we have this prefecture that we did not know yet about and we have to charge
density.
Now, the prefecture can, of course, be here because this does not affect any symmetry and it does
also not change the scalar or vector behavior here.
And only thing that we can do now is that we can determine the dimension or the unit of this prefecture.
So we can do this by looking at the unit of the charged density.
So this is a charge as the unit is Coulomb.
So this is Amper second.
And since it's a density, it's over the volume.
So it's divided by meters to the power of three.
So it's per second over meter three.
Now, the divergence of the electric field is the electric field, which is volt of a meter.
And this Nabala operator is essentially a vector of first order derivatives.
So the derivative with respect to the coordinate, so the unit of this is one of a meter.
So this is your volt of a meter square.
So if you want to relate these to these two terms here, then they must also have the same unit.
So there must be a prefecture which has the unit like this, what meter per second.
So this is why this makes total sense that we have had this prefecture.
However, we cannot tell anything about the numerical value of this prefecture and also not about to
sign.
Now we can do a very similar thing for the vectors that have the same symmetry behavior in both of these
operations, and this means the time derivative of B and the rotation of E both can enter the same maximal
equation.
And you actually, when you look at the unit, we see that they already have the same unit, so there
is not or there must not be a factor that can be put.
There must not be a factor.
As you can see in the Maxwell equation, the prefecture is actually a minus one.
So this we cannot explain by the symmetry and analysis here.
But I can tell you, if there's a minus sign, wouldn't it be there then all of the physics would be
wrong.
So that's unfortunately something that our method to you cannot explain.
But OK, it is then related to experiments, you could say.
Now, on this slide here, I show you all of the quantities that are anti symmetric with respect to
time inversion and space inversion.
So these are the current density, the time derivative of the electric field and the rotation of the
magnetic field.
And you can see here that these are all vectors, of course, and they have all different units.
So there must be three.
And when we look at the result, which is one of our Maxwell equations, we see that the rotation of
B is equal to this prefecture here zero times Epsilon zero, which is second square meter square and
the unit times.
There's one here time derivative of it.
And then we have another term, which is zero times, Jane.
And then the very last one is actually the shortest one, we only have a single entry for our objects
that are scalar and that aren't anti symmetric with respect to time and space.
So this is just a divergence of the magnetic field.
And so in terms of first order derivatives of B and E, there is not a single other term that has the
same behavior under these transformations and there's a scalar.
So this means just this divergence of B must be zero.
So that's that's the reason why there cannot be any magnetic monopoles.
So once again, let me admit that this lecture was probably kind of hard to understand because this
is a very different approach, but this is really, I would say, theoretical physics approach to all
of this topic.