So in the lecture so far we've dealt with invariable theories several times including

those inequalities at the start.

But now I want to switch to more usual quantum mechanical phenomena which however still are

very important because they are somehow strange and different from classical aspects and the

first that I want to discuss now will concern geometric phases like the Ergonov-Bohm effect

which is also in some sense a non-local effect and then I will deal in the rest of the summer

term with things like relativistic quantum mechanics, quantum electrodynamics and the

statistics of particles.

So that is the...

Okay so this chapter will be about what we call geometrical phases and that means phase

vectors that you accumulate that only depend on the path you take in configuration space

space.

So the first and oldest example is the Ergonov-Bohm effect that tells us about the fact that somehow

in quantum mechanics gauge potentials are to be taken more serious than in classical

physics.

And so in addition to this being a geometrical phase it's even a typological phase so it's

a special case of a geometrical phase and we will come to that.

But for now I just want to remind you of the classical description of a charged particle

moving in an electromagnetic field and then we will switch to the quantum description

and see what develops.

So if I consider electromagnetic fields the way even to define operationally what is an

electric field and what is a magnetic field is to imagine you have a point charge, a test

particle moving through the field with some velocity and then it will be deflected, it

will experience a force and this force turns out to have two components.

One of the components only depends on the position of the particle and the other also

depends on its velocity.

And so then the force always comes out to be of this form some pre-factor which defines

the charge of the particles times the electric field at the current position of the particle

plus V cross B and B is the magnetic field.

So that is the Lorentz force.

So I should say that in the following I am really trying to stick to SI units.

Okay, now that is nice.

In particular it means if you take several test particles with different velocities you

can figure out what is the electric and the magnetic field distribution throughout all

of space.

And one thing we know is that the electric and magnetic fields then fulfill Maxwell's

equations.

And in particular they fulfill the so-called homogeneous Maxwell equations that do not

depend on the charge distribution or the charge current distribution.

So that would be that the divergence of the magnetic field vanishes meaning the magnetic

field never has any sources or sinks.

Things like this do not occur because that would be the source of the magnetic field.

The magnetic field rather is a field that has vorticity.

And then the second homogeneous Maxwell equation is that if I take the curve of the electric

field it is given by the time derivative of the magnetic field.

So changes in magnetic field because you suddenly ramp up your current that is producing the

magnetic field will also for a short time at least produce an electric field that has

some vorticity.

So these are the homogeneous Maxwell equations.

And we know from studying electromagnetism that because these equations hold there are