So we are just discussing quantum electrodynamics and in particular the properties of the vacuum.

Let me just start by reminding you of what we discussed so far.

So we have the electromagnetic field or any other quantum field which is just a collection

of harmonic oscillators essentially.

So you would decompose it into normal modes.

These are waves of different waves number k and then each of these normal modes would

have an associated harmonic oscillator.

Some of them have very high frequency, some of them have very low frequency and each of

them is just a quantized harmonic oscillator which means that the excitation energy is

h by omega k.

But in particular what is of interest to us is the ground state energy and this ground

state energy is h by omega k.

So if I write down the Hamiltonian and I already have decomposed everything in normal modes

I would sum over k h by omega k and then the number of photons or the number of excitation

in general plus one half.

And so initially we spent some time discussing what this plus one half term in the energy

means and that you can drop it provided all your frequencies are just constant but once

your frequencies change then this term becomes important and that led us to discuss the Casimir

effect where changes of the boundary conditions seen by the field will change the frequencies

and will change this energy and that is a real effect because if the energy of any system

changes as a function of any parameter then there is a force associated with it.

So that was the Casimir effect.

And then we went on to discuss a few more properties of the quantum vacuum.

In particular we made the point last time that it is always extremely important to keep

the vacuum fluctuations even in situations where classically you could easily imagine

the income shortcut.

And one example was the beam splitter where classically you would only need to consider

the incoming field and the reflected field as well as the transmitted field.

You would not really need to consider this other part of the beam splitter unless there

is a light beam in there.

And we found that in quantum mechanics it is crucially important to take into account

this extra part of the beam splitter because it is there that the vacuum noise enters and

if you were to forget this extra part, you were to forget the vacuum noise then you would

fail at several levels.

You would fail formally because it would suddenly turn out that the commutators of the fields

calculated here would no longer be correct so to speak.

It is like the commutator of x and p, so they have a different value than ih bar.

And also physically it would make for nonsensical predictions because suddenly you take the

vacuum noise that is in this beam and you split it into these two beams so each of them

has a diminished amount of vacuum noise and that would lead to all kinds of predictions

that are made.

In a similar manner we looked at the cavity where you would have an incoming beam, light

circles around inside the cavity and light is emitted from the cavity because it decays

again through the semi-transparent mirror.

And again if you were to forget the vacuum fluctuations that also enter through the incoming

part then you would immediately arrive at contradictions like the commutator of a and

a dagger would start to evolve in time and not be unity.

So these pictures already tell us that it is important to keep the vacuum everywhere

and this picture was made more concrete when we tried a kind of semi-classical theory of

what might go on if the vacuum field couples to a charged particle, for example a bound