So we've started talking about quantum electrodynamics and maybe the most important difference to

classical electrodynamics is that there are these vacuum fluctuations.

So that is because of course you can decompose the field into a set of normal modes, each

of those normal modes is an harmonic oscillator, you describe it quantum mechanically so you

get these energy levels and there is a finite ground state energy h power omega.

And in the ground state there is a wave function with finite extent so whatever this coordinate

here means, electric field or magnetic field or vector potential it doesn't matter, you

find that you have finite fluctuations even in the ground state, even at zero temperature.

So today I want to start by discussing one of the effects where these vacuum fluctuations

show up, which is the Casimir effect.

I start with the Casimir effect because there the details of the interaction with matter

do not yet enter.

Matter only becomes important in surprise of six boundary conditions for the field.

So if h power omega half is the energy per field mode then you can also define energy

density and the energy per volume and frequency interval and calculate the energy density.

So what is the energy density of the vacuum fluctuations of the electromagnetic field?

I will call this epsilon for the energy density, epsilon depending on omega and for me this

will be the energy per volume and frequency interval.

Now if you think about it, I just need to count the number of modes in that frequency

interval, that number will grow if I consider larger volume so I divide by the volume to

get something that does not depend on the volume and then I have to multiply by the

energy per mode which for the vacuum fluctuations is just h power omega half.

So that would be h power omega half times the density of states and that I will call

d of omega.

Now this is a little exercise to calculate the density of states of the electromagnetic

field in three dimensions or of any field with a linear dispersion relation and it turns

out in three dimensions this just goes like omega squared.

So overall we find that the energy density of the vacuum fluctuations goes like the power

of frequency.

And there of course once again we understand why we get into trouble if we calculate the

fluctuations of the electric field at a single point because then we get contributions from

all those frequencies which simply diverge.

Now since that is the case obviously you will run into trouble if you want to calculate

not the energy per frequency interval but really just the total energy, that is if you

want to integrate over all of the frequencies you will get an infinite area.

The question is does that matter, should we be worried?

So if I write down the Hamiltonian I would say sum over all modes, sum over k h power

omega k a k dagger a k that is the number of photons in that mode but if we are in vacuum

there are zero photons plus one half.

So the one half would be the non-state energy.

The question is should we care about that?

Now as long as the omega k's are all constant, this is just a constant, it happens to be

infinite if you sum over all the frequencies but still it is fixed and so you could probably

remove it as a constant energy offset.

Now that is true as long as the omega k are fixed but if for some reason you start changing

the omega k because you change the boundary conditions then you can no longer argue like

this.

So the argument of subtracting the offset will be okay as long as the omega k don't

change.

Now we will come back to that but first I want to ask the question more generally, is