Okay, hello. We're just discussing the properties of the electromagnetic vacuum and in particular

we had been discussing an attempt at a local hidden variable theory that obviously cannot

work but still provides some interesting physical insights and that is termed stochastic electrodynamics.

And the idea is very simple. I told you last time you just assume that in classical Maxwell

electrodynamics you don't need to start with initial conditions where the field is just

zero everywhere but you can start instead with some arbitrary fluctuating field. Then

you can ask which properties does this fluctuating field has to have in terms of its spectrum

in order to be Lorentz invariance so that it does not select the preferred frame of

reference and it turns out there's exactly one and only one energy density spectrum that

gives you Lorentz invariance which is if the energy density spectrum goes like omega to

the third power. And so since you know the density of states in 3D is omega squared that

means the energy per mode automatically goes linear in frequency just as it does in quantum

electrodynamics simply because h bar omega half is the ground state energy of each mode.

And so starting from this h bar would be a free parameter which you take from experiment

and then you ask how does this affect say the motion of charges and we calculated the

simplest case which is a charge that is harmonically bound and interacts with this fluctuating

noise field. If there were only the noise then of course the charge would heat up and

increase its energy beyond any bound but also there is damping necessarily because it's

coupled to this radiation field and this damping is known as radiation damping and it has this

funny third time derivative of the position. Okay so that's the model we discussed. So

this is the fluctuating field plus our charge and we solved an equation which is the Newton's

equation of motion but the force would be the restoring force plus this radiation damping

force which goes like the third time derivative and so for dimensional reasons it should be

something like have a pre-factor with something like mass times the time plus the noisy field

and this particular time can be expressed in terms of the charge and fundamental constants.

It would be Q squared over M divided by 6 pi x down 0 C to the third. Okay and so since

this is a linear problem you can easily solve it. You'll find the response of the stamped

oscillator and in doing so we introduced a certain approximation. We said oh instead

of writing the third time derivative I can write down the second time derivative and

solve to lowest order this equation by equating it to x itself. So we said that x third time

derivative is about minus omega squared x dot so that automatically makes things easier

to solve. It tells you that this is indeed a damping force and it avoids some other problems

as well. So what is the spectrum we found? Here I'm plotting the fluctuation spectrum

for my position as a function of frequency. Of course it's strongly peaked at the resonance

frequency. The peak is of width gamma where gamma would be omega squared tau and then

it has some tails for example here it grows like omega to the third, here it goes down

like one over omega and so the variance of the position fluctuations will be just the

area under this curve. And it turns out this is h bar over 2 omega if you choose h bar

as the one adjustable parameter that would come from experiment and characterizes your

zero point fluctuations. Plus there are corrections. They will depend on the cutoff frequency

because the one over omega integral doesn't converge but these are like the corrections

in quantum electrodynamics. Again then you would have to introduce some randomization

procedure and so on. But this is obviously the ground state width of your quantum harmonic

oscillator and we also discussed that since the fluctuations of the field are Gaussian,

the fluctuations of x are Gaussian as well so even the full probability density just

looks exactly like in the quantum case. And so at least for this linear system there is

a nice correspondence between what quantum mechanics predicts and what you get from purely

classical electrodynamics assuming that you have such a Lorentz invariant zero point spectrum.

Now that means we have a very nice picture and it's good to compare the thermodynamics.

So in classical thermodynamics or statistical physics we would have the picture that I have