We have discussed Wohm's theory and Nelson's theory. These are both examples of hidden

variable theories that work in the sense that if you calculate the probability density at

any point in time of your particles coordinate or of your many particles various coordinates,

then it coincides with quantum mechanics. Now the first thing that is a little bit strange

is that there are two such theories and actually infinitely many more. The second thing that

we already have mentioned is that they are necessarily non-local in order to be able

to reproduce quantum mechanical predictions. After all, this is already what Bell's theorem

tells us, but we can also check it explicitly for these theories. Now, there is something

more because the non-locality really arises only at the level of these unobservable trajectories.

So I apply a force to particle number one and suddenly I see some change in the trajectory

of particle number two, which is very far away. This is strange on its own, but after

all these trajectories are unobservable and still you reproduce the correct statistical

predictions of quantum mechanics and we convinced ourselves that you cannot use this for signaling.

So things seem to be nevertheless alright. There is another problem though, which I very

briefly mentioned at the end of last lecture when we discussed Nelson's theory, but it

also applies to Bohm's theory, which is when you start measuring at the two different particles

at different times. Because, for example, looking at Bohm's theory and equally Nelson's

theory, you are only guaranteed to reproduce quantum mechanical statistical predictions

if you do all the measurements at the same time. And indeed you run into trouble when

you measure at different times. Now, this is quite strange. If it were not for relativity,

then you could at least say, oh, for some funny reason I am not allowed to ask this

question about measurements at different times. I should always only ask about simultaneous

measurements. But of course in the theory of relativity, whether something is simultaneous

or not depends on your frame of reference. So we come to the question, apparently Bohm's

and Nelson's theory cannot coincide with quantum mechanical predictions in all frames of reference.

Are there any hidden variable theories that do coincide with quantum mechanics in all

frames of reference? Now, when I am asking the question this way, can they be Lorentz

invariant, I mean precisely the following, whether I can construct a hidden variable

theory such that it coincides with quantum mechanical statistical predictions in all

frames of reference. So I can draw a picture, space-time diagram, and I would have trajectories,

say this is the trajectory of particle one, and that is the trajectory of particle number

two. And now Bohm's theory or Nelson's theory for example will guarantee me that if I suddenly

measure all those two positions at a given time, and I repeat the experiment many times,

I will correctly reproduce the quantum mechanical prediction psi of x1, x2 squared. But according

to relativity, if I am moving with respect to this frame of reference, then a time slice,

a slice of simultaneous time is just some line which is tilted in this old frame of

reference space-time diagram. And so I could look at those points where the trajectories

cross that new space axis. And again I could take an ensemble of many trajectories and

again ask the same question. So these are different space-like hyper-surfaces if you

like. And I want to make sure that quantum mechanics reproduces my prediction, that the

hidden variable theory reproduces my quantum mechanical predictions.

Okay, so it turns out that this will work for a single particle, not surprisingly. Also

for a single particle it's very easy to expand, for example Bohm's theory to the Dirac equation,

taking the Dirac equation instead of the Schrodinger equation. But we have to see what happens

with two particles. And so I'll be telling you about the Duncan experiment, which in

the version I'm going to tell it to you is actually due to a paper in 1996 by Daniel

Goldstein and Sange. But it was originally inspired by an earlier paper of Hardy in 1992,

in which he was explaining that the Hardy paper dealt with particles and antiparticles

that can annihilate. And so you might think, oh I need a new really relativistic quantum

field theory and variable theory to deal with that. So I prefer the version given here.