Last time we've been discussing geometric phases and then we've discussed statistics.

Now I first want to correct a sign mistake I made last time.

So when we discussed the very phase, that is the phase you accumulate during adiabatic

transport, we also checked whether this really gives the iron-off bone phase under the appropriate

circumstances.

And so you remember the setting, you have a flux enclosed in a solenoid, outside there's

no magnetic field, you take a wave function that is trapped in some potential and then

you take this potential around a path.

And the parameter we're talking about, which is changed adiabatically, would be Dundas,

this is Dundas at some time t prime, that would be Dundas at time 0.

And so if you work it out correctly, you find that indeed, as it should be for our definitions,

the very phase is equal to the iron-off bone phase.

So we have a plus sign here.

And briefly let me tell you where the calculation should be changed.

So if you want to write down the wave function, the adiabatic eigenstate, psi 0 as we called

it, in dependence on Dundas, you would say oh this is psi 0 evaluated for 0 vector potential.

Which is a cube, which you can take to be a purely real function.

And then you have to multiply with a phase factor.

But you have to do it correctly.

So here's e to the i cubed over h bar integral a ds.

But the point is you have to take the correct path and the correct path here would be to

start from the center of the wave packet and then proceed to the point r where you look

at the wave function.

And so you can convince yourself this is a correct solution of the instantaneous time

independent Schrodinger equation.

And if you take now the gradient with respect to lambda, since lambda really is the lower

boundary, you get a minus sign.

And so this makes this correct.

Now we started discussing particle statistics and first I remind you of Thamuels and Bosons,

which you get simply by considering the Hamiltonian of identical particles, which is symmetric

under exchange of particle coordinates and then considering the possible symmetries of

the wave function.

Then we found out, well, there are two possibilities in general.

Either you have a completely symmetric wave function or an anti-symmetric wave function.

So either when you interchange two coordinates, nothing changes.

The sign remains the same or you get a minus sign for each exchange.

But then we thought that maybe these things can be more flexible and indeed they can be,

especially in two dimensions, because it turns out that in two dimensions you can have a

situation where physically if you move the particle around this way, you can accumulate

a different phase than if you move it around in the other way.

And so we had the curious case of Anyons where you could accumulate quite an arbitrary phase

under particle exchange.

And you could build a consistent theory based on that.

And I remarked that there is actually a physical system where this is realized, which is the

fraction of quantum point.

Now this rested very much on the situation being two dimensional, because it rested on

the fact that you cannot contract this loop down to a point without going through this

other particle.

And maybe this is forbidden because of strongly repulsive interactions.

So physically you cannot contract the loop.