Hello. We were just discussing how you actually describe how you calculate the evolution under

the action of decoherence. And we started very general. We said that if I start with

a system density matrix at time zero, and I want to know the system density matrix at

time t, and I can assume that at time zero I had say independent initial states of system

and bath, then the mapping between initial system density matrix and final system density

matrix would simply be a linear map from density matrix to density matrix. And that led us

to this concept of chaos operators. And then we went one step further and said, well, now

let's go from the most general setting to the most special or simple setting, that is

when you can write on a Markov master equation, a Markov evolution equation for the density

matrix so that time derivative at some time is just linear operator applied to the density

matrix at that time. And so that led us to the concept of the Lindblad master equation,

and then we went through the different examples of Lindblad master equations. And so now what

I want to discuss next is what happens if things are not quite that simple. If, for

example, there are memory effects, if the noise is not simply white noise but something

more complicated, if the coupling is not weak anymore so that the derivation of the master

equation will fail. And the most general calculation approach in that case is when you use path

integrals. And of course, this is not a lecture about decoherence, so I will not discuss all

the technical details of these path integrals approaches because that would need a whole

lecture. But at least conceptually, it's very interesting because much of what we discussed

already in simple cases occurs there in the completely general setting.

So I want to start by reminding you of how a path integral works in principle. So these

are path integral approaches to describing decoherence or dissipative quantum systems.

But let's first go a step back and say how does a path integral work. And the idea is

that it's just another alternative way of, in principle, calculating the time evolution

of a wave function, alternative to simply applying the Schrödinger equation. And the

idea, as you may know, is say if I am dotting x and t and I'm given the wave function at

time zero, so that is still needed as an initial condition, the amplitude for the particle

to be here or there, then the evolution of the wave function can also be found by starting

from every possible point at time zero and then going to every possible point at time

t along every possible trajectory. And typically these trajectories would somehow look like

this. And then, to find one time as well, psi of x, t, or let's be a little bit more

specific, let's call this x subscript t, to say this is some of the final position to

which I want to go. That would be x zero, the initial position. And then I would say,

so this is an integral over all possible trajectories, which I call x, without specifying the argument,

just to imply that this is really an integral over all possible trajectories, not an integral

over the value of the trajectory at a particular time point, point in time. So this is a path

integral and then you would have the amplitude to find the particle at some point x zero

initially, at time zero, and you would multiply that, so to speak, with the amplitude, go

from here to there, given the specific system, you have to specify what is the mass, what

is the potential and so on, and all of this enters as a phase factor, e to the i over

h bar times s, and s would simply be the action, the classical action along this trajectory.

So s would be the integral zero to t, l, dt prime, l is simply the derivation, so it would

be, if I just had a simple particle, it would be the kinetic energy minus the potential

energy. And of course you see that this has to be interpreted in the proper manner, so

in actual practice you would be a discretized time, you have many different time slices,

in any time slice you have a coordinate for your path, and then this integral really means

you have multi-dimensional integral over all these different positions, if there are n

time slices and positions, plus the initial position and the way I've written it down,

and then there are certain normalization factors that have to be just fixed in order to say

make the result correct even for the free particle, and then also things like the time