The following content has been provided by the University of Erlangen-Nürnberg.

The following content has been provided by the University of Erlangen-Nürnberg.

The following content has been provided by the University of Erlangen-Nürnberg.

The following content has been provided by the University of Erlangen-Nürnberg.

The following content has been provided by the University of Erlangen-Nürnberg.

So then I write down an elementary amplitude like this.

e to the minus i h hat t x i.

So these x's, they are kind of eigenstates to replace improper eigenstates of the position operator.

So these are eigenstates of the self-adjoint operator x hat.

And this here is now an amplitude for a particle to move from initial position x i in the time interval t to a final position x f.

So that's the typical expression I'd like to compute.

And let me now show you that this can be written indeed as a path integral.

So how does this standard construction go?

Well, we cannot really evaluate.

I mean, if this is just a free series, if v is equal to zero, one can kind of straight away write down the solution.

If there is some non-trivial potential, it's less clear what to do.

But one thing one can look at is, so this is an evolution operator.

This object here.

And if h is a self-adjoint operator, then this is a unitary evolution operator.

So what I can do, or what is easier to handle, is this evolution operator for small times, now small delta t.

Evolution operator for small times delta t.

So I can show that this approximates, this object one is looking for, when delta t becomes small.

So this is e to the minus i.

This is, of course, a quantized operator, h hat delta t.

And so about this object, which approximates what I'm looking for, I can actually compute in an easy way the matrix elements.

So between two eigenstates of the position operator again.

So, and just let me write down what it is.

It's m, mouth is a particle, as it appears here in the Hamiltonian, divided by 2 pi i delta t exponential i,

imaginary i, m divided by 2 delta t, x prime minus x squared, minus a term which refers to the potential capital V,

i delta t divided by 2, Vx plus Vx prime.

Okay, so this is about, this is kind of an average for the potential there.

So that's what it is.

Now, the idea is now just to iterate, so to multiply this together, until you've reached this as a full time t.

So basically, what one imagines is that one has split up the time little t into n chunks,

and each of them has size delta t.

And then one can in simple cases actually prove that this evolution operator for the entire time interval

can actually be obtained as the limit n going to infinity of iterating this object n times u delta t to the nth power.

And of course, if one wants to do this somewhat clearly, one has to look exactly at what the operators are,

the constituents of this age, what boundary conditions are, etc.

So in sufficiently simple cases, I can prove a formula like this.

As you'd see, sometimes, of course, you have just no independent idea of what this is going to be.

Certainly later in the case when we're looking at gravity and field theory,

then you work, you start with an object like this and define through that limit, try to define what your unitary evolution is.

So that's then the logic.

But for the time being, if this is a sufficiently simple potential, one can kind of more or less argue that this formula is true.

Now, I put this now into my original amplitude here and try to evaluate it.

So how does that go?

Xf e to the minus i h hat t, xi.

So that was my original expression.

Now, I'm using this, I'm substituting this by the limit of the product of these evolution operators of small delta t.