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[SQUEAKING] [RUSTLING] [CLICKING] HONG LIU: OK. Good. So last lecture, we talked about the property of this propagator, which is defined to be x, x prime which is also the same as phi x, phi x prime here. OK? So this object, we discussed. So conceptually, it can be considered-- so you have two interpretations. So one is from here. So this roughly can be heuristically interpreted as the transition amplitude for a particle from x to x prime. OK, so heuristically, you can think of it, say, if you have a point-- say, if you have a space-time point, x prime has another points x. And then you ask, what's the transition amplitude from x prime to x? OK, so one way to interpret this object is that. And as I emphasize, this is only heuristically, because this does not really describe a generally localized state. It's only a approximately localized state, OK? And similarly, the second interpretation is the correlation function. It describes correlations of phi between, say, ...

[SQUEAKING] [RUSTLING] [CLICKING] HONG LIU: Good. Yeah. So last lecture, we discussed Noether's theorem. For every symmetry, for every continuous symmetry, there's a conserved current. And then we also started talking about relativistic quantum mechanics. How we want to unify special relativity in quantum mechanics. So the most immediate idea for that is what's called the relativistic quantum mechanics. And the most immediate generalization of the Schrodinger equation-- so if you have-- so at the end of last lecture, we talked about, say, the most immediate generalization of the Schrodinger equation, which-- so if I have E equal to, say, p squared divided by 2m, and then you go to non-relativistic quantum mechanics Schrodinger equation. And now if you have E squared equal to p squared plus m squared for relativistic particle, and then you get what's called the Klein-Gordon equation. And again, this side has the implication of the wave function. So this des...

[SQUEAKING] [RUSTLING] [CLICKING] HONG LIU: Let us start. So last lecture, we discussed the Compton scattering. So with that example essentially we covered all the tree-level diagrams in QED. So tree-level diagrams means those not involving loop. OK so far encountered diagrams, which are called the tree-level -- without loops. So these are called just tree-level diagrams. And so now you should be able to actually treat all possible tree-level diagrams. You already have all the tools to treat all possible tree-level diagrams. But tree-level diagrams, actually there are many important things missing just in the tree-level diagrams. So now today we will talk about how to go beyond that, so we talk about quantum fluctuations. So this is a very big subject and actually will consist of a big chunk of quantum field theory II. So here it's more just to give you a preview of the basic idea of the subject. So one feature of the tree-level diagram we have encountered. So if you ...