Introduction to conformal field theory, Lecture 5

In this seminar I will, over some 10 lectures, introduce the basics of conformal field theory. The emphasis will be on the physical content, however, there will be reference to mathematical formulations throughout.

The course is based on a mixture of Ginsparg's "Applied Conformal Field Theory", hep-th/9108028 and Schottenloher's "A mathematical introduction to conformal field theory".

Prerequisites for the course comprise: advanced QM, QFT, advanced QFT, and some familiarity with symplectic methods, which you can cover by watching my previous videos.

In the fifth lecture I introduce radial quantisation.

[abhisheknavhal]

22:11, 33:55, 45:06, 52:28, 58:25, 1:04:29, 1:10:33, 1:15:09, 1:21:36 oddly satisfying! Amazing lecture. The digression on what is "physical" was beautiful. 1:21:30 reflection in the board, this guy is tired :P

[kenkutube]

Thank you for posting your lectures! I've enjoyed all of your earlier courses! Looking forward to seeing more of your brilliant lectures.

[inspiredzozo]

What is the wiper you used on the blackboard at 21:50? I've never seen anything like it before

[jingyang]

Hi Tobias, very nice and clear lecture! One comment: When you wrote down the Ward identity at about 1:28:00, if you express the right-hand side in terms of operators, then the must be time-ordering there.

[wwsnake]

saying time is emergent phenomenon is interesting although we don't know the exact definition of time. Just imagine there is only one hydrogen atom in the universe, which is stationary in the ground state. At this moment, we can sort of say there is no time. As long as you add another one, then you will have dynamics and need something to record the 'event'. When I write to here, I realise light might be important for defining time.

[yan]

Great lecture! Thank you! I am quite late with that question, but I have been quite confused for some time about "local symmetries". I know and I convinced myself in a lot of cases, that gauge invariance is not a physical symmetry and can not be spontaneously broken (as wrongly suggested in a big chunk of literature as far is I know {like with the Higgs mechanism, where at no point ever gauge invariance is actually broken, the gauge freedom just reduces, because in the process we fix n gauge freedoms, corresponding to n massive gauge bosons in the end}).

But there is this suggested analogy in literature: global symmetry <-> infinitesimal transformation parameter is independent of the spacetime argument x. And local symmetry <-> gauge "symmetry" [rather invariance, but nevermind].

So I would like to know, are all local symmetries actually gauge "symmetries", or are there physical local symmetries? With physical I mean, that in contrast to a gauge transformation, a physical transformation actually does something to the physical system, but the equations of motion do not change, where for the "unphysical" gauge transformation, we just change a reference section in e.g. the complex line bundle, so we change the "basis" of the wavefunction, meaning we just change our description of the system, not the system itself.

[andrewhardy]

I'm wondering if there was a reason during the digression on reality that you chose not to talk about physical systems (like the Ising model) which has temperature dynamics and in which CFT is often applied to without time? These seems like more physical examples and more intuitive as to why we care about thermal field theories?

[Physics_Lad]

I am confused, so we (1) go to euclidean space with the space coordinate \sigma_1 -> \iota \sigma_1, then (2) compactify and (3) define new coordinates. Now, \sigma_0 and \sigma_1 both being are identical under rotational invariance. then how do we say time translation is \sigma_0 -> \sigma_0 + a?