Learning Conformal Field Theory after QM

[jasonwer]

I will be starting a Masters degree in physics next year - 30% of the assessment will be a thesis (review, not research). I have selected CFT as a topic and my future advisor has pointed me to Francesco's 'Conformal Field Theory' as the book to use.

Francesco's book is very good but I am having trouble bridging the gap between QM and CFT. Obviously the bridge has QFT written on it - but how much QFT do I need? All of Weinberg Vol 1.? I'm concerned I may spend too much time on a interesting but unneeded side path of QFT while I should spend more time on, say, Lie algebra for example.

Don't get me wrong - I would love to sit down and work my way through every page of Weinberg (can I hear laughter in the back row?). But with a family, and a finite amount of time, I need to find an efficient path towards understanding the basics of CFT.

Has anyone been down this road and could give me some advice? Any help would be much appreciated.

 

[Fredrik]

I haven't read this thread, but I suspect that you will find it useful. I also suspect that you will find chapter 2 in Weinberg (vol. 1) very useful and the rest of it not so useful.

 

[jasonwer]

Thanks Fredrik - I suspected such. I thought learning CFT would allow me to learn QFT along the way but it seems they are quite separate topics with just canonical quantization in common. Am I correct in this view?

 

[Fredrik]

I haven't actually studied CFT, so I can't answer that question. I actually assumed that you had studied some QFT when I said that the rest of the Weinberg's book would probably be "not so useful". If you haven't studied any QFT, you might have to read (the first chapters of) a book about it, but in that case, you should probably get an easier book, like Mandl & Shaw.

You might want to send Sam a PM if he doesn't show up in this thread. (The guy who started that other thread). I'm sure he can answer your question.

 

[jasonwer]

I haven't actually studied CFT, so I can't answer that question. I actually assumed that you had studied some QFT when I said that the rest of the Weinberg's book would probably be "not so useful". If you haven't studied any QFT, you might have to read (the first chapters of) a book about it, but in that case, you should probably get an easier book, like Mandl & Shaw.

I should have been clearer about my (minimal) QFT background. I am part way through Weinberg chapter 2 but the canonical quantization procedure makes me uneasy - I can understand operators acting on Kets (thanks to matrix representations and linear algebra theory) but operators acting on fields is taking some getting used to. I'll try Mandl & Shaw like you suggested.

 

[Fredrik]

I can understand operators acting on Kets (thanks to matrix representations and linear algebra theory) but operators acting on fields is taking some getting used to. I'll try Mandl & Shaw like you suggested.

The operators aren't acting on fields. The fields are operators that act on kets. That's what the canonical quantization accomplishes. It takes a classical field, expresses it as a Fourier series* and plugs it into the field equations to get relationships between the Fourier coefficients, and finally promotes the Fourier coefficients to creation and annihilation operators. This makes the field an operator.

*) or a Fourier transform.

Weinberg's approach is almost the opposite: Start by considering the implications of the postulates of QM and the fact that the isometry group of spacetime is Minkowski space. This gives us a natural definition of one-particle states. Those Hilbert spaces can be combined to form a Hilbert space that can describe states with an arbitrary number of particles. Then he introduces creation and annihilation operators as functions that change n-particle states to n+1- or n-1-particle states, and finally concludes that they can be used to express the generator of translations in time in a way that makes a theory with interactions Lorentz invariant.

Weinberg's approach is really cool, but much more difficult to follow.

 

[Avodyne]

Weinberg is definitely NOT the place to start for an intro to QFT. Mandl & Shaw is good. Srednicki (available free online) is good, too, but is mostly from the path-integral point of view, which is less useful for CFT.

 

[jasonwer]

The operators aren't acting on fields. The fields are operators that act on kets. That's what the canonical quantization accomplishes. It takes a classical field, expresses it as a Fourier series* and plugs it into the field equations to get relationships between the Fourier coefficients, and finally promotes the Fourier coefficients to creation and annihilation operators. This makes the field an operator.

Thanks again Fredrik. I can now see where I made a massive wrong assumption.

I thought the field was the ket, therefore I thought the state of the system was given by the field. This led me to expect the creation and annihilation operators to act on the field (resulting in a field with more or less particles). Then last week, when I read something like 'the field is now an operator' (just like you wrote) I knew I had taken a wrong turn.

Ok, now I just need to make friends with the kets. When I see a state ket in QFT should I expect that this can be expanded in some basis or should I let go of some of my QM baggage? Is it helpful to think of these kets as living in some abstract space?

 

[Fredrik]

Ok, now I just need to make friends with the kets. When I see a state ket in QFT should I expect that this can be expanded in some basis or should I let go of some of my QM baggage? Is it helpful to think of these kets as living in some abstract space?

They are just vectors in some Hilbert space H. Don't worry about which Hibert space that is. (I think it can be constructed as the set of solutions of the field equation, at least in a theory without interaction terms in the Lagrangian, but the books usually don't even mention that).

A bra is a linear function that takes kets to complex numbers, i.e. it's a member of the dual space H*. There's a theorem that guarantees that for each ket $|\alpha\rangle\in H$, there's a bra $\langle\alpha|\in H^*$ that takes an arbitrary ket $|\beta\rangle$ to the scalar product $(|\alpha\rangle,|\beta\rangle$).

Recall that when T is a linear function acting on x, it's conventional to write Tx instead of T(x). This convention is used with bras. Also, whenever two | symbols should appear next to each other, only one is written out. So we have e.g.

[tex](|\alpha\rangle,\beta\rangle)=\langle\alpha|(|\beta\rangle)=\langle\alpha||\beta\rangle=\langle\alpha|\beta\rangle[/itex]

This takes some getting used to. This is one place where it gets confusing: If $X$ is an operator, $X^\dagger$ is defined by

$$(|\alpha\rangle,X|\beta\rangle)=(X^\dagger|\alpha\rangle,|\beta\rangle)$$

but in bra-ket notation, this equation is just

$$\langle\alpha|(X|\beta\rangle)=(\langle\alpha|X)|\beta\rangle$$

 

[samalkhaiat]

I will be starting a Masters degree in physics next year - 30% of the assessment will be a thesis (review, not research). I have selected CFT as a topic and my future advisor has pointed me to Francesco's 'Conformal Field Theory' as the book to use.

Francesco's book is very good but I am having trouble bridging the gap between QM and CFT. Obviously the bridge has QFT written on it - but how much QFT do I need? All of Weinberg Vol 1.? I'm concerned I may spend too much time on a interesting but unneeded side path of QFT while I should spend more time on, say, Lie algebra for example.

Don't get me wrong - I would love to sit down and work my way through every page of Weinberg (can I hear laughter in the back row?). But with a family, and a finite amount of time, I need to find an efficient path towards understanding the basics of CFT.

Has anyone been down this road and could give me some advice? Any help would be much appreciated

Hi Jason, I got your PM.

1) you need to decide about the dimension of your spacetime! The mathematical (and therefore physical) nature of the conformal group C(1,n-1) changes drastically when n = 2(i.e., in 2-dimensional spacetime).
The book you mentioned is about conformal symmetry in 2-dimensional quantum field theories. Indeed, quantum C(1,1) field theory is the best framework for the study of ; systems at the critical point of a 2nd order phase transition, vacuum configurations of strings, braid group statistics, topological field theories, integrable systems, the fractional Hall effect and high teperature superconductivity.
The QCFT C(1,1) requires you to have a previous knowledge of; affine Lie algebras, Virasoro algebra, W-algebra and ofcourse all technical aspects of QFT such as correlation functions, Ward identities, operator product expansion, renormalization, anomalies etc.

2) you need to understand the following structures

Classical field theory(CFT) = Continuous Mechanics + Poicare group as a symmetry.

QFT = CFT + quantization rule.

Classical Conformal field theory(CCFT) = Continuous mechanics + C(1,n-1) as a symmetry group.

QCFT = CCFT + quantization rule

So, you need to start with; the conformal group, the conformal algebra and the Lagrangian formulation of classical field theory. More or less, this is the subject of the thread

www.physicsforums.com/showthread.php?t=172461

After all, my aim from that thread was to make the subject and some of its immediate consequences accessible to graduate students.

good luck with your work

regards

sam

 

[jasonwer]

They are just vectors in some Hilbert space H. Don't worry about which Hibert space that is. (I think it can be constructed as the set of solutions of the field equation, at least in a theory without interaction terms in the Lagrangian, but the books usually don't even mention that).

I realized too late that my last question was somewhat obvious. With the QFT operators given in terms of creation and annihilation operators it makes sense for the Kets to be given in terms of particle number. I spent the weekend with Mandl & Shaw (like you suggested, thanks) and it confirmed my guess. I can also finally see why Mandl & Shaw is recommended as a first QFT book.

I'm familiar with the Dirac notation, the harmonic oscillator etc. but wasn't sure how much carries across to QFT. For example - in QM we can determine the time evolution of a Ket but I suspect that we can't (or don't need to) determine the time evolution of the vacuum state Ket in QFT. Such questions are keeping things interesting and motivating further study so there is no need to spoil the punch line!

 

[jasonwer]

So, you need to start with; the conformal group, the conformal algebra and the Lagrangian formulation of classical field theory. More or less, this is the subject of the thread

www.physicsforums.com/showthread.php?t=172461

Thanks Sam, I'll definitely use your thread as a 'roadmap' for CFT and to check my understanding.

Indeed, quantum C(1,1) field theory is the best framework for the study of;
systems at the critical point of a 2nd order phase transition, vacuum configurations of strings, braid group statistics, topological field theories, integrable systems, the fractional Hall effect and high teperature superconductivity

CFT seems to be a really 'cool' topic given it's mathematical beauty and wide range of applications, but for some reason it does not seem very popular. Do you know why this may be? Is it because CFT may be seen as just a part of string theory?

The reasons why I have formed this view are...

'I have noticed that questions about this subject get either ignored or receive some confusing answers' .This quote is from your CFT thread (as you would know) and I've noticed the same situation.

I've also found that a Google or Arxiv search for CFT brings up mainly CFT in the context of string theory or AdS-CFT correspondence but not as an independent field of study.

 

[nrqed]

[

CFT seems to be a really 'cool' topic given it's mathematical beauty and wide range of applications, but for some reason it does not seem very popular. Do you know why this may be? Is it because CFT may be seen as just a part of string theory?

The reasons why I have formed this view are...

'I have noticed that questions about this subject get either ignored or receive some confusing answers' .This quote is from your CFT thread (as you would know) and I've noticed the same situation.

I've also found that a Google or Arxiv search for CFT brings up mainly CFT in the context of string theory or AdS-CFT correspondence but not as an independent field of study.

It is true that it is hard to get help on that topic. But this is always the case with advanced topics. There are few people around who know enough of the stuff to not only explain it, but explain it well.

You should look at books specifically on CFT. For example the book by Mathieu, Senechal et al.

Or there are several very good introductory papers on the archives.