CFT charge/transformation question

[da_willem]

Struggling my way through Conformal Field Theory before getting into strin theory I stumbled upon the following quantum mechanical relation:

$$\delta_{\epsilon} \phi (z, \bar{z}) = [Q_{\epsilon}, \phi (z, \bar{z})$$

Thus some conformal transformation in the field (with parameter epsilon) is equal to the commutator of the associated conserved charge with this field. It reminded me of the relation in QM of the time dependence of an operator and its relation to the commutator with the Hamiltonian.

Can somebody tell me how the above relation is founded in QM/Noethers theorem/CFT ?

 

[nonplus]

Struggling my way through Conformal Field Theory before getting into strin theory I stumbled upon the following quantum mechanical relation:

$$\delta_{\epsilon} \phi (z, \bar{z}) = [Q_{\epsilon}, \phi (z, \bar{z})]$$

Thus some conformal transformation in the field (with parameter epsilon) is equal to the commutator of the associated conserved charge with this field. It reminded me of the relation in QM of the time dependence of an operator and its relation to the commutator with the Hamiltonian.

Can somebody tell me how the above relation is founded in QM/Noethers theorem/CFT ?

It is explained (briefly) at page 74 in Zee's QFT book. Very roughly you know that for a conserved current the integral over space= the charge Q, is proportional to the canonical momentum conjugate to the field. Combine this with [x,p]=i and you are there.
And so the conserved charges are the generators of symmetries in field theory.

Good luck with the CFT's, it's hard work...

 

[da_willem]

I read your reply and the page in Zee you mentioned and now it makes sense! I already had some feeling for the relation, but to see where it comes is great, thanks! Also for the 'good luck' part, I'll need it.