Conservation of current induced by two-dimensional scale-invariance

Question

I'm writing my bachelor thesis in CFT (without ever having taken any courses in any field theory) and I'm trying to figure out why in two-dimensional scale-invariant theory the conservation of a current allways implies that its $z$ and $bar{z}$ components are seperately conserved. Where $z$ denotes the complex variable with positive imagenary part and $bar{z}$ the complex variable with negative imagenary part. Could someone derive this for me? Or give me an intuition?

Peter Kravchuk · Answer

Here is a fairly standard argument. In order for the charge operator to be dimensionless, we need $J$ to have dimension $1$. Consider then the two-point function of $J^+$. Poincare+scaling invariance implies
$$
langle 0| J^+(x)J^+(0) |0rangle = (x^-)^{-2}.
$$
This obviously implies
$$
langle 0| (partial_+J^+)(x)(partial_+J^+)(0) |0rangle = 0.
$$
Since the two-point function of the operator $partial_+J^+$ vanishes, it must be that this operator itself vanishes (this is a standard result in unitary QFT). Wick-rotation to Euclidean signature then leads to your statement.

Conservation of current induced by two-dimensional scale-invariance

One Answer

Add your own answers!

Ask a Question