I try to show the following relation

$$ [D,:\phi^n:]=-i\left(x^{\mu} \partial_{\mu} + \frac{n}{2}\right):\phi^n: $$

where $D$ is the Dilatation operator which is given by $$ D = -~i \sum_{l,m} \left(l +\frac{n}{2}\right) a_{l,m}^{\dagger}a_{l,m},$$ $\phi$ is a free massless field in 3 dimensions and $:\phi^n:$ means the normal ordered product of the fields.

My question is: Is there a compact nice expression for $:\phi^n:$ ? Because I only have the definition $$ :\phi^2:(x) = \lim_{y \to x} \phi(y) \phi(x) - \lim_{y \to x} \frac {1}{(d-2) \omega_d} |x-y|^{2-d}$$ with $d=3$ (dimension) and $\omega_d$ the volume of the sphere in $d$ dimensions. When I try to generalize this to the $n$-th power, I can't find a compact expression for this object.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.