I've always treated propagator in the momentum representation so when it diverges, we are on-shell. But what is the interpretation of light-cone divergences in position space? If it is something we have to renormalize, how can I do it? More precisely, if I have to renormalize light-cone divergences of the following Feynman propagator: \begin{equation*} G(s)=\frac{m}{8\pi s} H^{(1)}_1(m s)\,\,,\,\,s^2=(x^0-y^0)^2-|\vec{x}-\vec{y}|^2 \end{equation*} (where $H^{(1)}_1$ is a Hankel function of the first type) how should I proceed? Is there a method like dimensional regularization for this problem?


1 Answer 1


There is no need to renormalize here.

The interpretation is that a quantum field is not an operator-valued function $\hat{\phi}(x)$, but an operator-valued distribution, aka generalized function. That means that $\hat{\phi}$ is a linear functional from test functions on space-time (e.g. you can choose functions with compact support to get temporal distributions).

Intuitively, $$ \hat{\phi}(f) = \int d^4 x f(x) \hat{\phi}(x). $$

But this formula is only formal, because $\hat{\phi}(x)$ doesn't mathematically exist.

For any two test functions $f_{1,2}$, the quantity

$$ \left< \hat{\phi}(f_1) \hat{\phi}(f_2) \right> = \int d^4 x f_1(x) \int d^4 y f_2(y) \cdot G(x - y) $$

is finite and well defined, even when $G$ has a divergence on the light cone.


Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.