In this source, the author (Andrzej Pokraka, Solutions to problems from Peskin & Schroeder) is computing an integral related to scalar QED. In the step where the equation is labelled (29), the author is changing the expression $l_\mu l_\nu$ to $\frac{1}{d} g_{\mu\nu} l^2$ (where $d$ is the spacetime dimensions). How can you do that? I have tried playing with the indices, but I don't see yet a way to transfer indices from the metric to the $l$'s.

  • $\begingroup$ Maybe it is not that $l_{\mu} l_{\nu} = \frac{1}{d} g_{\mu \nu} l^2$, but rather $\int l_{\mu} l_{\nu} = \int \frac{1}{d} g_{\mu \nu} l^2$ ? $\endgroup$ – MBolin Jul 18 at 23:11
  • $\begingroup$ There’s the rest of that sentence top of the next pages that says they dropped terms. Trace of the metric will give you the part proportional to $d$. $\endgroup$ – Brick Jul 18 at 23:19
  • $\begingroup$ @Brick Yes they dropped terms linear in $l$, but isn't that referring to the previous steps? $\endgroup$ – Jxx Jul 18 at 23:21
  • $\begingroup$ @MBolin Mm that sounds quite plausible. I am starring at it with the integral now, and I am still stuck though. $\endgroup$ – Jxx Jul 18 at 23:22
  • $\begingroup$ @Jxx I am quite sure that's what happens. See eg. equation (10.79) in pages.physics.cornell.edu/~ajd268/Notes/UsefulFormulas.pdf. I can't give an explanation right now though. But at least you can check that if you contract with $g_{\mu \nu}$ you get the same thing on both sides. $\endgroup$ – MBolin Jul 18 at 23:31

The result of the integral is a rank-2 Lorentz tensor that depends only on the Lorentz scalar $q^2$. This means that it is a rank-2 isotropic Lorentz tensor; i.e., none of its components change under Lorentz transformations. The Lorentz group has only two fundamental isotropic tensors: $g_{\mu\nu}$ and $\epsilon_{\mu\nu\kappa\lambda}$. All others are built from these. Thus the integral must be proportional to $g_{\mu\nu}$ and the constant of proportionality is found by contracting.

This is the Minkowski-space analog of the Euclidean-space integration of products of unit vectors over all angles, such as

$$\int d\Omega\,n_i n_j \propto \delta_{ij},$$

$$\int d\Omega\,n_i n_j n_k n_l \propto \delta_{ij}\delta_{kl} + \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk},$$



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.