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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.

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  • $\begingroup$ 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
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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

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

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

etc.

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