# How to integrate over light-ray directions in a covariant way?

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Say we are in Minkowski. Choose a set $S$ of null vectors $k^\mu$ with $(k^x)^2+(k^y)^2 + (k^z)^2 = (k^0)^2=1$. In the spatial sense, the vectors of $S$ span a "cellestial sphere". For every of these initial null vectors, there is a unique light-ray coming from the origin -- a vector not in $S$ would generate a light-ray already in the set, just with a linearly rescaled affine parametrization. We can thus define a projective equivalence $k^\mu \sim l^\mu$ if $k^\mu = \lambda l ^\mu, \lambda >0$ and work with null vectors modulo this equivalence. This set is isomorphic to $S$ and can be called "the true cellestial sphere".

The question now is: Is there a Lorentz-invariant measure over this set of light-ray directions/ the cellestial sphere?

We could integrate with a euclidean measure over the sphere in $(k^x)^2+(k^y)^2 + (k^z)^2 =1$ but this would trivially fail under boosts. I.e., the problem seems to be somehow analogous to the problem of gauge fixing in el-mag field quantization because a part of the solution is to covariantly constrain an unphysical degree of freedom of a massless particle (at least in the sense of classical light-rays, not waves), and $k^0=1$ will not do.

I thought about the geometrical construction of spinors and how 2-spinors can be understood exactly as directions of light-rays but the $\xi^\alpha, \alpha=1,2$ representation is plagued with the very same $\lambda$-redundancy as the $k^\mu$s. The only representation not plagued by the $\lambda$-redundancy is the one obtained by a stereographic projection from $S$ into the complex plane, i.e. through a single c-number on the Riemann sphere $z=\xi^1/\xi^2$. Lorentz transformations are then represented by restricted Möbius transformations but Möbius-invariant measure is impossible, at least in a canonical sense.

Another approach would be to try to "factor out" the $\lambda$-infinity or regularize it in a renormalization-like manner but approaching from time-like or space-like vectors really doesn't help. Any ideas?

asked Apr 30, 2015

The closest thing to what you want might be the traditional invariant measure on the mass-shell for a massless particle: $\int {d^3k \over 2|k|}$. This is not an integral over the celestial sphere, it is an integral over the momentum-cone. Consistent with the impossibility of invariant measure on the sphere, this measure can't be reduced from the cone to the sphere, because the measure diverges if you integrate over |k| at fixed angle.
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