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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 in Theoretical Physics by Void (1,505 points) [ no revision ]

1 Answer

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The question is self-answering--- a Lorentz invariant measure has to be rotationally invariant, in which case it is the uniform measure on the sphere around the origin, and then, as you said, it can't be boost invariant. Equivalently, as you noticed (and as is discussed a lot in Penrose's "Spinors and Space Time"), the Lorentz transformations make a Mobius transformation on the directions, and there are no Mobius invariant measures.

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.

answered Apr 30, 2015 by Ron Maimon (7,535 points) [ no revision ]

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