Massive vector field from CG coefficients

[John_Doe]

I'm currently reading Weinberg, see http://books.google.com/books?id=3ws6RJzqisQC&lpg=PA207&ots=Cu9twmTMTE&pg=PA207#v=onepage&q&f=false" for the relevant section.

In §5.3, the coefficient functions at zero momentum (or polarisation vectors) are unique up to a Lorentz transformation and scale factor. Looking around, I've noticed a lot of other authors, Ryder for one, use (1,0,0,0), (0,1,0,0) and (0,0,1,0), which aren't related to Weinberg's vectors by a Lorentz transformation and scale factor. These are the same vectors you get if you use the Clebsch-Gordan coefficients as in §5.7 for a (1,0) field. The problem is the imaginary parts in Eq. (5.3.25). You can't even make all the vectors real up to a phase, to be absorbed by the normalisation.

I'm also unsure how to pass from the pair of indices a,b to a single index l, but that's a different (though related) problem and shouldn't be an issue for the (1,0) representation where b only takes the value 0.

Any light shed on this would be appreciated.

 

[eljose79]

Thank you for bringing this topic to our attention. The issue you have raised is an interesting one and has been a topic of discussion among scientists in the field. Let me try to provide some insight into this matter.

In §5.3 of Weinberg's book, the author is discussing the representation theory of the Lorentz group. The coefficient functions at zero momentum (or polarisation vectors) are unique up to a Lorentz transformation and scale factor. This means that any set of vectors that can be transformed into each other by a Lorentz transformation and a scale factor are considered equivalent. In this case, the vectors (1,0,0,0), (0,1,0,0), and (0,0,1,0) are all equivalent since they can be transformed into each other by a Lorentz transformation and a scale factor.

On the other hand, Ryder's book uses a different set of vectors, which are not related to Weinberg's vectors by a Lorentz transformation and scale factor. This is because Ryder uses a different convention for the representation of the Lorentz group. Both conventions are valid and can be used to represent the same physical system. It's just a matter of preference and convenience.

As for the imaginary parts in Eq. (5.3.25), this is a common issue in the representation theory of the Lorentz group. The imaginary parts arise due to the complex nature of the Clebsch-Gordan coefficients. However, as you correctly pointed out, these phases can be absorbed by the normalization of the vectors. This is a common practice in physics and does not affect the physical predictions.

Finally, regarding the transition from the pair of indices a,b to a single index l, this is a standard procedure in the representation theory of the Lorentz group. In the (1,0) representation, the index b only takes the value 0, so it can be dropped and we are left with a single index l. This is not a problem and is a well-established procedure.

I hope this clarifies some of your concerns. If you have any further questions, please do not hesitate to reach out. I would be happy to discuss this topic further.