# The asymptotic behavior of the propagator of a field

In Steven Weinberg's book "The Quantum Theory of Fields" vol. I, Section 12.1, page 500, he writes:

We will write the asymptotic behaviour of the propagator $\Delta_f(k)$ of a field of type $f$ in the form $$\Delta_f(k)\sim k^{-2+2s_f}$$ Looking back at Chapter 6, we see that $s_f=0$ for scalar fields, $s_f=\frac{1}{2}$ for Dirac fields, ans $s_f=1$ for massive vector fields. More generally, it can be shown that for massive fields of Lorentz transformation type $(A,B)$, we have $s_f=A+B$. Speaking loosely, we may call $s_f$ the 'spin'."

How can we show that $s_f=A+B$ holds for massive fields of type $(A,B)$? Does anyone have some ideas of the proof? Thanks a lot in advance!

• note (§6.2): in Weinberg's notation, it would suffice to prove that $P_{ab,\tilde a\tilde b}(p)=\mathcal O(p^{A+B})$. This seems to be proved in §5.7, which cites Phys. Rev. 181, 1893, section V. Aug 2, 2017 at 16:24

This does not seem to require a very complicated answer. One can seperate it into two steps. As a first step one can assume that the reason that Weinberg loosely calls $s_f$ the "spin" is because it come out to have the same numerical value as the spin.
The next step then is to understand why the spin of a field would be given by the sum of the two parts in its formulation as a irreducible representation of the proper Lorentz group. This is a purely group theory question. The Lie algebra for the proper Lorentz group SO(1,3) is composed of tensor products of pairs of Pauli matrices. So the algebra of SO(1,3) is the same as the algebra of a product of two SU(2) groups. (There is a subtlety in that SO(3,1) is not compact, but that does not concern us now.) The irreducible representations of SO(1,3) are given as a pair of irreducible representations of SU(2) and this is usually just denoted by $(A,B)$, the respective spins of the two representations. The combined spin of the representation SO(1,3) is obtained in the same way that spin combines for SU(2): $A+B$.
• It seems to me that the question is about why the asymptotic of the propagator is $k^{-2+2(A+B)}$, and not about why $A+B$ is the spin. I might be wrong though. Aug 4, 2017 at 8:15