Non-covariant parts of the propagator?

[ismaili]

I'm reading Weinberg's volume I.
I don't quite understand what's the origin of the non-covariant parts of the propagator.

The propagator can be calculated to be
$$\Delta_{\ell m}(2\pi)^{-4}\int d^4q\frac{P_{\ell m}(q)\,e^{iq\cdot(x-y)}}{q^2+m^2-i\epsilon}\quad\cdots(*)$$
where
$$P_{\ell m}(q)\equiv 2\sqrt{\mathbf{q}^2+m^2}\sum_{\sigma}u_\ell(\mathbf{p},\sigma)u^*_m(\mathbf{p},\sigma)\quad\cdots(**)$$ is the spin sum of the field coefficients (coefficients of annihilation operator).
Eq(*) is an integral over all the possible four momentum $$q$$, but the polynomial $$P_{\ell m}(q)$$ defined by eq(**) is "on-shell", i.e. $$q^2=-m^2$$; hence, we have to extend the definition of the polynomial $$P_{\ell m}$$. Note that if $$q$$ is on-shell, we can always write $$P_{\ell m}(q)$$ as a polynomial "linear" in $$q^0$$. Hence, we define the generalized polynomial as a linear function of $$q^0$$, and the polynomial transforms covariantly, i.e.
$$P^{(L)}_{\ell m}(\Lambda q) = D_{\ell\ell'}(\Lambda)D^*_{mm'}(\Lambda)P^{(L)}_{\ell'm'}(q)$$, where $$D(\Lambda)$$ is certain representation of the Lorentz group. Now consider the polynomial of a massive vector field, $$P^{(L)}_{\ell m}(p) = \eta_{\mu\nu} + m^{-2}q_\mu q_\nu$$, since there is a quadratic term in $$q^0$$, hence actually the correct polynomial should be $$P^{(L)}_{\mu\nu} = \eta_{\mu\nu} + m^{-2}\left[q_\mu q_\nu - \delta^0_\mu\delta^0_\nu(q^2+m^2)\right]$$.
We substitute this polynomial into eq(*), hence we get the propagator for massive vector field is
$$(2\pi)^{-4}\int d^4q\frac{(\eta_{\mu\nu} + m^{-2}q_\mu q_\nu)e^{iq\cdot(x-y)}}{q^2+m^2-i\epsilon} + m^{-2}\delta^{(4)}(x-y)\delta^0_\mu\delta^0_\nu$$, the second term is the non-covariant part of the propagator.

What I don't understand is, why should we define the generalized polynomial in a way like what he did? Why the linear in $$q^0$$ so important? I'm confused and I don't get the logic and the reasoning, is there someone who can instruct me? These stuff is contained in p.277 or so basically. Any ideas would be appreciated.

 

[makehc]

You've probably already resolved this, but:

the extension of the polynomial doesn't need to be defined to be linear in q^0. Weinberg uses that form purely to simplify the manipulation of moving the derivative operators to the left of the theta functions in (6.2.12) (Note the second term's dependence on P^(1)_{lm}.) Then, he rewrites the result in terms of a covariant polynomial and a non-covariant term in (6.2.20,.21).