I'm reading Weinberg's volume I.(adsbygoogle = window.adsbygoogle || []).push({});

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

The propagator can be calculated to be

[tex]\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(*)[/tex]

where

[tex]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(**)[/tex] is the spin sum of the field coefficients (coefficients of annihilation operator).

Eq(*) is an integral over all the possible four momentum [tex]q[/tex], but the polynomial [tex]P_{\ell m}(q) [/tex] defined by eq(**) is "on-shell", i.e. [tex]q^2=-m^2[/tex]; hence, we have to extend the definition of the polynomial [tex]P_{\ell m}[/tex]. Note that if [tex]q[/tex] is on-shell, we can always write [tex]P_{\ell m}(q)[/tex] as a polynomial "linear" in [tex]q^0[/tex]. Hence, we define the generalized polynomial as a linear function of [tex]q^0[/tex], and the polynomial transforms covariantly, i.e.

[tex]P^{(L)}_{\ell m}(\Lambda q) = D_{\ell\ell'}(\Lambda)D^*_{mm'}(\Lambda)P^{(L)}_{\ell'm'}(q)[/tex], where [tex]D(\Lambda)[/tex] is certain representation of the Lorentz group. Now consider the polynomial of a massive vector field, [tex]P^{(L)}_{\ell m}(p) = \eta_{\mu\nu} + m^{-2}q_\mu q_\nu[/tex], since there is a quadratic term in [tex]q^0[/tex], hence actually the correct polynomial should be [tex]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][/tex].

We substitute this polynomial into eq(*), hence we get the propagator for massive vector field is

[tex](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[/tex], 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 [tex]q^0[/tex] 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.

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# Non-covariant parts of the propagator?

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