Non-covariance of the higher rank propagator (from Weinberg's QFT textbook)

Question

In chapter 6.2 of Weinberg's QFT Vol.1, he gave the general form of Wick contractions of all possible fields(scalar, spinor, vector, etc.), he showed

$$\Delta_{lm}(x,y)=\theta(x-y)P^{(L)}_{lm}\left(-i\frac{\partial}{\partial x}\right)\Delta_+(x-y) +\theta(y-x)P^{(L)}_{lm}\left(-i\frac{\partial}{\partial x}\right)\Delta_+(y-x)\tag{6.2.8}$$

where $$\Delta_+(x)=\int d^3p(2p^0)^{-1}e^{ip\cdot x},$$ and $P^{(L)}_{lm}$ is a covariant polynomial if its argument is a on-shell 4-momentum(i.e. $P^{(L)}_{lm}(\sqrt{\mathbf{p^2}+m^2},\mathbf{p})$), but may not be covariant for off-shell 4-momentum.

Then by a series of mathematical identities, he was able to show $\Delta_{mn}(x,y)$ can be written in a 4-momentum integral:

$$\Delta_{lm}(x,y)=\int d^4q\frac{P^{(L)}_{lm}(q)\exp(iq\cdot (x-y))}{q^2+m^2-i\epsilon}\tag{6.2.18}$$

From here he argued since the 4-momentum $q$ in $(6.2.18)$ is not always on-shell, $P^{(L)}_{lm}$ may not be covariant, and in turn the $\Delta_{mn}(x,y)$ may not be covariant. It's indeed not for a vector field as what he showed immediately after, and that's why we need to add a non-covariant term in the Hamiltonian to make the propagtor covariant ect.etc.

I did follow the steps he showed, but something strange occured to me when I looked back at the expression $(6.2.8)$: $(6.2.8)$ looks completely covariant, since $\theta$ is invariant, $\Delta_+$ is invariant and is written in 3-momentum integral so that $P^{(L)}_{lm}(-i\frac{\partial}{\partial x})\Delta_+(x-y)$ must be covariant, yet $(6.2.18)$-which is supposed to be equivalent to $(6.2.8)$- is not covariant. I am wondering if there's any subtlety I missed.

Also there's another mystery to me: in chapter 3.5 (page 144), he gave a perturbative proof of Lorentz invariance based on Dyson series, the invariance of Hamiltonian density and micro-causality condition, and he did mention Lorentz invariance could be disturbed by the reasonings given in chapter 6.2, but the proof in chap 3.5 is completely formal and I can't see exactly how the reasonings in 6.2 can jeopardize it.

(PS: I wish I could have formulated my question in a more self-contained way, but I couldn't unless I copied some whole pages of Weinberg, so I apologize for the potential vagueness in advance)

Answer 1

I don't know if the poster is still interested in this question. Anyway here are my two cents.

Without introducing $P^{(L)}_{\ell m}$, just consider the massive spin one case and use E.(6.2.6) into Eq.(6.2.8).

Eq.(6.2.6) can be written as $P_{\mu\nu} = \sum_{n=0}^2 P_{\mu\nu}^{(n)}$ where $P_{\mu\nu}^{(i)}$ is proportional to $q_0^n$.

Follow Weinberg steps and you'll get an equation similar to (6.2.12) with terms proportional to $P_{\mu\nu}$, $\delta(x_0-y_0) P_{\mu\nu}^{(1)}$ and $\delta^\prime (x_0-y_0) P_{\mu\nu}^{(2)}$.

The $\delta$ term vanishes but the $\delta^\prime$ doesn't and gives the last term in Eq.(6.2.21).

Bottom line: Eq.(6.2.8) is not covariant because $\theta(x_0-y_0) \partial_\mu f(x-y)$ is not covariant (the $x_0$ dependence in f results in $\delta^{(n)} terms).