- #1
ismaili
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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.
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.