We can build propagators (in this case, for massive vector fields) with at least two approaches:
(1) from canonical quantization (or exact solutions, if you like)
$$\left[[(i\partial)^2-m^2]\delta_{\space\space\nu}^\mu + (1-\frac{1}{\alpha})(i\partial^\mu)(i\partial_\nu)\right]W_{\nu}(x)= 0$$
$$D_{\mu\nu}(x-x') = \sum_{\vec{p},\lambda}W_{\mu,\lambda}^{(+)}(x) W_{\nu,\lambda}^{(+)*}(x')\theta(t-t') + W_{\mu,\lambda}^{(-)}(x) W_{\nu,\lambda}^{(-)*}(x')\theta(t'-t)$$ (2) or from path integral's propagator equation: $$\left[[(i\partial)^2-m^2]\delta_{\space\space\nu}^\mu + (1-\frac{1}{\alpha})(i\partial^\mu)(i\partial_\nu)\right]D_{\space\space\rho}^\nu(x-x')=\delta_{\space\space\rho}^\nu \delta(x-x')$$
When $\alpha\neq\infty$ there is 4th polarization's contribution to (2), i.e. terms involving $\alpha$. It is possible to solve equation in (1) for 4th polarization but it wouldn't have positive normalization (as long as this is not a physical solution that's OK). But I want to use this 4th polarization to build the propagator which will be identical to the one obtained by method (2).
Any ideas on how to incorporate (normalize, add/substract) the solution for 4th polarization into sum over the momenta and polarizations in (1)?
P.S. Is it somehow related to ghosts?
