We know that if we try to quantize the free electromagnetic field without a gauge fixing term added to the Lagrangian, then one of the conjugate momentum density $\Pi^0$ vanishes. We also find that the propagator doesn't exist, the reason of which is usually attributed to the operator $(g^{\lambda\mu}\square-\partial^\mu\partial^\lambda)$ being non-invertible (singular).

My question is whether the non-existence of the propagator is related to the vanishing of $\Pi^0$? I think it is related. The reason I think so is that this problem of non-existence of propagator doesn't arise in case of scalar field or Dirac field where $\Pi^0\neq 0$. If I am on the right track, then where is really the connection between these two problems? Or is it just a coincident?

EDIT : The vanishing of $\Pi^0$ is obviously a problem with canonical quantization because the commutation relations cannot be satisfied. Does the non-existence of the propagator also pose any problem as far as the canonical quantization procedure is concerned?

If we try to quantize the free electromagnetic field without a gauge fixing term added to the Lagrangian density $\mathscr{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, (i)zeroth component of the conjugate momentum density $\pi^0=$ vanishes, and (ii)also the propagator doesn't exist where the reason for its non-existence is usually attributed to the operator $(g^{\lambda\mu}\square-\partial^\mu\partial^\lambda)$ being non-invertible (singular).

My question is whether (i)the vanishing of $\pi^0$ and (ii) the non-existence of the propagator is related? 

I think they are related because the problem of non-existence of the propagator doesn't arise in the case of a scalar field or Dirac field where $\pi^0\neq 0$. Moreover, the fixation of gauge solves both the problems at one shot. But I'm not sure where is the relation connection these two problems.

We know that if we try to quantize the free electromagnetic field without a gauge fixing term added to the Lagrangian, then one of the conjugate momentum density $\Pi^0$ vanishes. We also find that the propagator doesn't exist, the reason of which is usually attributed to the operator $(g^{\lambda\mu}\square-\partial^\mu\partial^\lambda)$ being non-invertible (singular).

My question is whether the non-existence of the propagator is related to the vanishing of $\Pi^0$? I think it is related. The reason I think so is that this problem of non-existence of propagator doesn't arise in case of scalar field or Dirac field where $\Pi^0\neq 0$. If I am on the right track, then where is really the connection between these two problems? Or is it just a coincident?

We know that if we try to quantize the free electromagnetic field without a gauge fixing term added to the Lagrangian, then one of the conjugate momentum density $\Pi^0$ vanishes. We also find that the propagator doesn't exist, the reason of which is usually attributed to the operator $(g^{\lambda\mu}\square-\partial^\mu\partial^\lambda)$ being non-invertible (singular).

My question is whether the non-existence of the propagator is related to the vanishing of $\Pi^0$? I think it is related. The reason I think so is that this problem of non-existence of propagator doesn't arise in case of scalar field or Dirac field where $\Pi^0\neq 0$. If I am on the right track, then where is really the connection between these two problems? Or is it just a coincident?

EDIT : The vanishing of $\Pi^0$ is obviously a problem with canonical quantization because the commutation relations cannot be satisfied. Does the non-existence of the propagator also pose any problem as far as the canonical quantization procedure is concerned?