In the standard classical Maxwell theory, we use the following arguments to claim that there are only two propagating degrees of freesom

- $A_\mu$ has 4 components

- $A_0$ is non-dynamical (-1)

- $\mathcal{L}_\text{Maxwell}$  enjoys gauge symmetry and should be removed (-1)

So in total, we have $4 - 1 - 1 = 2$ dofs.


However, in a quantization procedure as discussed in David Tong's Quantum Field Theory lecture note (section 6.2.2 Lorentz Gauge), the Lagrangian is modified to
$$
\mathcal{L} = - \frac{1}{4} F_{\mu \nu}F^{\mu \nu} - \frac{1}{2} (\partial_\mu A^\mu)^2 \ .
$$

- $A_\mu$ has 4 components.

- $A_0$ is dynamical again

- there is no gauge symmetry left.

**Question: at this stage, can we claim that there are only 2 propagating dofs?** (I feel we cannot.)

Of course, subsequently we introduced the **Gupta-Bleuler condition**
$$
\partial^\mu A_\mu^{(+)} |\text{phy}\rangle = 0
$$
to make the observation (or rather, to ***decide***) that only the two transverse polarisation are "physical". **I feel, only at this stage, that we are in position to fix #dof to 2**.

**Please correct me if my understanding is incorrect** (the counting of dofs has been fuzzy in my head, although I know what words to say to get the right number).