I'm going over Greiner's field quantization book.
In chapter 7 quantization of a massless spin 1 field, the book says in the Lorenz gauge, we have equation 7.26:
$$\hat{A}^\mu(x)=\int\frac{d^3k}{\sqrt{2\omega_k(2\pi)^3}}\sum_{\lambda=0}^3\big(\hat{a}_{\boldsymbol{k}\lambda}\epsilon^\mu(\boldsymbol{k},\lambda)e^{-ik\cdot x}+\hat{a}_{\boldsymbol{k}\lambda}^\dagger\epsilon^\mu(\boldsymbol{k},\lambda)e^{ik\cdot x}\big).\tag{7.26}$$
here $\epsilon^\mu(\boldsymbol{k},\lambda)$ are certain chosen polarization vectors.
In the Coulomb gauge, we have $A_0(\boldsymbol{x},t)=0$ and equation 7.105
$$\hat{\boldsymbol{A}}(\boldsymbol{x},t)=\int\frac{d^3k}{\sqrt{2\omega_k(2\pi)^3}}\sum_{\lambda=1}^2\boldsymbol{\epsilon}(k,\lambda)(\hat{a}_{\boldsymbol{k}\lambda}e^{-ik\cdot x}+\hat{a}_{\boldsymbol{k}\lambda}^\dagger e^{ik\cdot x}\big).\tag{7.105}$$
Here $\epsilon^\mu(\boldsymbol{k},\lambda)$ are another set of polarization vectors.
My questions are: which one should I use in general? How do we see that they give the same answer?
Best Answer
One may show that gauge-invariant physical observables of a gauge theory (in this case QED) does not depend on the specific gauge-fixing condition (e.g. Lorenz gauge, Coulomb gauge, etc). See e.g. my related Phys.SE answer here.