My aim is to derive the photon propagator in an Coulomb gauge following Pokorski's book method.(adsbygoogle = window.adsbygoogle || []).push({});

In this book the photon propagator in Lorenz gauge was obtained as follows:

Now, in Coulomb gauge,

- Lorenz gauge: ##\partial_{\mu}A^{\mu}=0##
- It's proved that ##\delta_{\mu}A^{\mu}_T=0##, where ##A^{\mu}_T=(g^{\mu\nu}-\frac{\partial^{\mu}\partial{\nu}}{\partial^2})A^{\mu}## is the transverse field.
- Then, ##\partial^2A^T_{\mu}=0\rightarrow (\partial^2-i\epsilon)D_{\mu\nu}(x-y)=-(g_{\mu\nu}-\frac{\partial_{\mu}\partial_{\nu}}{\partial^2})\delta(x-y)##, is the equation for the corresponding the Green's function in the transverse space.
- After a Fourien transformations this becomes ##(-k^2-i\epsilon)\tilde{D}_{\mu\nu}(k)=-(g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^2})##.

$$\tilde{D}^{\alpha\beta}_{\mu\nu}=\frac{\delta^{\alpha\beta}}{k^2+i\epsilon}\left[g_{\mu\nu}-\frac{k\cdot n(k_{\mu}n_{\nu}+k_{\nu}n_{\mu})-k_{\nu}k_{\mu}}{(k\cdot n)^2-k^2}\right]$$.

- Coulomb gauge: ##\partial_{\mu}A^{\mu}-(n_{\mu}\partial^{\mu})(n_{\mu}A^{\mu})=0, \; n_{\mu}(1,0,0,0)##

- I've tried to do the same program as before but i'm stuck. It's supose the propagator we have to obtain is:

The reference,

Gauge Field Theories, 2000. Stefan Pokorski. Pages: 129-132.

I'll appreciate any help.

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# A Photon propagator in Coulomb gauge

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