# Where is the Yennie gauge useful in Gupta-Bleuer formalism (or QED in general)?

Consider the Lagrangian of the Gupta-Bleuer formalism given by: $$\mathcal{L} =-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} -\frac{1}{2\xi}(\partial A)^2.$$ I understand the necessity of the gauge fixing term: Without it, we would have $$\Pi^\kappa=F^{\kappa 0}$$ and therefore $$\Pi^0=F^{00}=0$$ due to the antisymmetry of the electromagnetic tensor, resulting in a contradiction with the commutator $$[A^0(t,x),\Pi^0(t,x')]=i\delta^3(x-x')$$ given by canonical quantization.

The Wikipedia page of gauge fixing claims, that the Feynman gauge $$\xi=1$$ and the Yennie gauge $$\xi=3$$ are often the most useful ones for calculations. I understand this for the Feynman gauge, as for example the Gupta-Bleuer propagator is (compare with equation 4.95 here for example) given by: $$\widetilde{\Delta}_{\mathrm{GB},\mu\nu}(k) =-\frac{i}{k^2+i\epsilon}\left(\eta_{\mu\nu}+(\xi-1)\frac{k_\mu k_\nu}{k^2}\right).$$ But where is the Yennie gauge useful? So far, I have not seen an example or found one. ("Yennie" is only mentioned in eight posts in total on PSE.) What is an example or a sketch of a derivation in Gupta-Bleuer formalism or QED, where the Yennie gauge is useful, or an explanation why it is special enough to deserve its own name?

• Did you do your research before posting this question? Literally the first result upon Google searching “Yennie gauge” is the article sciencedirect.com/science/article/pii/0003491680903309”. The very first paragraph of this paper answers your question. Jun 7 at 14:32
• @Prahar You can be assured I did do my research. (Given a seemingly simple question like this, it would be rather inappropriate not to do so.) But as my question indicates, I have searched mostly with "Gupta-Bleuer formalism" added or through scripts about it, that have only mentioned, but not explained the existence of the Yennie gauge - leaving me to assume, that there must be a deeper reason. Thank you for the link, the paper is not too long to not give it a full read. Jun 7 at 15:30

• Thank you! It seems like footnote 13 on page 430 about "the gauge, in which the renormalized photon propagator has the form $\delta_\nu^{\;\mu}/k^2$ for small $k^2$" is a simple answer, that doesn't even require the need to understand most of or all of the paper. Jun 7 at 17:37