gauge fixing

List of gauges

$\partial_\mu A^\mu = 0\,$ .

$A_0 = 0\,$

Also known as the Arnowitt-Fickler gauge. Choose

$n^\mu A_\mu = 0\,$

where $n^\mu\,$ is some spacelike vector. For example,

$A_3 = 0\,$ .

Sometimes $A_- = 0\,$ is used. The axial gauge is useful in that it decouples Faddeev-Popov ghosts. The photon propagator is

$-\frac{i}{k^2} \left( g^{\mu\nu} - \frac{k^\mu k^\nu}{(k\cdot n)^2} - \frac{k^\mu n^\nu + k^\nu n^\mu}{k \cdot n} \right)\,$

If $A_\mu\,$ is non-singular at the origin, then we may impose^[1]

$x^\mu A_\mu = 0\,$ ,

from which $x^\mu F_{\mu\nu} = x^\mu \partial_\mu A_\nu - x^\mu \partial_\nu A_\mu = (1 + x^\mu \partial_\mu) A_\nu\,$ . Therefore

$\alpha x^\mu F_{\mu\nu} (\alpha x)\,$	$= (1 + \alpha x^\mu \partial_\mu) A_\nu(\alpha x)\,$ ,
	$=\frac{d}{d\alpha}\left[ \alpha A_\nu(\alpha x)\right]\,$ ,

so that we may write

$A_\mu(x) = \int_0^1\!d\alpha\,\alpha x^\sigma F_{\sigma\mu}(\alpha x)\,$ .

$(\partial_\mu - i g A_\mu)A^\mu = 0\,$ .

↑ ^1.0 ^1.1 Stephan Narison (2004). QCD as a Theory of Hadrons: From Partons to Confinement. Cambridge University Press. ISBN 978-0521811644.
↑ J. L. Gervais and A. Neveu (1972). "Feynman rules for massive gauge fields with dual diagram topology". Nucl. Phys. B 46 (2): 381-401. DOI:10.1016/0550-3213(72)90071-5.
↑ Mark Srednicki (2007). Quantum Field Theory. Cambridge University Press, Cambridge, 486. ISBN 978-0521864497.