# Fock-Schwinger gauge in pure Yang-Mills theory and coordinate dependence of equations of motion

Let $$(\Omega^\bullet (\mathbb{R}^n,\mathfrak{g}),d_A)$$ be the Yang-Mills cochain complex on $$\mathbb{R}^n$$, where $$d_A$$ is the gauge covariant derivative, $$d_A \circ d_A=0$$. I was wondering: if we place ourselves in the Fock-Schwinger gauge $$x \cdot A=0$$, doesn't it make the equation of motion $$d_A \star F=0$$ coordinate dependent when we express $$A$$ with $$F$$ via the inversion formula?

• Do you mean origin dependent? – user263715 Mar 1 at 9:34
• I mean that the equations of motion depend explicitly on the coordinates variables since by the inversion formula $A$ depends explicitly on $x$. – Jeanbaptiste Roux Mar 1 at 9:37
• I would agree @user263715, you should not get a dependence of the coordinate system but of a reference point. Since you'r concerned with $\mathbb{R^n}$ a natural choice would be the origin. Rather than a space point on the manifold, the $x$ in the formula $x\cdot A = 0$ than denotes the vector pointing from your reference $\vec 0$ to the space point you're concerned with. This is because a scalar product is an operation on vectors but not spacepoints. So we do not need any coordinate system to make this expression meaningful. – Johnny Longsom Mar 1 at 10:19