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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?

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  • $\begingroup$ Do you mean origin dependent? $\endgroup$ – user263715 Mar 1 at 9:34
  • $\begingroup$ I mean that the equations of motion depend explicitly on the coordinates variables since by the inversion formula $A$ depends explicitly on $x$. $\endgroup$ – Jeanbaptiste Roux Mar 1 at 9:37
  • $\begingroup$ 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. $\endgroup$ – Johnny Longsom Mar 1 at 10:19

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