In QED, the photon field has the following commutation relations: \begin{equation} [A^{\mu}(t,\vec{x}),A^{\nu}(t,\vec{y})]=0, \tag{1} \end{equation} where $A^{\mu}(t,\vec{x})$ is the photon filed. As for QCD, I can not find similar relations for gluon, can we have \begin{equation} [A_a^{\mu}(t,\vec{x}),A_a^{\nu}(t,\vec{y})]=0 \, \, \, \, \, (2) \end{equation} for gluon field with color index $a$? Or do we have the following equation \begin{equation} [A_a^{\mu}(t,\vec{x}),A_a^{\nu}(t,\vec{x})]=0 \, \, \, \, \, (3) \end{equation} for gluon at same $x$? From the definition of gluon field-strength tensor \begin{equation} F^{\mu \nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}-ig[A^{\mu},A^{\nu}], \, \, \, \, \, (4) \end{equation} where $A^{\mu}=A_a^{\mu}t^a$, the last term of Eq.(4) is often reexpressed as \begin{equation} [A^{\mu},A^{\nu}]=A_a^{\mu}A_b^{\nu}t^a t^b-A_b^{\nu}A_a^{\mu}t^bt^a=A_a^{\mu}A_b^{\nu} [t^a, t^b].\, \, \, \, \, (5) \end{equation} Does Eq(5) implies that \begin{equation} A_a^{\mu}A_b^{\nu}=A_b^{\nu}A_a^{\mu} \, \, \, \, \, (6) \end{equation} thus gluon commutes at the same position. Could someone tell me which of the above equations are not correct, or how we understand the commutation relations of gluon?

In QED, the photon field has the following commutation relations: \begin{equation} [A^{\mu}(t,\vec{x}),A^{\nu}(t,\vec{y})]=0, \tag{1} \end{equation} where $A^{\mu}(t,\vec{x})$ is the photon filed. As for QCD, I can not find similar relations for gluon, can we have \begin{equation} [A_a^{\mu}(t,\vec{x}),A_a^{\nu}(t,\vec{y})]=0 \, \, \, \, \, (2) \end{equation} for gluon field with color index $a$? Or do we have the following equation \begin{equation} [A_a^{\mu}(t,\vec{x}),A_a^{\nu}(t,\vec{x})]=0 \, \, \, \, \, (3) \end{equation} for gluon at same $x$? From the definition of gluon field-strength tensor \begin{equation} F^{\mu \nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}-ig[A^{\mu},A^{\nu}], \, \, \, \, \, (4) \end{equation} where $A^{\mu}=A_a^{\mu}t^a$, the last term of Eq.(4) is often reexpressed as \begin{equation} [A^{\mu},A^{\nu}]=A_a^{\mu}A_b^{\nu}t^a t^b-A_b^{\nu}A_a^{\mu}t^bt^a=A_a^{\mu}A_b^{\nu} [t^a, t^b].\, \, \, \, \, (5) \end{equation} Does Eq.(5) implies that \begin{equation} A_a^{\mu}A_b^{\nu}=A_b^{\nu}A_a^{\mu} \, \, \, \, \, (6) \end{equation} thus gluon commutes at the same position. Could someone tell me which of the above equations are not correct, or how do we understand the commutation relations of gluon?

In QED, the photon field has the following commutation relations: \begin{equation} [A^{\mu}(t,\vec{x}),A^{\nu}(t,\vec{y})]=0, \, \, \, \, \, (1) \end{equation} where $A^{\mu}(t,\vec{x})$ is the photon filed. As for QCD, I can not find similar relations for gluon, can we have \begin{equation} [A_a^{\mu}(t,\vec{x}),A_a^{\nu}(t,\vec{y})]=0 \, \, \, \, \, (2) \end{equation} for gluon field with color index $a$? Or do we have the following equation \begin{equation} [A_a^{\mu}(t,\vec{x}),A_a^{\nu}(t,\vec{x})]=0 \, \, \, \, \, (3) \end{equation} for gluon at same $x$? From the difinition of gluon field-strength tensor \begin{equation} F^{\mu \nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}-ig[A^{\mu},A^{\nu}], \, \, \, \, \, (4) \end{equation} where $A^{\mu}=A_a^{\mu}t^a$, the last term of Eq.(4) is often reexpressed as \begin{equation} [A^{\mu},A^{\nu}]=A_a^{\mu}A_b^{\nu}t^a t^b-A_b^{\nu}A_a^{\mu}t^bt^a=A_a^{\mu}A_b^{\nu} [t^a, t^b].\, \, \, \, \, (5) \end{equation} Does Eq(5) implies that \begin{equation} A_a^{\mu}A_b^{\nu}=A_b^{\nu}A_a^{\mu} \, \, \, \, \, (6) \end{equation} thus gluon commutes at the same postion. Could someone tell me which of the above equations are not correct, or how we understand the commutation relations of gluon?

In QED, the photon field has the following commutation relations: \begin{equation} [A^{\mu}(t,\vec{x}),A^{\nu}(t,\vec{y})]=0, \tag{1} \end{equation} where $A^{\mu}(t,\vec{x})$ is the photon filed. As for QCD, I can not find similar relations for gluon, can we have \begin{equation} [A_a^{\mu}(t,\vec{x}),A_a^{\nu}(t,\vec{y})]=0 \, \, \, \, \, (2) \end{equation} for gluon field with color index $a$? Or do we have the following equation \begin{equation} [A_a^{\mu}(t,\vec{x}),A_a^{\nu}(t,\vec{x})]=0 \, \, \, \, \, (3) \end{equation} for gluon at same $x$? From the definition of gluon field-strength tensor \begin{equation} F^{\mu \nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}-ig[A^{\mu},A^{\nu}], \, \, \, \, \, (4) \end{equation} where $A^{\mu}=A_a^{\mu}t^a$, the last term of Eq.(4) is often reexpressed as \begin{equation} [A^{\mu},A^{\nu}]=A_a^{\mu}A_b^{\nu}t^a t^b-A_b^{\nu}A_a^{\mu}t^bt^a=A_a^{\mu}A_b^{\nu} [t^a, t^b].\, \, \, \, \, (5) \end{equation} Does Eq(5) implies that \begin{equation} A_a^{\mu}A_b^{\nu}=A_b^{\nu}A_a^{\mu} \, \, \, \, \, (6) \end{equation} thus gluon commutes at the same position. Could someone tell me which of the above equations are not correct, or how we understand the commutation relations of gluon?