Non-abelian Yang-Mills in 1+1 dimensions

Question

1. Abelian electrodynamics in 1+1 dimensions is solvable, in the sense that we can find the space of solutions for the equation of motions $\partial_\mu F^{\mu\nu}=0$. To see this, one first notice that the field strength has only one component: $F_{10}=-F_{01}$. The equations of motion is then: $$\partial_0F_{10} = \partial_1F_{10} = 0.$$ So, the solution is a constant $F$.

2. My question is: Can we solve the equation of motion when we consider non-abelian Yang-Mills theory? The equation of motion is now: $$ D_0 F_{01} = \partial_0F_{01} + [A_0,F_{01}]=0 \\ D_1 F_{01} = \partial_1F_{01} + [A_1,F_{01}]=0.$$ Can we solve for $F$?

Answer 1

1. Let us choose the axial gauge $$A_1~=~0.\tag{A}$$

2. Then the chromo-electric field is $$F_{10}~=~\partial_1A_0-\partial_0A_1+[A_1,A_0]~\stackrel{(A)}{=}~\partial_1A_0.\tag{B}$$ NB: Be aware that $F_{10}$ is not gauge invariant.

3. From $$0~=~[D_1,F_{10}]~\stackrel{(A)}{=}~\partial_1F_{10}\tag{C}$$ follows that the chromo-electric field $F_{10}(t)$ does not depend on $x$.

4. Moreover the chromo-electric potential $$A_0(x,t)~\stackrel{(B)+(C)}{=}~a_0(t)+xF_{10}(t)\tag{D}$$ is an affine function of $x$.

5. Then $${\rm ad}a_0~\equiv~[a_0,\cdot]\tag{E}$$ plays the role of an Hamiltonian for a Schrödinger-like equation: $$0~=~[D_0,F_{10}]~\stackrel{(D)+(E)}{=}~\left(\partial_0+{\rm ad}a_0\right)F_{10}.\tag{F}$$

6. The solution is an (anti)time-ordered exponential $$F_{10}(t_2)~\stackrel{(F)}{=}~\begin{Bmatrix}T \cr AT\end{Bmatrix}\exp\left[-\int_{t_1}^{t_2}\!dt~{\rm ad}a_0(t) \right] F_{10}(t_1)\quad\text{for}\quad\begin{Bmatrix}t_1\leq t_2 \cr t_1\geq t_2\end{Bmatrix},\tag{G}$$ cf. e.g. this Phys.SE post.

7. If we instead choose the temporal gauge $$A_0~=~0.\tag{H}$$ there is an analogous solution with the roles of $x\leftrightarrow t$ exchanged.